From 4618a2a7eef2b692012f3f7348de71745cbd2caa Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Fri, 22 Oct 2021 07:55:23 +0200 Subject: [PATCH] added more text --- doc/pub/week42/html/._week42-bs000.html | 66 ++++---- doc/pub/week42/html/._week42-bs001.html | 66 ++++---- doc/pub/week42/html/._week42-bs002.html | 66 ++++---- doc/pub/week42/html/._week42-bs003.html | 66 ++++---- doc/pub/week42/html/._week42-bs004.html | 66 ++++---- doc/pub/week42/html/._week42-bs005.html | 66 ++++---- doc/pub/week42/html/._week42-bs006.html | 66 ++++---- doc/pub/week42/html/._week42-bs007.html | 66 ++++---- doc/pub/week42/html/._week42-bs008.html | 66 ++++---- doc/pub/week42/html/._week42-bs009.html | 66 ++++---- doc/pub/week42/html/._week42-bs010.html | 66 ++++---- doc/pub/week42/html/._week42-bs011.html | 66 ++++---- doc/pub/week42/html/._week42-bs012.html | 66 ++++---- doc/pub/week42/html/._week42-bs013.html | 66 ++++---- doc/pub/week42/html/._week42-bs014.html | 66 ++++---- 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'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
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  • A more efficient way of coding the above Convolution
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  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
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  • Principle of Superposition
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  • Simple Code Example
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  • Wrapping up Fourier transforms
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  • Finding the Coefficients
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  • The CIFAR01 data set
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  • Set up the model
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  • Add Dense layers on top
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  • Compile and train the model
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  • Finally, evaluate the model
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  • Key Idea
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  • Mathematics of CNNs
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  • Convolution Examples: Polynomial multiplication
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  • Efficient Polynomial Multiplication
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  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
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  • Simple Code Example
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  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
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  • Cross-Correlation
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  • More on Dimensionalities
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  • Further Dimensionality Remarks
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
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  • Setting it up
    • +
  • The MNIST dataset again
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  • Strong correlations
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  • Layers of a CNN
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  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
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  • The CIFAR01 data set
    • +
  • Verifying the data set
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  • Set up the model
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  • Add Dense layers on top
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  • Compile and train the model
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    • @@ -447,7 +449,7 @@ MathJax.Hub.Config({
  • 9
  • 10
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    • diff --git a/doc/pub/week42/html/._week42-bs001.html b/doc/pub/week42/html/._week42-bs001.html index 28ce7fee6..bf047098d 100644 --- a/doc/pub/week42/html/._week42-bs001.html +++ b/doc/pub/week42/html/._week42-bs001.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
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  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -472,7 +474,7 @@ MathJax.Hub.Config({
  • 10
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    • -
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    • diff --git a/doc/pub/week42/html/._week42-bs002.html b/doc/pub/week42/html/._week42-bs002.html index 3325980bc..49e013f03 100644 --- a/doc/pub/week42/html/._week42-bs002.html +++ b/doc/pub/week42/html/._week42-bs002.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -428,7 +430,7 @@ we will also study the usage of 11
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    • diff --git a/doc/pub/week42/html/._week42-bs003.html b/doc/pub/week42/html/._week42-bs003.html index 317403358..c952c0e62 100644 --- a/doc/pub/week42/html/._week42-bs003.html +++ b/doc/pub/week42/html/._week42-bs003.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -445,7 +447,7 @@ and output layer to any given precision.
  • 12
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    • diff --git a/doc/pub/week42/html/._week42-bs004.html b/doc/pub/week42/html/._week42-bs004.html index 23bb79b38..dc750b3ee 100644 --- a/doc/pub/week42/html/._week42-bs004.html +++ b/doc/pub/week42/html/._week42-bs004.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
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  • Transforming images
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    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
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  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
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  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -448,7 +450,7 @@ for the solution to be unique.
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  • A more efficient way of coding the above Convolution
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    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
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  • Cross-Correlation
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  • More on Dimensionalities
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  • Further Dimensionality Remarks
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
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  • Strong correlations
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  • Layers of a CNN
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  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
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  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -453,7 +455,7 @@ As described previously, an optimization method could be used to minimize the pa
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    • diff --git a/doc/pub/week42/html/._week42-bs006.html b/doc/pub/week42/html/._week42-bs006.html index 5a115942a..3c7f47705 100644 --- a/doc/pub/week42/html/._week42-bs006.html +++ b/doc/pub/week42/html/._week42-bs006.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
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  • Transforming images
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    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
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  • Wrapping up Fourier transforms
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  • Finding the Coefficients
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  • Verifying the data set
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  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -455,7 +457,7 @@ The neural net should then find the parameters \( P \) that minimizes the cost f
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    • diff --git a/doc/pub/week42/html/._week42-bs007.html b/doc/pub/week42/html/._week42-bs007.html index 32201d76b..c2c86e56a 100644 --- a/doc/pub/week42/html/._week42-bs007.html +++ b/doc/pub/week42/html/._week42-bs007.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
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  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
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  • Convolution Examples: Polynomial multiplication
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  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
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  • Simple Code Example
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  • Wrapping up Fourier transforms
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  • Finding the Coefficients
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  • Final words on Fourier Transforms
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  • Two-dimensional Objects
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  • The MNIST dataset again
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  • Strong correlations
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  • Layers of a CNN
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  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
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  • Running with Keras
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    • -
  • Final visualization
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  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -437,7 +439,7 @@ Automatic differentiation is a method of finding the derivatives numerically wit
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    • diff --git a/doc/pub/week42/html/._week42-bs008.html b/doc/pub/week42/html/._week42-bs008.html index 2af4895fa..5d945049d 100644 --- a/doc/pub/week42/html/._week42-bs008.html +++ b/doc/pub/week42/html/._week42-bs008.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
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  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
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  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
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  • Simple Code Example
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  • Wrapping up Fourier transforms
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  • Finding the Coefficients
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  • Two-dimensional Objects
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    • -
  • Finally, evaluate the model
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  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -455,7 +457,7 @@ Having an analytical solution at hand, it is possible to use it to compare how w
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    • diff --git a/doc/pub/week42/html/._week42-bs009.html b/doc/pub/week42/html/._week42-bs009.html index 4157cb03a..8f68afbbc 100644 --- a/doc/pub/week42/html/._week42-bs009.html +++ b/doc/pub/week42/html/._week42-bs009.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
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    • -
  • A more efficient way of coding the above Convolution
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    • -
  • Principle of Superposition
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  • Simple Code Example
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  • Wrapping up Fourier transforms
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  • Finding the Coefficients
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    • -
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    • -
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  • Compile and train the model
    • -
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    • +
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    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
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  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -446,7 +448,7 @@ In this example, \( \gamma = 2 \) and \( g_0 = 10 \).
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    • diff --git a/doc/pub/week42/html/._week42-bs010.html b/doc/pub/week42/html/._week42-bs010.html index 5f124c78b..36b31fef4 100644 --- a/doc/pub/week42/html/._week42-bs010.html +++ b/doc/pub/week42/html/._week42-bs010.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
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  • A more efficient way of coding the above Convolution
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  • Simple Code Example
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  • Wrapping up Fourier transforms
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  • Finding the Coefficients
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  • Final words on Fourier Transforms
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  • Layers of a CNN
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  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
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  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
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  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
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  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
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  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
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  • Running with Keras
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    • +
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    • +
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    • +
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    • @@ -440,7 +442,7 @@ with \( h_1(x) \) ensuring that \( g_t(x) \) satisfies some conditions and \( h_
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    • diff --git a/doc/pub/week42/html/._week42-bs011.html b/doc/pub/week42/html/._week42-bs011.html index e26a043c4..19e5c36e9 100644 --- a/doc/pub/week42/html/._week42-bs011.html +++ b/doc/pub/week42/html/._week42-bs011.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
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  • Efficient Polynomial Multiplication
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  • A more efficient way of coding the above Convolution
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  • Principle of Superposition
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  • Simple Code Example
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  • Finding the Coefficients
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  • Two-dimensional Objects
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  • The MNIST dataset again
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  • Strong correlations
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  • Layers of a CNN
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  • Systematic reduction
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    • -
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    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
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    • +
  • Finally, evaluate the model
    • @@ -452,7 +454,7 @@ $$
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    • diff --git a/doc/pub/week42/html/._week42-bs012.html b/doc/pub/week42/html/._week42-bs012.html index 3aa31bf90..407450f9c 100644 --- a/doc/pub/week42/html/._week42-bs012.html +++ b/doc/pub/week42/html/._week42-bs012.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
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  • Efficient Polynomial Multiplication
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  • A more efficient way of coding the above Convolution
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  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
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  • Principle of Superposition
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  • Simple Code Example
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  • Finding the Coefficients
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  • Two-dimensional Objects
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  • The MNIST dataset again
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  • Strong correlations
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  • Layers of a CNN
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  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
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  • Importing Keras and Tensorflow
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  • Running with Keras
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    • -
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    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
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    • +
  • Verifying the data set
    • +
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    • +
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    • +
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    • @@ -461,7 +463,7 @@ is fulfilled as best as possible.
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    • diff --git a/doc/pub/week42/html/._week42-bs013.html b/doc/pub/week42/html/._week42-bs013.html index aad34f8d4..b0b6c753d 100644 --- a/doc/pub/week42/html/._week42-bs013.html +++ b/doc/pub/week42/html/._week42-bs013.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
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  • Mathematics of CNNs
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  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
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  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
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  • Principle of Superposition
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  • Simple Code Example
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  • Wrapping up Fourier transforms
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  • Finding the Coefficients
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  • Two-dimensional Objects
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  • Cross-Correlation
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  • Further Dimensionality Remarks
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  • Strong correlations
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  • Layers of a CNN
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  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
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  • Importing Keras and Tensorflow
    • -
  • Running with Keras
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  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
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    • +
  • Finally, evaluate the model
    • @@ -457,7 +459,7 @@ for an input value \( x \).
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    • diff --git a/doc/pub/week42/html/._week42-bs014.html b/doc/pub/week42/html/._week42-bs014.html index a77e75262..7ac864711 100644 --- a/doc/pub/week42/html/._week42-bs014.html +++ b/doc/pub/week42/html/._week42-bs014.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
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  • A more efficient way of coding the above Convolution
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  • Principle of Superposition
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  • Simple Code Example
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  • Wrapping up Fourier transforms
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  • Finding the Coefficients
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  • Final words on Fourier Transforms
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  • Two-dimensional Objects
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  • Cross-Correlation
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  • Setting it up
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  • The MNIST dataset again
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  • Strong correlations
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  • Layers of a CNN
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  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
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  • Running with Keras
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    • -
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  • The CIFAR01 data set
    • -
  • Verifying the data set
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  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
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    • +
  • Finally, evaluate the model
    • @@ -455,7 +457,7 @@ $$
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    • diff --git a/doc/pub/week42/html/._week42-bs015.html b/doc/pub/week42/html/._week42-bs015.html index 435f48455..1b4a82ce3 100644 --- a/doc/pub/week42/html/._week42-bs015.html +++ b/doc/pub/week42/html/._week42-bs015.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
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  • Mathematics of CNNs
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    • -
  • A more efficient way of coding the above Convolution
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  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
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  • Principle of Superposition
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  • Simple Code Example
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  • Wrapping up Fourier transforms
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  • Finding the Coefficients
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  • Final words on Fourier Transforms
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  • Layers of a CNN
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  • Verifying the data set
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  • Set up the model
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  • Mathematics of CNNs
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  • Convolution Examples: Polynomial multiplication
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  • Efficient Polynomial Multiplication
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  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
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  • Set up the model
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    • @@ -440,7 +442,7 @@ The input layer will consist of \( N_{\text{input} } \) neurons, passing its ele
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    • diff --git a/doc/pub/week42/html/._week42-bs016.html b/doc/pub/week42/html/._week42-bs016.html index ec3cecf7a..ed2e51ff1 100644 --- a/doc/pub/week42/html/._week42-bs016.html +++ b/doc/pub/week42/html/._week42-bs016.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
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  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
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  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
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  • Principle of Superposition
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  • Simple Code Example
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  • Wrapping up Fourier transforms
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  • Finding the Coefficients
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  • Final words on Fourier Transforms
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  • Two-dimensional Objects
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  • Strong correlations
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  • Layers of a CNN
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  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
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  • Running with Keras
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    • -
  • Verifying the data set
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  • Set up the model
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  • Add Dense layers on top
    • -
  • Compile and train the model
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  • Finally, evaluate the model
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  • Key Idea
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  • Mathematics of CNNs
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  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
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  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
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  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
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  • Final part
    • +
  • Final visualization
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  • The CIFAR01 data set
    • +
  • Verifying the data set
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  • Set up the model
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    • @@ -449,7 +451,7 @@ $$
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    • diff --git a/doc/pub/week42/html/._week42-bs017.html b/doc/pub/week42/html/._week42-bs017.html index 9dcbbfbe2..b8e7924c2 100644 --- a/doc/pub/week42/html/._week42-bs017.html +++ b/doc/pub/week42/html/._week42-bs017.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
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  • Efficient Polynomial Multiplication
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  • A more efficient way of coding the above Convolution
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  • Principle of Superposition
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  • Simple Code Example
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  • Wrapping up Fourier transforms
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  • Finding the Coefficients
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  • Final words on Fourier Transforms
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  • Two-dimensional Objects
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  • Cross-Correlation
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  • Strong correlations
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  • Layers of a CNN
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  • Prerequisites: Collect and pre-process data
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  • Importing Keras and Tensorflow
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  • Running with Keras
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  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -450,7 +452,7 @@ $$
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  • A more efficient way of coding the above Convolution
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    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
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  • More on Dimensionalities
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  • Further Dimensionality Remarks
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
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  • Layers of a CNN
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  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
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    • +
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    • @@ -465,7 +467,7 @@ it is assumes that the number of neurons in the output layer is one.
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  • A more efficient way of coding the above Convolution
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    • -
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    • -
  • Finally, evaluate the model
    • +
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  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
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  • Efficient Polynomial Multiplication
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  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
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  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
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  • More on Dimensionalities
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  • Further Dimensionality Remarks
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
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  • Layers of a CNN
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  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
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    • +
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    • +
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    • +
  • Finally, evaluate the model
    • @@ -448,7 +450,7 @@ $$
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    • diff --git a/doc/pub/week42/html/._week42-bs020.html b/doc/pub/week42/html/._week42-bs020.html index d6c983d1d..e518e0cb2 100644 --- a/doc/pub/week42/html/._week42-bs020.html +++ b/doc/pub/week42/html/._week42-bs020.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
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  • A more efficient way of coding the above Convolution
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  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
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  • Principle of Superposition
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  • Simple Code Example
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  • Finding the Coefficients
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    • -
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    • +
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  • Mathematics of CNNs
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  • Convolution Examples: Polynomial multiplication
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  • Efficient Polynomial Multiplication
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  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
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  • More on Dimensionalities
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  • Further Dimensionality Remarks
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -449,7 +451,7 @@ In this case we seek a continuous range of values since we are approximating a f
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    • diff --git a/doc/pub/week42/html/._week42-bs021.html b/doc/pub/week42/html/._week42-bs021.html index eb3e69c89..aa9697774 100644 --- a/doc/pub/week42/html/._week42-bs021.html +++ b/doc/pub/week42/html/._week42-bs021.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
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    • -
  • A more efficient way of coding the above Convolution
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  • Principle of Superposition
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  • Simple Code Example
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  • Mathematics of CNNs
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  • Efficient Polynomial Multiplication
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  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
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  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
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  • Cross-Correlation
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  • More on Dimensionalities
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  • Further Dimensionality Remarks
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
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    • +
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    • +
  • Verifying the data set
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    • +
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    • +
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    • +
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    • @@ -448,7 +450,7 @@ Here, gradient descent with a constant step size has been chosen.
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    • diff --git a/doc/pub/week42/html/._week42-bs022.html b/doc/pub/week42/html/._week42-bs022.html index 84f499895..469e68618 100644 --- a/doc/pub/week42/html/._week42-bs022.html +++ b/doc/pub/week42/html/._week42-bs022.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
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  • A more efficient way of coding the above Convolution
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  • Mathematics of CNNs
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  • Convolution Examples: Polynomial multiplication
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  • Efficient Polynomial Multiplication
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  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
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  • Principle of Superposition
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  • Simple Code Example
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  • Wrapping up Fourier transforms
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  • Finding the Coefficients
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  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
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  • More on Dimensionalities
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  • Further Dimensionality Remarks
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
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  • Final part
    • +
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    • +
  • Verifying the data set
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    • @@ -470,7 +472,7 @@ $$
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    • diff --git a/doc/pub/week42/html/._week42-bs023.html b/doc/pub/week42/html/._week42-bs023.html index 30ef1043e..aaee7ef0d 100644 --- a/doc/pub/week42/html/._week42-bs023.html +++ b/doc/pub/week42/html/._week42-bs023.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
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  • A more efficient way of coding the above Convolution
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  • A more efficient way of coding the above Convolution
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  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
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  • Principle of Superposition
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  • Simple Code Example
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  • Wrapping up Fourier transforms
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  • Finding the Coefficients
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  • Final words on Fourier Transforms
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  • Two-dimensional Objects
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  • Cross-Correlation
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  • More on Dimensionalities
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
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  • Setting it up
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  • The MNIST dataset again
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  • Strong correlations
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  • Layers of a CNN
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  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
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    • @@ -580,7 +582,7 @@ MathJax.Hub.Config({
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  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
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  • Convolution Examples: Polynomial multiplication
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  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
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  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
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  • More on Dimensionalities
    • -
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    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
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  • Layers of a CNN
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  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
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    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -600,7 +602,7 @@ The number of neurons within each hidden layer are given as a list of integers i
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    • diff --git a/doc/pub/week42/html/._week42-bs025.html b/doc/pub/week42/html/._week42-bs025.html index 9cf9e8926..e323360a8 100644 --- a/doc/pub/week42/html/._week42-bs025.html +++ b/doc/pub/week42/html/._week42-bs025.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
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  • Transforming images
  • CNNs in brief
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  • Mathematics of CNNs
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  • Convolution Examples: Polynomial multiplication
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  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
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  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
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    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
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  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
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    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -452,7 +454,7 @@ Here, we stay with a more simple approach and implement for comparison, the simp
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    • diff --git a/doc/pub/week42/html/._week42-bs026.html b/doc/pub/week42/html/._week42-bs026.html index 099aae0f5..6328a1a9f 100644 --- a/doc/pub/week42/html/._week42-bs026.html +++ b/doc/pub/week42/html/._week42-bs026.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
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  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
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  • Convolution Examples: Polynomial multiplication
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  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
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  • Finding the Coefficients
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  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
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  • More on Dimensionalities
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  • Further Dimensionality Remarks
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -448,7 +450,7 @@ In this example, we let \( \alpha = 2 \), \( A = 1 \), and \( g_0 = 1.2 \).
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    • diff --git a/doc/pub/week42/html/._week42-bs027.html b/doc/pub/week42/html/._week42-bs027.html index 1837e922f..14b34f9ae 100644 --- a/doc/pub/week42/html/._week42-bs027.html +++ b/doc/pub/week42/html/._week42-bs027.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
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  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
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  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
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  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
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  • More on Dimensionalities
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  • Further Dimensionality Remarks
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
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  • Setting it up
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  • The MNIST dataset again
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  • Strong correlations
    • -
  • Layers of a CNN
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  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -453,7 +455,7 @@ $$
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    • diff --git a/doc/pub/week42/html/._week42-bs028.html b/doc/pub/week42/html/._week42-bs028.html index 3362ec6c4..efba5ac8c 100644 --- a/doc/pub/week42/html/._week42-bs028.html +++ b/doc/pub/week42/html/._week42-bs028.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
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  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
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  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
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  • More on Dimensionalities
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  • Further Dimensionality Remarks
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    • -
  • Setting it up
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  • The MNIST dataset again
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  • Strong correlations
    • -
  • Layers of a CNN
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  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -602,7 +604,7 @@ The network will be the similar as for the exponential decay example, but with s
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    • diff --git a/doc/pub/week42/html/._week42-bs029.html b/doc/pub/week42/html/._week42-bs029.html index e62d03422..bc9885db7 100644 --- a/doc/pub/week42/html/._week42-bs029.html +++ b/doc/pub/week42/html/._week42-bs029.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -556,7 +558,7 @@ extending the program that uses the network using Autograd:
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    • diff --git a/doc/pub/week42/html/._week42-bs030.html b/doc/pub/week42/html/._week42-bs030.html index 2b0828a14..d5878301a 100644 --- a/doc/pub/week42/html/._week42-bs030.html +++ b/doc/pub/week42/html/._week42-bs030.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
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  • Transforming images
  • CNNs in brief
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  • Mathematics of CNNs
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  • Convolution Examples: Polynomial multiplication
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  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -458,7 +460,7 @@ In addition, it could be interesting to see how a typical method for numerically
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  • A more efficient way of coding the above Convolution
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  • Simple Code Example
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    • -
  • Verifying the data set
    • -
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    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
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  • Further Dimensionality Remarks
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -465,7 +467,7 @@ $$
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    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
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  • Finding the Coefficients
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  • The MNIST dataset again
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  • Strong correlations
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  • Layers of a CNN
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  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • -
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  • Running with Keras
    • -
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    • -
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    • -
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    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
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    • +
  • Finally, evaluate the model
    • @@ -586,7 +588,7 @@ MathJax.Hub.Config({
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    • diff --git a/doc/pub/week42/html/._week42-bs033.html b/doc/pub/week42/html/._week42-bs033.html index 662009697..3b602a5ae 100644 --- a/doc/pub/week42/html/._week42-bs033.html +++ b/doc/pub/week42/html/._week42-bs033.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
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  • Transforming images
  • CNNs in brief
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  • Mathematics of CNNs
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  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
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  • Simple Code Example
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  • Wrapping up Fourier transforms
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  • Finding the Coefficients
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  • Two-dimensional Objects
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  • Cross-Correlation
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    • -
  • Setting it up
    • -
  • The MNIST dataset again
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  • Strong correlations
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  • Layers of a CNN
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  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
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    • -
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    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
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  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -525,7 +527,7 @@ which makes it possible to solve for the vector \( \boldsymbol{g} \).
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  • A more efficient way of coding the above Convolution
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  • Principle of Superposition
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  • Simple Code Example
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  • Finding the Coefficients
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  • Final words on Fourier Transforms
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    • -
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    • -
  • Set up the model
    • -
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    • -
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    • -
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    • +
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  • Mathematics of CNNs
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  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
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  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
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  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -629,7 +631,7 @@ We can then compare the result from this numerical scheme with the output from o
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  • Efficient Polynomial Multiplication
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  • A more efficient way of coding the above Convolution
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  • Principle of Superposition
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  • Simple Code Example
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  • Finding the Coefficients
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  • Strong correlations
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    • -
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    • -
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    • +
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  • Mathematics of CNNs
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  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
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  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
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  • More on Dimensionalities
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  • Further Dimensionality Remarks
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
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  • Final part
    • +
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    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
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    • @@ -450,7 +452,7 @@ where \( f \) is an expression involving all kinds of possible mixed derivatives
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  • Simple Code Example
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  • Finding the Coefficients
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  • Final words on Fourier Transforms
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  • Cross-Correlation
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  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -450,7 +452,7 @@ The role of the function \( h_2(x_1,\dots,x_N,N(x_1,\dots,x_N,P)) \), is to ensu
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  • Layers used to build CNNs
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  • Mathematics of CNNs
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  • Convolution Examples: Polynomial multiplication
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  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
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    • -
  • Further Dimensionality Remarks
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    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
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  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -450,7 +452,7 @@ $$
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    • diff --git a/doc/pub/week42/html/._week42-bs038.html b/doc/pub/week42/html/._week42-bs038.html index 4d1705fe5..f4805001c 100644 --- a/doc/pub/week42/html/._week42-bs038.html +++ b/doc/pub/week42/html/._week42-bs038.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -446,7 +448,7 @@ $$
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    • diff --git a/doc/pub/week42/html/._week42-bs039.html b/doc/pub/week42/html/._week42-bs039.html index e283f3029..6670c424e 100644 --- a/doc/pub/week42/html/._week42-bs039.html +++ b/doc/pub/week42/html/._week42-bs039.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -452,7 +454,7 @@ with \( u(x) \) being some given function.
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    • diff --git a/doc/pub/week42/html/._week42-bs040.html b/doc/pub/week42/html/._week42-bs040.html index ffb8ef3d5..329afc221 100644 --- a/doc/pub/week42/html/._week42-bs040.html +++ b/doc/pub/week42/html/._week42-bs040.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -459,7 +461,7 @@ First, we will look into how Autograd could be used in a network tailored to sol
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    • diff --git a/doc/pub/week42/html/._week42-bs041.html b/doc/pub/week42/html/._week42-bs041.html index 7636c883a..a46c87c92 100644 --- a/doc/pub/week42/html/._week42-bs041.html +++ b/doc/pub/week42/html/._week42-bs041.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -489,7 +491,7 @@ network at each possible pair \( (x,t) \), given an array for the desired
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    • diff --git a/doc/pub/week42/html/._week42-bs042.html b/doc/pub/week42/html/._week42-bs042.html index 0cb85a586..919f8a172 100644 --- a/doc/pub/week42/html/._week42-bs042.html +++ b/doc/pub/week42/html/._week42-bs042.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -456,7 +458,7 @@ since \( (0) = u(1) = 0 \) and \( u(x) = \sin(\pi x) \).
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    • diff --git a/doc/pub/week42/html/._week42-bs043.html b/doc/pub/week42/html/._week42-bs043.html index ab889cc02..7b377a71c 100644 --- a/doc/pub/week42/html/._week42-bs043.html +++ b/doc/pub/week42/html/._week42-bs043.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
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  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
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  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -494,7 +496,7 @@ mixed derivatives of \( g(x,t) \).
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    • diff --git a/doc/pub/week42/html/._week42-bs044.html b/doc/pub/week42/html/._week42-bs044.html index 2de2ec37c..80ddce559 100644 --- a/doc/pub/week42/html/._week42-bs044.html +++ b/doc/pub/week42/html/._week42-bs044.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
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  • A more efficient way of coding the above Convolution
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    • -
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  • Finding the Coefficients
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    • -
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    • -
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    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -678,7 +680,7 @@ Using TensorFlow results in a much better execution time. Try it!
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    • diff --git a/doc/pub/week42/html/._week42-bs045.html b/doc/pub/week42/html/._week42-bs045.html index fae109e5f..2bd8d5adb 100644 --- a/doc/pub/week42/html/._week42-bs045.html +++ b/doc/pub/week42/html/._week42-bs045.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
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  • Mathematics of CNNs
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  • Convolution Examples: Polynomial multiplication
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  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
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  • Finding the Coefficients
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  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
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  • Further Dimensionality Remarks
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  • Setting it up
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  • The MNIST dataset again
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  • Strong correlations
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    • -
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    • -
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    • -
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    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -456,7 +458,7 @@ where \( \frac{\partial g(x,t)}{\partial t} \Big |_{t = 0} \) means the derivati
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    • diff --git a/doc/pub/week42/html/._week42-bs046.html b/doc/pub/week42/html/._week42-bs046.html index 7b2cc3eff..2b82b3bd1 100644 --- a/doc/pub/week42/html/._week42-bs046.html +++ b/doc/pub/week42/html/._week42-bs046.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
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  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
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  • Principle of Superposition
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  • Simple Code Example
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  • Finding the Coefficients
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  • Two-dimensional Objects
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  • Cross-Correlation
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  • Further Dimensionality Remarks
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  • Setting it up
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  • The MNIST dataset again
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  • Strong correlations
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  • Layers of a CNN
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  • Running with Keras
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    • -
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  • Verifying the data set
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    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -457,7 +459,7 @@ In this example, let \( c = 1 \) and \( u(x) = \sin(\pi x) \) and \( v(x) = -\pi
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    • diff --git a/doc/pub/week42/html/._week42-bs047.html b/doc/pub/week42/html/._week42-bs047.html index cb9622f76..de2c00839 100644 --- a/doc/pub/week42/html/._week42-bs047.html +++ b/doc/pub/week42/html/._week42-bs047.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
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    • -
  • A more efficient way of coding the above Convolution
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  • Simple Code Example
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  • Finding the Coefficients
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  • Two-dimensional Objects
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    • -
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    • -
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    • +
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  • Mathematics of CNNs
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  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
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  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -451,7 +453,7 @@ Note that this trial solution satisfies the conditions only if \( u(0) = v(0) =
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    • diff --git a/doc/pub/week42/html/._week42-bs048.html b/doc/pub/week42/html/._week42-bs048.html index 853aca8ad..7f14c74de 100644 --- a/doc/pub/week42/html/._week42-bs048.html +++ b/doc/pub/week42/html/._week42-bs048.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
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  • A more efficient way of coding the above Convolution
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  • Principle of Superposition
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  • Simple Code Example
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  • Finding the Coefficients
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  • Strong correlations
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  • Set up the model
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  • Add Dense layers on top
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  • Compile and train the model
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  • Mathematics of CNNs
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  • Convolution Examples: Polynomial multiplication
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  • Efficient Polynomial Multiplication
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  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
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  • Layers of a CNN
    • +
  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
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  • Final part
    • +
  • Final visualization
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  • The CIFAR01 data set
    • +
  • Verifying the data set
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  • Set up the model
    • +
  • Add Dense layers on top
    • +
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    • +
  • Finally, evaluate the model
    • @@ -439,7 +441,7 @@ $$
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    • diff --git a/doc/pub/week42/html/._week42-bs049.html b/doc/pub/week42/html/._week42-bs049.html index 37330200c..0f856b329 100644 --- a/doc/pub/week42/html/._week42-bs049.html +++ b/doc/pub/week42/html/._week42-bs049.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
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  • A more efficient way of coding the above Convolution
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  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
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  • Principle of Superposition
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  • Simple Code Example
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  • Wrapping up Fourier transforms
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  • Finding the Coefficients
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  • Final words on Fourier Transforms
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  • Two-dimensional Objects
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  • Cross-Correlation
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  • More on Dimensionalities
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  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -656,7 +658,7 @@ MathJax.Hub.Config({
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  • Mathematics of CNNs
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  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -438,7 +440,7 @@ MathJax.Hub.Config({
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  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -451,7 +453,7 @@ Networks still apply (back propagation, gradient descent etc etc).
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    • diff --git a/doc/pub/week42/html/._week42-bs052.html b/doc/pub/week42/html/._week42-bs052.html index 57176a088..31a3c45cb 100644 --- a/doc/pub/week42/html/._week42-bs052.html +++ b/doc/pub/week42/html/._week42-bs052.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
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  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -447,7 +449,7 @@ Another good read is the article here 61
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    • diff --git a/doc/pub/week42/html/._week42-bs053.html b/doc/pub/week42/html/._week42-bs053.html index 177c2b5ea..ce57ef778 100644 --- a/doc/pub/week42/html/._week42-bs053.html +++ b/doc/pub/week42/html/._week42-bs053.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
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  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -441,7 +443,7 @@ before the transformation.
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    • diff --git a/doc/pub/week42/html/._week42-bs054.html b/doc/pub/week42/html/._week42-bs054.html index 42e6f590b..3d0385190 100644 --- a/doc/pub/week42/html/._week42-bs054.html +++ b/doc/pub/week42/html/._week42-bs054.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
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  • Mathematics of CNNs
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  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
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  • More on Dimensionalities
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  • Further Dimensionality Remarks
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
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  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
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    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -455,7 +457,7 @@ in the input).
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    • diff --git a/doc/pub/week42/html/._week42-bs055.html b/doc/pub/week42/html/._week42-bs055.html index 4dc04b012..f3ee2c131 100644 --- a/doc/pub/week42/html/._week42-bs055.html +++ b/doc/pub/week42/html/._week42-bs055.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
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  • Further Dimensionality Remarks
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
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    • -
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    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
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  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -455,7 +457,7 @@ would quickly lead to possible overfitting.
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  • Convolution Examples: Polynomial multiplication
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  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -467,7 +469,7 @@ dimension.
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    • diff --git a/doc/pub/week42/html/._week42-bs057.html b/doc/pub/week42/html/._week42-bs057.html index 7af17b854..b3da4653b 100644 --- a/doc/pub/week42/html/._week42-bs057.html +++ b/doc/pub/week42/html/._week42-bs057.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
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  • Convolution Examples: Polynomial multiplication
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  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
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  • Wrapping up Fourier transforms
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  • Finding the Coefficients
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    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
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  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -450,7 +452,7 @@ A simple CNN for image classification could have the architecture:
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    • diff --git a/doc/pub/week42/html/._week42-bs058.html b/doc/pub/week42/html/._week42-bs058.html index 7ff476452..6393e977e 100644 --- a/doc/pub/week42/html/._week42-bs058.html +++ b/doc/pub/week42/html/._week42-bs058.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
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  • Mathematics of CNNs
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  • Convolution Examples: Polynomial multiplication
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  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
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  • Simple Code Example
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  • Wrapping up Fourier transforms
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  • Finding the Coefficients
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  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -446,7 +448,7 @@ are consistent with the labels in the training set for each image.
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    • diff --git a/doc/pub/week42/html/._week42-bs059.html b/doc/pub/week42/html/._week42-bs059.html index 103b92733..98901ee5e 100644 --- a/doc/pub/week42/html/._week42-bs059.html +++ b/doc/pub/week42/html/._week42-bs059.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
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  • Convolution Examples: Polynomial multiplication
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  • A more efficient way of coding the above Convolution
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  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
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  • Simple Code Example
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  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -422,6 +424,9 @@ the course IN5400 – Machine Learning for Image Analysis and the slides of CS231 which is taught at Stanford University (consistently ranked as one of the top computer science programs in the world). Michael Nielsen's book is a must read, in particular chapter 6 which deals with CNNs. +

    +The textbook by Goodfellow et al, see chapter 9 contains an in depth discussion as well. +

    @@ -448,7 +453,7 @@ and the slides of 68

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    • diff --git a/doc/pub/week42/html/._week42-bs060.html b/doc/pub/week42/html/._week42-bs060.html index c7596585c..c7faaffad 100644 --- a/doc/pub/week42/html/._week42-bs060.html +++ b/doc/pub/week42/html/._week42-bs060.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
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  • CNNs in brief
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  • Mathematics of CNNs
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  • Convolution Examples: Polynomial multiplication
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  • Efficient Polynomial Multiplication
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  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
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  • Principle of Superposition
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  • Simple Code Example
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  • Finding the Coefficients
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  • Final words on Fourier Transforms
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    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -404,41 +406,17 @@ MathJax.Hub.Config({ -

    Mathematics of CNNs

    +

    Key Idea

    -The mathematics of CNNs is based on the mathematical operation of -convolution. In mathematics (in particular in functional analysis), -convolution is represented by matheematical operation (integration, -summation etc) on two function in order to produce a third function -that expresses how the shape of one gets modified by the other. -Convolution has a plethora of applications in a variety of disciplines, spanning from statistics to signal processing, computer vision, solutions of differential equations,linear algebra, engineering, and yes, machine learning. +A dense neural network is representd by an affine operation (like matrix-matrix multiplication) where all parameters are included.

    -Mathematically, convolution is defined as follows (one-dimensional example): -Let us define a continuous function \( y(t) \) given by -$$ -y(t) = \int x(a) w(t-a) da, -$$ - -where \( x(a) \) represents a so-called input and \( w(t-a) \) is normally called the weight function or kernel. +The key idea in CNNs for say imaging is that in images neighbor pixels tend to be related! So we connect +only neighboring neurons in the input instead of connecting all with the first hidden layer.

    -The above integral is written in a more compact form as -$$ -y(t) = \left(x * w\right)(t). -$$ - -

    -The discretized version reads -$$ -y(t) = \sum_{a=-\infty}^{a=\infty}x(a)w(t-a). -$$ - -Computing the inverse of the above convolution operations is known as deconvolution. - -

    -How can we use this? And what does it mean? Let us study some familiar examples first. +We say we perform a filtering (convolution is the mathematical operation).

    @@ -466,7 +444,7 @@ How can we use this? And what does it mean? Let us study some familiar examples

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  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
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  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
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  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
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  • Cross-Correlation
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  • More on Dimensionalities
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  • Further Dimensionality Remarks
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
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  • The MNIST dataset again
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  • Strong correlations
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  • Layers of a CNN
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  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
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    • -
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    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -404,28 +406,42 @@ MathJax.Hub.Config({ -

    Convolution Examples: Polynomial multiplication

    +

    Mathematics of CNNs

    -We have already met such an example in project 1 when we tried to set -up the design matrix for a two-dimensional function. This was an -example of polynomial multiplication. Let us recast such a problem in terms of the convolution operation. -Let us look a the following polynomials to second and third order, respectively: +The mathematics of CNNs is based on the mathematical operation of +convolution. In mathematics (in particular in functional analysis), +convolution is represented by matheematical operation (integration, +summation etc) on two function in order to produce a third function +that expresses how the shape of one gets modified by the other. +Convolution has a plethora of applications in a variety of disciplines, spanning from statistics to signal processing, computer vision, solutions of differential equations,linear algebra, engineering, and yes, machine learning. + +

    +Mathematically, convolution is defined as follows (one-dimensional example): +Let us define a continuous function \( y(t) \) given by $$ -p(t) = \alpha_0+\alpha_1 t+\alpha_2 t^2, +y(t) = \int x(a) w(t-a) da, $$ -and +where \( x(a) \) represents a so-called input and \( w(t-a) \) is normally called the weight function or kernel. + +

    +The above integral is written in a more compact form as $$ -s(t) = \beta_0+\beta_1 t+\beta_2 t^2+\beta_3 t^3. +y(t) = \left(x * w\right)(t). $$

    -The polynomial multiplication gives us a new polynomial of degree \( 5 \) +The discretized version reads $$ -z(t) = \delta_0+\delta_1 t+\delta_2 t^2+\delta_3 t^3+\delta_4 t^4+\delta_5 t^5. +y(t) = \sum_{a=-\infty}^{a=\infty}x(a)w(t-a). $$ +Computing the inverse of the above convolution operations is known as deconvolution. + +

    +How can we use this? And what does it mean? Let us study some familiar examples first. +

    @@ -452,7 +468,7 @@ $$

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  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -404,40 +406,28 @@ MathJax.Hub.Config({ -

    Efficient Polynomial Multiplication

    +

    Convolution Examples: Polynomial multiplication

    -Computing polynomial products can be implemented efficiently if we rewrite the more brute force multiplications using convolution. -We note first that the new coefficients are given as - +We have already met such an example in project 1 when we tried to set +up the design matrix for a two-dimensional function. This was an +example of polynomial multiplication. Let us recast such a problem in terms of the convolution operation. +Let us look a the following polynomials to second and third order, respectively: $$ -\begin{split} -\delta_0=&\alpha_0\beta_0\\ -\delta_1=&\alpha_1\beta_0+\alpha_1\beta_0\\ -\delta_2=&\alpha_0\beta_2+\alpha_1\beta_1+\alpha_2\beta_0\\ -\delta_3=&\alpha_1\beta_2+\alpha_2\beta_1+\alpha_0\beta_3\\ -\delta_4=&\alpha_2\beta_2+\alpha_1\beta_3\\ -\delta_5=&\alpha_2\beta_3.\\ -\end{split} +p(t) = \alpha_0+\alpha_1 t+\alpha_2 t^2, +$$ + +and +$$ +s(t) = \beta_0+\beta_1 t+\beta_2 t^2+\beta_3 t^3. $$

    -We note that \( \alpha_i=0 \) except for \( i\in \left\{0,1,2\right\} \) and \( \beta_i=0 \) except for \( i\in\left\{0,1,2,3\right\} \). - -

    -We can then rewrite the coefficients \( \delta_j \) using a discrete convolution as +The polynomial multiplication gives us a new polynomial of degree \( 5 \) $$ -\delta_j = \sum_{i=-\infty}^{i=\infty}\alpha_i\beta_{j-i}=(\alpha * \beta)_j, +z(t) = \delta_0+\delta_1 t+\delta_2 t^2+\delta_3 t^3+\delta_4 t^4+\delta_5 t^5. $$ -or as a double sum with restriction \( l=i+j \) -$$ -\delta_l = \sum_{ij}\alpha_i\beta_{j}. -$$ - -

    -Do you see a potential drawback with these equations? -

    @@ -464,7 +454,7 @@ Do you see a potential drawback with these equations?

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  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -404,43 +406,39 @@ MathJax.Hub.Config({ -

    A more efficient way of coding the above Convolution

    +

    Efficient Polynomial Multiplication

    -Since we only have a finite number of \( \alpha \) and \( \beta \) values -which are non-zero, we can rewrite the above convolution expressions -as a matrix-vector multiplication +Computing polynomial products can be implemented efficiently if we rewrite the more brute force multiplications using convolution. +We note first that the new coefficients are given as $$ -\boldsymbol{\delta}=\begin{bmatrix}\alpha_0 & 0 & 0 & 0 \\ - \alpha_1 & \alpha_0 & 0 & 0 \\ - \alpha_2 & \alpha_1 & \alpha_0 & 0 \\ - 0 & \alpha_2 & \alpha_1 & \alpha_0 \\ - 0 & 0 & \alpha_2 & \alpha_1 \\ - 0 & 0 & 0 & \alpha_2 - \end{bmatrix}\begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \\ \beta_3\end{bmatrix}. +\begin{split} +\delta_0=&\alpha_0\beta_0\\ +\delta_1=&\alpha_1\beta_0+\alpha_1\beta_0\\ +\delta_2=&\alpha_0\beta_2+\alpha_1\beta_1+\alpha_2\beta_0\\ +\delta_3=&\alpha_1\beta_2+\alpha_2\beta_1+\alpha_0\beta_3\\ +\delta_4=&\alpha_2\beta_2+\alpha_1\beta_3\\ +\delta_5=&\alpha_2\beta_3.\\ +\end{split} $$

    -The process is commutative and we can easily see that we can rewrite the multiplication in terms of a matrix holding \( \beta \) and a vector holding \( \alpha \). -In this case we have +We note that \( \alpha_i=0 \) except for \( i\in \left\{0,1,2\right\} \) and \( \beta_i=0 \) except for \( i\in\left\{0,1,2,3\right\} \). + +

    +We can then rewrite the coefficients \( \delta_j \) using a discrete convolution as $$ -\boldsymbol{\delta}=\begin{bmatrix}\beta_0 & 0 & 0 \\ - \beta_1 & \beta_0 & 0 \\ - \beta_2 & \beta_1 & \beta_0 \\ - \beta_3 & \beta_2 & \beta_1 \\ - 0 & \beta_3 & \beta_2 \\ - 0 & 0 & \beta_3 - \end{bmatrix}\begin{bmatrix} \alpha_0 \\ \alpha_1 \\ \alpha_2\end{bmatrix}. +\delta_j = \sum_{i=-\infty}^{i=\infty}\alpha_i\beta_{j-i}=(\alpha * \beta)_j, +$$ + +or as a double sum with restriction \( l=i+j \) +$$ +\delta_l = \sum_{ij}\alpha_i\beta_{j}. $$

    -Note that the use of these matrices is for mathematical purposes only and not implementation purposes. -When implementing the above equation we do not encode (and allocate memory) the matrices explicitely. -We rather code the convolutions in the minimal memory footprint that they require. - -

    -Does the number of floating point operations change here when we use the commutative property? +Do you see a potential drawback with these equations?

    @@ -468,7 +466,7 @@ Does the number of floating point operations change here when we use the commuta

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  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
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  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -404,27 +406,43 @@ MathJax.Hub.Config({ -

    Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)

    +

    A more efficient way of coding the above Convolution

    -For problems with so-called harmonic oscillations, given by for example the following differential equation -$$ -m\frac{d^2x}{dt^2}+\eta\frac{dx}{dt}+x(t)=F(t), -$$ +Since we only have a finite number of \( \alpha \) and \( \beta \) values +which are non-zero, we can rewrite the above convolution expressions +as a matrix-vector multiplication -where \( F(t) \) is an applied external force acting on the system (often called a driving force), one can use the theory of Fourier transformations to find the solutions of this type of equations. +$$ +\boldsymbol{\delta}=\begin{bmatrix}\alpha_0 & 0 & 0 & 0 \\ + \alpha_1 & \alpha_0 & 0 & 0 \\ + \alpha_2 & \alpha_1 & \alpha_0 & 0 \\ + 0 & \alpha_2 & \alpha_1 & \alpha_0 \\ + 0 & 0 & \alpha_2 & \alpha_1 \\ + 0 & 0 & 0 & \alpha_2 + \end{bmatrix}\begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \\ \beta_3\end{bmatrix}. +$$

    -If one has several driving forces, \( F(t)=\sum_n F_n(t) \), one can find -the particular solution to each \( F_n \), \( x_{pn}(t) \), and the particular -solution for the entire driving force is then given by a series like +The process is commutative and we can easily see that we can rewrite the multiplication in terms of a matrix holding \( \beta \) and a vector holding \( \alpha \). +In this case we have +$$ +\boldsymbol{\delta}=\begin{bmatrix}\beta_0 & 0 & 0 \\ + \beta_1 & \beta_0 & 0 \\ + \beta_2 & \beta_1 & \beta_0 \\ + \beta_3 & \beta_2 & \beta_1 \\ + 0 & \beta_3 & \beta_2 \\ + 0 & 0 & \beta_3 + \end{bmatrix}\begin{bmatrix} \alpha_0 \\ \alpha_1 \\ \alpha_2\end{bmatrix}. +$$ -$$ -\begin{equation} -x_p(t)=\sum_nx_{pn}(t). -\tag{21} -\end{equation} -$$ +

    +Note that the use of these matrices is for mathematical purposes only and not implementation purposes. +When implementing the above equation we do not encode (and allocate memory) the matrices explicitely. +We rather code the convolutions in the minimal memory footprint that they require. + +

    +Does the number of floating point operations change here when we use the commutative property?

    @@ -452,7 +470,7 @@ $$

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  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
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  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -404,32 +406,27 @@ MathJax.Hub.Config({ -

    Principle of Superposition

    +

    Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)

    -This is known as the principle of superposition. It only applies when -the homogenous equation is linear. If there were an anharmonic term -such as \( x^3 \) in the homogenous equation, then when one summed various -solutions, \( x=(\sum_n x_n)^2 \), one would get cross -terms. Superposition is especially useful when \( F(t) \) can be written -as a sum of sinusoidal terms, because the solutions for each -sinusoidal (sine or cosine) term is analytic. - -

    -Driving forces are often periodic, even when they are not -sinusoidal. Periodicity implies that for some time \( \tau \) - +For problems with so-called harmonic oscillations, given by for example the following differential equation $$ -\begin{eqnarray} -F(t+\tau)=F(t). -\end{eqnarray} +m\frac{d^2x}{dt^2}+\eta\frac{dx}{dt}+x(t)=F(t), $$ +where \( F(t) \) is an applied external force acting on the system (often called a driving force), one can use the theory of Fourier transformations to find the solutions of this type of equations. +

    -One example of a non-sinusoidal periodic force is a square wave. Many -components in electric circuits are non-linear, e.g. diodes, which -makes many wave forms non-sinusoidal even when the circuits are being -driven by purely sinusoidal sources. +If one has several driving forces, \( F(t)=\sum_n F_n(t) \), one can find +the particular solution to each \( F_n \), \( x_{pn}(t) \), and the particular +solution for the entire driving force is then given by a series like + +$$ +\begin{equation} +x_p(t)=\sum_nx_{pn}(t). +\tag{21} +\end{equation} +$$

    @@ -457,7 +454,7 @@ driven by purely sinusoidal sources.

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    • diff --git a/doc/pub/week42/html/._week42-bs066.html b/doc/pub/week42/html/._week42-bs066.html index 9f40ef9e7..ebd030362 100644 --- a/doc/pub/week42/html/._week42-bs066.html +++ b/doc/pub/week42/html/._week42-bs066.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -404,46 +406,33 @@ MathJax.Hub.Config({ -

    Simple Code Example

    +

    Principle of Superposition

    -The code here shows a typical example of such a square wave generated using the functionality included in the scipy Python package. We have used a period of \( \tau=0.2 \). +This is known as the principle of superposition. It only applies when +the homogenous equation is linear. If there were an anharmonic term +such as \( x^3 \) in the homogenous equation, then when one summed various +solutions, \( x=(\sum_n x_n)^2 \), one would get cross +terms. Superposition is especially useful when \( F(t) \) can be written +as a sum of sinusoidal terms, because the solutions for each +sinusoidal (sine or cosine) term is analytic.

    - - -

    import numpy as np
-import math
-from scipy import signal
-import matplotlib.pyplot as plt
-
-# number of points                                                                                       
-n = 500
-# start and final times                                                                                  
-t0 = 0.0
-tn = 1.0
-# Period                                                                                                 
-t = np.linspace(t0, tn, n, endpoint=False)
-SqrSignal = np.zeros(n)
-SqrSignal = 1.0+signal.square(2*np.pi*5*t)
-plt.plot(t, SqrSignal)
-plt.ylim(-0.5, 2.5)
-plt.show()
-
    -

    -For the sinusoidal example the -period is \( \tau=2\pi/\omega \). However, higher harmonics can also -satisfy the periodicity requirement. In general, any force that -satisfies the periodicity requirement can be expressed as a sum over -harmonics, +Driving forces are often periodic, even when they are not +sinusoidal. Periodicity implies that for some time \( \tau \) $$ -\begin{equation} -F(t)=\frac{f_0}{2}+\sum_{n>0} f_n\cos(2n\pi t/\tau)+g_n\sin(2n\pi t/\tau). -\tag{22} -\end{equation} +\begin{eqnarray} +F(t+\tau)=F(t). +\end{eqnarray} $$ +

    +One example of a non-sinusoidal periodic force is a square wave. Many +components in electric circuits are non-linear, e.g. diodes, which +makes many wave forms non-sinusoidal even when the circuits are being +driven by purely sinusoidal sources. +

    @@ -470,7 +459,7 @@ $$

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    • diff --git a/doc/pub/week42/html/._week42-bs067.html b/doc/pub/week42/html/._week42-bs067.html index 2a595368d..a0ad453a1 100644 --- a/doc/pub/week42/html/._week42-bs067.html +++ b/doc/pub/week42/html/._week42-bs067.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -404,37 +406,46 @@ MathJax.Hub.Config({ -

    Wrapping up Fourier transforms

    +

    Simple Code Example

    -We can write down the answer for -\( x_{pn}(t) \), by substituting \( f_n/m \) or \( g_n/m \) for \( F_0/m \). By -writing each factor \( 2n\pi t/\tau \) as \( n\omega t \), with \( \omega\equiv -2\pi/\tau \), +The code here shows a typical example of such a square wave generated using the functionality included in the scipy Python package. We have used a period of \( \tau=0.2 \). + +

    + + +

    import numpy as np
+import math
+from scipy import signal
+import matplotlib.pyplot as plt
+
+# number of points                                                                                       
+n = 500
+# start and final times                                                                                  
+t0 = 0.0
+tn = 1.0
+# Period                                                                                                 
+t = np.linspace(t0, tn, n, endpoint=False)
+SqrSignal = np.zeros(n)
+SqrSignal = 1.0+signal.square(2*np.pi*5*t)
+plt.plot(t, SqrSignal)
+plt.ylim(-0.5, 2.5)
+plt.show()
+
    +

    +For the sinusoidal example the +period is \( \tau=2\pi/\omega \). However, higher harmonics can also +satisfy the periodicity requirement. In general, any force that +satisfies the periodicity requirement can be expressed as a sum over +harmonics, $$ \begin{equation} -\tag{23} -F(t)=\frac{f_0}{2}+\sum_{n>0}f_n\cos(n\omega t)+g_n\sin(n\omega t). +F(t)=\frac{f_0}{2}+\sum_{n>0} f_n\cos(2n\pi t/\tau)+g_n\sin(2n\pi t/\tau). +\tag{22} \end{equation} $$ -

    -The solutions for \( x(t) \) then come from replacing \( \omega \) with -\( n\omega \) for each term in the particular solution, - -$$ -\begin{eqnarray} -x_p(t)&=&\frac{f_0}{2k}+\sum_{n>0} \alpha_n\cos(n\omega t-\delta_n)+\beta_n\sin(n\omega t-\delta_n),\\ -\nonumber -\alpha_n&=&\frac{f_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\ -\nonumber -\beta_n&=&\frac{g_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\ -\nonumber -\delta_n&=&\tan^{-1}\left(\frac{2\beta n\omega}{\omega_0^2-n^2\omega^2}\right). -\end{eqnarray} -$$ -

    @@ -461,7 +472,7 @@ $$

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    • diff --git a/doc/pub/week42/html/._week42-bs068.html b/doc/pub/week42/html/._week42-bs068.html index 9006fba9e..0fd29e84b 100644 --- a/doc/pub/week42/html/._week42-bs068.html +++ b/doc/pub/week42/html/._week42-bs068.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -404,73 +406,37 @@ MathJax.Hub.Config({ -

    Finding the Coefficients

    +

    Wrapping up Fourier transforms

    -Because the forces have been applied for a long time, any non-zero -damping eliminates the homogenous parts of the solution, so one need -only consider the particular solution for each \( n \). - -

    -The problem is considered solved if one can find expressions for the -coefficients \( f_n \) and \( g_n \), even though the solutions are expressed -as an infinite sum. The coefficients can be extracted from the -function \( F(t) \) by - -$$ -\begin{eqnarray} -\tag{24} -f_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\cos(2n\pi t/\tau),\\ -\nonumber -g_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\sin(2n\pi t/\tau). -\end{eqnarray} -$$ - -

    -To check the consistency of these expressions and to verify -Eq. (24), one can insert the expansion of \( F(t) \) in -Eq. (23) into the expression for the coefficients in -Eq. (24) and see whether - -$$ -\begin{eqnarray} -f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~\left\{ -\frac{f_0}{2}+\sum_{m>0}f_m\cos(m\omega t)+g_m\sin(m\omega t) -\right\}\cos(n\omega t). -\end{eqnarray} -$$ - -

    -Immediately, one can throw away all the terms with \( g_m \) because they -convolute an even and an odd function. The term with \( f_0/2 \) -disappears because \( \cos(n\omega t) \) is equally positive and negative -over the interval and will integrate to zero. For all the terms -\( f_m\cos(m\omega t) \) appearing in the sum, one can use angle addition -formulas to see that \( \cos(m\omega t)\cos(n\omega -t)=(1/2)(\cos[(m+n)\omega t]+\cos[(m-n)\omega t] \). This will integrate -to zero unless \( m=n \). In that case the \( m=n \) term gives +We can write down the answer for +\( x_{pn}(t) \), by substituting \( f_n/m \) or \( g_n/m \) for \( F_0/m \). By +writing each factor \( 2n\pi t/\tau \) as \( n\omega t \), with \( \omega\equiv +2\pi/\tau \), $$ \begin{equation} -\int_{-\tau/2}^{\tau/2}dt~\cos^2(m\omega t)=\frac{\tau}{2}, -\tag{25} +\tag{23} +F(t)=\frac{f_0}{2}+\sum_{n>0}f_n\cos(n\omega t)+g_n\sin(n\omega t). \end{equation} $$

    -and +The solutions for \( x(t) \) then come from replacing \( \omega \) with +\( n\omega \) for each term in the particular solution, $$ \begin{eqnarray} -f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~f_n/2\\ +x_p(t)&=&\frac{f_0}{2k}+\sum_{n>0} \alpha_n\cos(n\omega t-\delta_n)+\beta_n\sin(n\omega t-\delta_n),\\ \nonumber -&=&f_n~\checkmark. +\alpha_n&=&\frac{f_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\ +\nonumber +\beta_n&=&\frac{g_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\ +\nonumber +\delta_n&=&\tan^{-1}\left(\frac{2\beta n\omega}{\omega_0^2-n^2\omega^2}\right). \end{eqnarray} $$ -

    -The same method can be used to check for the consistency of \( g_n \). -

    @@ -497,7 +463,7 @@ The same method can be used to check for the consistency of \( g_n \).

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    • diff --git a/doc/pub/week42/html/._week42-bs069.html b/doc/pub/week42/html/._week42-bs069.html index 69064c3d9..877a6ebbf 100644 --- a/doc/pub/week42/html/._week42-bs069.html +++ b/doc/pub/week42/html/._week42-bs069.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -404,50 +406,73 @@ MathJax.Hub.Config({ -

    Final words on Fourier Transforms

    +

    Finding the Coefficients

    -The code here uses the Fourier series applied to a -square wave signal. The code here -visualizes the various approximations given by Fourier series compared -with a square wave with period \( T=0.2 \) (dimensionless time), width \( 0.1 \) and max value of the force \( F=2 \). We -see that when we increase the number of components in the Fourier -series, the Fourier series approximation gets closer and closer to the -square wave signal. +Because the forces have been applied for a long time, any non-zero +damping eliminates the homogenous parts of the solution, so one need +only consider the particular solution for each \( n \).

    +The problem is considered solved if one can find expressions for the +coefficients \( f_n \) and \( g_n \), even though the solutions are expressed +as an infinite sum. The coefficients can be extracted from the +function \( F(t) \) by - -

    import numpy as np
-import math
-from scipy import signal
-import matplotlib.pyplot as plt
+$$
+\begin{eqnarray}
+\tag{24}
+f_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\cos(2n\pi t/\tau),\\
+\nonumber
+g_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\sin(2n\pi t/\tau).
+\end{eqnarray}
+$$
+
+
    
+To check the consistency of these expressions and to verify
+Eq. (24), one can insert the expansion of \( F(t) \) in
+Eq. (23) into the expression for the coefficients in
+Eq. (24) and see whether
+
+$$
+\begin{eqnarray}
+f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~\left\{
+\frac{f_0}{2}+\sum_{m>0}f_m\cos(m\omega t)+g_m\sin(m\omega t)
+\right\}\cos(n\omega t).
+\end{eqnarray}
+$$
+
+
    
+Immediately, one can throw away all the terms with \( g_m \) because they
+convolute an even and an odd function. The term with \( f_0/2 \)
+disappears because \( \cos(n\omega t) \) is equally positive and negative
+over the interval and will integrate to zero. For all the terms
+\( f_m\cos(m\omega t) \) appearing in the sum, one can use angle addition
+formulas to see that \( \cos(m\omega t)\cos(n\omega
+t)=(1/2)(\cos[(m+n)\omega t]+\cos[(m-n)\omega t] \). This will integrate
+to zero unless \( m=n \). In that case the \( m=n \) term gives
+
+$$
+\begin{equation}
+\int_{-\tau/2}^{\tau/2}dt~\cos^2(m\omega t)=\frac{\tau}{2},
+\tag{25}
+\end{equation}
+$$
+
+
    
+and
+
+$$
+\begin{eqnarray}
+f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~f_n/2\\
+\nonumber
+&=&f_n~\checkmark.
+\end{eqnarray}
+$$
+
+
    
+The same method can be used to check for the consistency of \( g_n \).
 
-# number of points                                                                                       
-n = 500
-# start and final times                                                                                  
-t0 = 0.0
-tn = 1.0
-# Period                                                                                                 
-T =0.2
-# Max value of square signal                                                                             
-Fmax= 2.0
-# Width of signal   
-Width = 0.1
-t = np.linspace(t0, tn, n, endpoint=False)
-SqrSignal = np.zeros(n)
-FourierSeriesSignal = np.zeros(n)
-SqrSignal = 1.0+signal.square(2*np.pi*5*t+np.pi*Width/T)
-a0 = Fmax*Width/T
-FourierSeriesSignal = a0
-Factor = 2.0*Fmax/np.pi
-for i in range(1,500):
-    FourierSeriesSignal += Factor/(i)*np.sin(np.pi*i*Width/T)*np.cos(i*t*2*np.pi/T)
-plt.plot(t, SqrSignal)
-plt.plot(t, FourierSeriesSignal)
-plt.ylim(-0.5, 2.5)
-plt.show()
-

    @@ -474,7 +499,7 @@ plt.show()

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    • diff --git a/doc/pub/week42/html/._week42-bs070.html b/doc/pub/week42/html/._week42-bs070.html index efba17262..de0345dcf 100644 --- a/doc/pub/week42/html/._week42-bs070.html +++ b/doc/pub/week42/html/._week42-bs070.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -404,26 +406,50 @@ MathJax.Hub.Config({ -

    Two-dimensional Objects

    +

    Final words on Fourier Transforms

    -We often use convolutions over more than one dimension at a time. If -we have a two-dimensional image \( I \) as input, we can have a filter -defined by a two-dimensional kernel \( K \). This leads to an output \( S \) - -$$ -S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(m,n)K(i-m,j-n). -$$ +The code here uses the Fourier series applied to a +square wave signal. The code here +visualizes the various approximations given by Fourier series compared +with a square wave with period \( T=0.2 \) (dimensionless time), width \( 0.1 \) and max value of the force \( F=2 \). We +see that when we increase the number of components in the Fourier +series, the Fourier series approximation gets closer and closer to the +square wave signal.

    -Convolution is a commutatitave process, which means we can rewrite this equation as -$$ -S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(i-m,j-n)K(m,n). -$$ -

    -Normally the latter is more straightforward to implement in a machine elarning library since there is less variation in the range of values of \( m \) and \( n \). + +

    import numpy as np
+import math
+from scipy import signal
+import matplotlib.pyplot as plt
 
+# number of points                                                                                       
+n = 500
+# start and final times                                                                                  
+t0 = 0.0
+tn = 1.0
+# Period                                                                                                 
+T =0.2
+# Max value of square signal                                                                             
+Fmax= 2.0
+# Width of signal   
+Width = 0.1
+t = np.linspace(t0, tn, n, endpoint=False)
+SqrSignal = np.zeros(n)
+FourierSeriesSignal = np.zeros(n)
+SqrSignal = 1.0+signal.square(2*np.pi*5*t+np.pi*Width/T)
+a0 = Fmax*Width/T
+FourierSeriesSignal = a0
+Factor = 2.0*Fmax/np.pi
+for i in range(1,500):
+    FourierSeriesSignal += Factor/(i)*np.sin(np.pi*i*Width/T)*np.cos(i*t*2*np.pi/T)
+plt.plot(t, SqrSignal)
+plt.plot(t, FourierSeriesSignal)
+plt.ylim(-0.5, 2.5)
+plt.show()
+

    @@ -450,7 +476,7 @@ Normally the latter is more straightforward to implement in a machine elarning

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    • diff --git a/doc/pub/week42/html/._week42-bs071.html b/doc/pub/week42/html/._week42-bs071.html index 980c00a3b..7419cca74 100644 --- a/doc/pub/week42/html/._week42-bs071.html +++ b/doc/pub/week42/html/._week42-bs071.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -404,14 +406,26 @@ MathJax.Hub.Config({ -

    Cross-Correlation

    +

    Two-dimensional Objects

    -Many deep learning libraries implement cross-correlation instead of convolution +We often use convolutions over more than one dimension at a time. If +we have a two-dimensional image \( I \) as input, we can have a filter +defined by a two-dimensional kernel \( K \). This leads to an output \( S \) + $$ -S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(i+m,j-+)K(m,n). +S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(m,n)K(i-m,j-n). $$ +

    +Convolution is a commutatitave process, which means we can rewrite this equation as +$$ +S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(i-m,j-n)K(m,n). +$$ + +

    +Normally the latter is more straightforward to implement in a machine elarning library since there is less variation in the range of values of \( m \) and \( n \). +

    @@ -438,7 +452,7 @@ $$

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    • diff --git a/doc/pub/week42/html/._week42-bs072.html b/doc/pub/week42/html/._week42-bs072.html index 73f63ea26..0ff84b9ef 100644 --- a/doc/pub/week42/html/._week42-bs072.html +++ b/doc/pub/week42/html/._week42-bs072.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -404,34 +406,14 @@ MathJax.Hub.Config({ -

    More on Dimensionalities

    +

    Cross-Correlation

    -In feilds like signal processing (and imaging as well), one designs -so-called filters. These filters are defined by the convolutions and -are often hand-crafted. One may specify filters for smoothing, edge -detection, frequency reshaping, and similar operations. However with -neural networks the idea is to automatically learn the filters and use -many of them in conjunction with non-linear operations (activation -functions). - -

    -As an example consider a neural network operating on sound sequence -data. Assume that we an input vector \( \boldsymbol{x} \) of length \( d=10^6 \). We -construct then a neural network with onle hidden layer only with -\( 10^4 \) nodes. This means that we will have a weight matrix with -\( 10^4\times 10^6=10^{10} \) weights to be determined, together with \( 10^4 \) biases. - -

    -Assume furthermore that we have an output layer which is meant to train whether the sound sequence represents a human voice (true) or something else (false). -It means that we have only one output node. But since this output node connects to \( 10^4 \) nodes in the hidden layer, there are in total \( 10^4 \) weights to be determined for the output layer, plus one bias. In total we have - +Many deep learning libraries implement cross-correlation instead of convolution $$ -\mathrm{NumberParameters}=10^{10}+10^4+10^4+1 \approx 10^{10}, +S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(i+m,j-+)K(m,n). $$ -that is ten billion parameters to determine. -

    @@ -458,7 +440,7 @@ that is ten billion parameters to determine.

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    • diff --git a/doc/pub/week42/html/._week42-bs073.html b/doc/pub/week42/html/._week42-bs073.html index 9a87f8ee5..225067b61 100644 --- a/doc/pub/week42/html/._week42-bs073.html +++ b/doc/pub/week42/html/._week42-bs073.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -404,25 +406,33 @@ MathJax.Hub.Config({ -

    Further Dimensionality Remarks

    +

    More on Dimensionalities

    -In today’s architecture one can train such neural networks, however -this is a huge number of parameters for the task at hand. In general, -it is a very wasteful and inefficient use of dense matrices as -parameters. Just as importantly, such trained network parameters are -very specific for the type of input data on which they were trained -and the network is not likely to generalize easily to variations in -the input. +In feilds like signal processing (and imaging as well), one designs +so-called filters. These filters are defined by the convolutions and +are often hand-crafted. One may specify filters for smoothing, edge +detection, frequency reshaping, and similar operations. However with +neural networks the idea is to automatically learn the filters and use +many of them in conjunction with non-linear operations (activation +functions).

    -The main principles that justify convolutions is locality of -information and repetion of patterns within the signal. Sound samples -of the input in adjacent spots are much more likely to affect each -other than those that are very far away. Similarly, sounds are -repeated in multiple times in the signal. While slightly simplistic, -reasoning about such a sound example demonstrates this. The same -principles then apply to images and other similar data. +As an example consider a neural network operating on sound sequence +data. Assume that we an input vector \( \boldsymbol{x} \) of length \( d=10^6 \). We +construct then a neural network with onle hidden layer only with +\( 10^4 \) nodes. This means that we will have a weight matrix with +\( 10^4\times 10^6=10^{10} \) weights to be determined, together with \( 10^4 \) biases. + +

    +Assume furthermore that we have an output layer which is meant to train whether the sound sequence represents a human voice (true) or something else (false). +It means that we have only one output node. But since this output node connects to \( 10^4 \) nodes in the hidden layer, there are in total \( 10^4 \) weights to be determined for the output layer, plus one bias. In total we have + +$$ +\mathrm{NumberParameters}=10^{10}+10^4+10^4+1 \approx 10^{10}, +$$ + +that is ten billion parameters to determine.

    @@ -450,7 +460,7 @@ principles then apply to images and other similar data.

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    • diff --git a/doc/pub/week42/html/._week42-bs074.html b/doc/pub/week42/html/._week42-bs074.html index a3e698681..0e6295d40 100644 --- a/doc/pub/week42/html/._week42-bs074.html +++ b/doc/pub/week42/html/._week42-bs074.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -404,18 +406,25 @@ MathJax.Hub.Config({ -

    CNNs in more detail, building convolutional neural networks in Tensorflow and Keras

    +

    Further Dimensionality Remarks

    -As discussed above, CNNs are neural networks built from the assumption that the inputs -to the network are 2D images. This is important because the number of features or pixels in images -grows very fast with the image size, and an enormous number of weights and biases are needed in order to build an accurate network. +In today’s architecture one can train such neural networks, however +this is a huge number of parameters for the task at hand. In general, +it is a very wasteful and inefficient use of dense matrices as +parameters. Just as importantly, such trained network parameters are +very specific for the type of input data on which they were trained +and the network is not likely to generalize easily to variations in +the input.

    -As before, we still have our input, a hidden layer and an output. What's novel about convolutional networks -are the convolutional and pooling layers stacked in pairs between the input and the hidden layer. -In addition, the data is no longer represented as a 2D feature matrix, instead each input is a number of 2D -matrices, typically 1 for each color dimension (Red, Green, Blue). +The main principles that justify convolutions is locality of +information and repetion of patterns within the signal. Sound samples +of the input in adjacent spots are much more likely to affect each +other than those that are very far away. Similarly, sounds are +repeated in multiple times in the signal. While slightly simplistic, +reasoning about such a sound example demonstrates this. The same +principles then apply to images and other similar data.

    @@ -443,7 +452,7 @@ matrices, typically 1 for each color dimension (Red, Green, Blue).

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    • diff --git a/doc/pub/week42/html/._week42-bs075.html b/doc/pub/week42/html/._week42-bs075.html index bd2f874b0..1f72f3c2e 100644 --- a/doc/pub/week42/html/._week42-bs075.html +++ b/doc/pub/week42/html/._week42-bs075.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -404,14 +406,18 @@ MathJax.Hub.Config({ -

    Setting it up

    +

    CNNs in more detail, building convolutional neural networks in Tensorflow and Keras

    -It means that to represent the entire -dataset of images, we require a 4D matrix or tensor. This tensor has the dimensions: -$$ -(n_{inputs},\, n_{pixels, width},\, n_{pixels, height},\, depth) . -$$ +As discussed above, CNNs are neural networks built from the assumption that the inputs +to the network are 2D images. This is important because the number of features or pixels in images +grows very fast with the image size, and an enormous number of weights and biases are needed in order to build an accurate network. + +

    +As before, we still have our input, a hidden layer and an output. What's novel about convolutional networks +are the convolutional and pooling layers stacked in pairs between the input and the hidden layer. +In addition, the data is no longer represented as a 2D feature matrix, instead each input is a number of 2D +matrices, typically 1 for each color dimension (Red, Green, Blue).

    @@ -439,7 +445,7 @@ $$

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    • diff --git a/doc/pub/week42/html/._week42-bs076.html b/doc/pub/week42/html/._week42-bs076.html index 85a98270f..d191be2b8 100644 --- a/doc/pub/week42/html/._week42-bs076.html +++ b/doc/pub/week42/html/._week42-bs076.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -404,20 +406,14 @@ MathJax.Hub.Config({ -

    The MNIST dataset again

    +

    Setting it up

    -The MNIST dataset consists of grayscale images with a pixel size of -\( 28\times 28 \), meaning we require \( 28 \times 28 = 724 \) weights to each -neuron in the first hidden layer. - -

    -If we were to analyze images of size \( 128\times 128 \) we would require -\( 128 \times 128 = 16384 \) weights to each neuron. Even worse if we were -dealing with color images, as most images are, we have an image matrix -of size \( 128\times 128 \) for each color dimension (Red, Green, Blue), -meaning 3 times the number of weights \( = 49152 \) are required for every -single neuron in the first hidden layer. +It means that to represent the entire +dataset of images, we require a 4D matrix or tensor. This tensor has the dimensions: +$$ +(n_{inputs},\, n_{pixels, width},\, n_{pixels, height},\, depth) . +$$

    @@ -445,7 +441,7 @@ single neuron in the first hidden layer.

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    • diff --git a/doc/pub/week42/html/._week42-bs077.html b/doc/pub/week42/html/._week42-bs077.html index fb2de04dd..35793cdf2 100644 --- a/doc/pub/week42/html/._week42-bs077.html +++ b/doc/pub/week42/html/._week42-bs077.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -404,20 +406,20 @@ MathJax.Hub.Config({ -

    Strong correlations

    +

    The MNIST dataset again

    -Images typically have strong local correlations, meaning that a small -part of the image varies little from its neighboring regions. If for -example we have an image of a blue car, we can roughly assume that a -small blue part of the image is surrounded by other blue regions. +The MNIST dataset consists of grayscale images with a pixel size of +\( 28\times 28 \), meaning we require \( 28 \times 28 = 724 \) weights to each +neuron in the first hidden layer.

    -Therefore, instead of connecting every single pixel to a neuron in the -first hidden layer, as we have previously done with deep neural -networks, we can instead connect each neuron to a small part of the -image (in all 3 RGB depth dimensions). The size of each small area is -fixed, and known as a receptive. +If we were to analyze images of size \( 128\times 128 \) we would require +\( 128 \times 128 = 16384 \) weights to each neuron. Even worse if we were +dealing with color images, as most images are, we have an image matrix +of size \( 128\times 128 \) for each color dimension (Red, Green, Blue), +meaning 3 times the number of weights \( = 49152 \) are required for every +single neuron in the first hidden layer.

    @@ -445,7 +447,7 @@ fixed, and known as a 86

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    • diff --git a/doc/pub/week42/html/._week42-bs078.html b/doc/pub/week42/html/._week42-bs078.html index 582f564b7..36a5d8a13 100644 --- a/doc/pub/week42/html/._week42-bs078.html +++ b/doc/pub/week42/html/._week42-bs078.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -402,26 +404,22 @@ MathJax.Hub.Config({

     

     

     

    - + -

    Layers of a CNN

    -The layers of a convolutional neural network arrange neurons in 3D: width, height and depth. -The input image is typically a square matrix of depth 3. +

    Strong correlations

    -A convolution is performed on the image which outputs -a 3D volume of neurons. The weights to the input are arranged in a number of 2D matrices, known as filters. +Images typically have strong local correlations, meaning that a small +part of the image varies little from its neighboring regions. If for +example we have an image of a blue car, we can roughly assume that a +small blue part of the image is surrounded by other blue regions.

    -Each filter slides along the input image, taking the dot product -between each small part of the image and the filter, in all depth -dimensions. This is then passed through a non-linear function, -typically the Rectified Linear (ReLu) function, which serves as the -activation of the neurons in the first convolutional layer. This is -further passed through a pooling layer, which reduces the size of the -convolutional layer, e.g. by taking the maximum or average across some -small regions, and this serves as input to the next convolutional -layer. +Therefore, instead of connecting every single pixel to a neuron in the +first hidden layer, as we have previously done with deep neural +networks, we can instead connect each neuron to a small part of the +image (in all 3 RGB depth dimensions). The size of each small area is +fixed, and known as a receptive.

    @@ -449,7 +447,7 @@ layer.

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    • diff --git a/doc/pub/week42/html/._week42-bs079.html b/doc/pub/week42/html/._week42-bs079.html index 5042ed1cd..a3f4c3e4a 100644 --- a/doc/pub/week42/html/._week42-bs079.html +++ b/doc/pub/week42/html/._week42-bs079.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -402,19 +404,26 @@ MathJax.Hub.Config({

     

     

     

    - + -

    Systematic reduction

    +

    Layers of a CNN

    +The layers of a convolutional neural network arrange neurons in 3D: width, height and depth. +The input image is typically a square matrix of depth 3.

    -By systematically reducing the size of the input volume, through -convolution and pooling, the network should create representations of -small parts of the input, and then from them assemble representations -of larger areas. The final pooling layer is flattened to serve as -input to a hidden layer, such that each neuron in the final pooling -layer is connected to every single neuron in the hidden layer. This -then serves as input to the output layer, e.g. a softmax output for -classification. +A convolution is performed on the image which outputs +a 3D volume of neurons. The weights to the input are arranged in a number of 2D matrices, known as filters. + +

    +Each filter slides along the input image, taking the dot product +between each small part of the image and the filter, in all depth +dimensions. This is then passed through a non-linear function, +typically the Rectified Linear (ReLu) function, which serves as the +activation of the neurons in the first convolutional layer. This is +further passed through a pooling layer, which reduces the size of the +convolutional layer, e.g. by taking the maximum or average across some +small regions, and this serves as input to the next convolutional +layer.

    @@ -442,7 +451,7 @@ classification.

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    • -
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  • »
    • diff --git a/doc/pub/week42/html/._week42-bs080.html b/doc/pub/week42/html/._week42-bs080.html index 73a7d11e6..208acf973 100644 --- a/doc/pub/week42/html/._week42-bs080.html +++ b/doc/pub/week42/html/._week42-bs080.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -404,51 +406,18 @@ MathJax.Hub.Config({ -

    Prerequisites: Collect and pre-process data

    +

    Systematic reduction

    +

    +By systematically reducing the size of the input volume, through +convolution and pooling, the network should create representations of +small parts of the input, and then from them assemble representations +of larger areas. The final pooling layer is flattened to serve as +input to a hidden layer, such that each neuron in the final pooling +layer is connected to every single neuron in the hidden layer. This +then serves as input to the output layer, e.g. a softmax output for +classification. - -

    # import necessary packages
-import numpy as np
-import matplotlib.pyplot as plt
-from sklearn import datasets
-
-
-# ensure the same random numbers appear every time
-np.random.seed(0)
-
-# display images in notebook
-%matplotlib inline
-plt.rcParams['figure.figsize'] = (12,12)
-
-
-# download MNIST dataset
-digits = datasets.load_digits()
-
-# define inputs and labels
-inputs = digits.images
-labels = digits.target
-
-# RGB images have a depth of 3
-# our images are grayscale so they should have a depth of 1
-inputs = inputs[:,:,:,np.newaxis]
-
-print("inputs = (n_inputs, pixel_width, pixel_height, depth) = " + str(inputs.shape))
-print("labels = (n_inputs) = " + str(labels.shape))
-
-
-# choose some random images to display
-n_inputs = len(inputs)
-indices = np.arange(n_inputs)
-random_indices = np.random.choice(indices, size=5)
-
-for i, image in enumerate(digits.images[random_indices]):
-    plt.subplot(1, 5, i+1)
-    plt.axis('off')
-    plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')
-    plt.title("Label: %d" % digits.target[random_indices[i]])
-plt.show()
-

    @@ -475,7 +444,7 @@ plt.show()

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    • diff --git a/doc/pub/week42/html/._week42-bs081.html b/doc/pub/week42/html/._week42-bs081.html index 75dc2d339..a86192f84 100644 --- a/doc/pub/week42/html/._week42-bs081.html +++ b/doc/pub/week42/html/._week42-bs081.html @@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source 'layers-used-to-build-cnns'), ('Transforming images', 2, None, 'transforming-images'), ('CNNs in brief', 2, None, 'cnns-in-brief'), + ('Key Idea', 2, None, 'key-idea'), ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'), ('Convolution Examples: Polynomial multiplication', 2, @@ -358,37 +359,38 @@ MathJax.Hub.Config({
  • Layers used to build CNNs
  • Transforming images
  • CNNs in brief
    • -
  • Mathematics of CNNs
    • -
  • Convolution Examples: Polynomial multiplication
    • -
  • Efficient Polynomial Multiplication
    • -
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
    • -
  • Two-dimensional Objects
    • -
  • Cross-Correlation
    • -
  • More on Dimensionalities
    • -
  • Further Dimensionality Remarks
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • The CIFAR01 data set
    • -
  • Verifying the data set
    • -
  • Set up the model
    • -
  • Add Dense layers on top
    • -
  • Compile and train the model
    • -
  • Finally, evaluate the model
    • +
  • Key Idea
    • +
  • Mathematics of CNNs
    • +
  • Convolution Examples: Polynomial multiplication
    • +
  • Efficient Polynomial Multiplication
    • +
  • A more efficient way of coding the above Convolution
    • +
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • +
  • Principle of Superposition
    • +
  • Simple Code Example
    • +
  • Wrapping up Fourier transforms
    • +
  • Finding the Coefficients
    • +
  • Final words on Fourier Transforms
    • +
  • Two-dimensional Objects
    • +
  • Cross-Correlation
    • +
  • More on Dimensionalities
    • +
  • Further Dimensionality Remarks
    • +
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • +
  • Setting it up
    • +
  • The MNIST dataset again
    • +
  • Strong correlations
    • +
  • Layers of a CNN
    • +
  • Systematic reduction
    • +
  • Prerequisites: Collect and pre-process data
    • +
  • Importing Keras and Tensorflow
    • +
  • Running with Keras
    • +
  • Final part
    • +
  • Final visualization
    • +
  • The CIFAR01 data set
    • +
  • Verifying the data set
    • +
  • Set up the model
    • +
  • Add Dense layers on top
    • +
  • Compile and train the model
    • +
  • Finally, evaluate the model
    • @@ -404,32 +406,50 @@ MathJax.Hub.Config({ -

    Importing Keras and Tensorflow

    +

    Prerequisites: Collect and pre-process data

    -

    from tensorflow.keras import datasets, layers, models
-from tensorflow.keras.layers import Input
-from tensorflow.keras.models import Sequential      #This allows appending layers to existing models
-from tensorflow.keras.layers import Dense           #This allows defining the characteristics of a particular layer
-from tensorflow.keras import optimizers             #This allows using whichever optimiser we want (sgd,adam,RMSprop)
-from tensorflow.keras import regularizers           #This allows using whichever regularizer we want (l1,l2,l1_l2)
-from tensorflow.keras.utils import to_categorical   #This allows using categorical cross entropy as the cost function
-#from tensorflow.keras import Conv2D
-#from tensorflow.keras import MaxPooling2D
-#from tensorflow.keras import Flatten
+
    # import necessary packages
+import numpy as np
+import matplotlib.pyplot as plt
+from sklearn import datasets
 
-from sklearn.model_selection import train_test_split
 
-# representation of labels
-labels = to_categorical(labels)
+# ensure the same random numbers appear every time
+np.random.seed(0)
 
-# split into train and test data
-# one-liner from scikit-learn library
-train_size = 0.8
-test_size = 1 - train_size
-X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,
-                                                    test_size=test_size)
+# display images in notebook
+%matplotlib inline
+plt.rcParams['figure.figsize'] = (12,12)
+
+
+# download MNIST dataset
+digits = datasets.load_digits()
+
+# define inputs and labels
+inputs = digits.images
+labels = digits.target
+
+# RGB images have a depth of 3
+# our images are grayscale so they should have a depth of 1
+inputs = inputs[:,:,:,np.newaxis]
+
+print("inputs = (n_inputs, pixel_width, pixel_height, depth) = " + str(inputs.shape))
+print("labels = (n_inputs) = " + str(labels.shape))
+
+
+# choose some random images to display
+n_inputs = len(inputs)
+indices = np.arange(n_inputs)
+random_indices = np.random.choice(indices, size=5)
+
+for i, image in enumerate(digits.images[random_indices]):
+    plt.subplot(1, 5, i+1)
+    plt.axis('off')
+    plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')
+    plt.title("Label: %d" % digits.target[random_indices[i]])
+plt.show()
 
    
 
    
 
    
@@ -456,6 +476,8 @@ X_train, X_test, Y_train, Y_test = train_tes
   
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+  
  • ...
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+  
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    • 
 
 
diff --git a/doc/pub/week42/html/._week42-bs082.html b/doc/pub/week42/html/._week42-bs082.html
index b0fef9b9a..8340a943b 100644
--- a/doc/pub/week42/html/._week42-bs082.html
+++ b/doc/pub/week42/html/._week42-bs082.html
@@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source
                'layers-used-to-build-cnns'),
               ('Transforming images', 2, None, 'transforming-images'),
               ('CNNs in brief', 2, None, 'cnns-in-brief'),
+              ('Key Idea', 2, None, 'key-idea'),
               ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'),
               ('Convolution Examples: Polynomial multiplication',
                2,
@@ -358,37 +359,38 @@ MathJax.Hub.Config({
       
  • Layers used to build CNNs
    • 
       
  • Transforming images
    • 
       
  • CNNs in brief
    • 
-      
  • Mathematics of CNNs
    • 
-      
  • Convolution Examples: Polynomial multiplication
    • 
-      
  • Efficient Polynomial Multiplication
    • 
-      
  • A more efficient way of coding the above Convolution
    • 
-      
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • 
-      
  • Principle of Superposition
    • 
-      
  • Simple Code Example
    • 
-      
  • Wrapping up Fourier transforms
    • 
-      
  • Finding the Coefficients
    • 
-      
  • Final words on Fourier Transforms
    • 
-      
  • Two-dimensional Objects
    • 
-      
  • Cross-Correlation
    • 
-      
  • More on Dimensionalities
    • 
-      
  • Further Dimensionality Remarks
    • 
-      
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • 
-      
  • Setting it up
    • 
-      
  • The MNIST dataset again
    • 
-      
  • Strong correlations
    • 
-      
  • Layers of a CNN
    • 
-      
  • Systematic reduction
    • 
-      
  • Prerequisites: Collect and pre-process data
    • 
-      
  • Importing Keras and Tensorflow
    • 
-      
  • Running with Keras
    • 
-      
  • Final part
    • 
-      
  • Final visualization
    • 
-      
  • The CIFAR01 data set
    • 
-      
  • Verifying the data set
    • 
-      
  • Set up  the model
    • 
-      
  • Add Dense layers on top
    • 
-      
  • Compile and train the model
    • 
-      
  • Finally, evaluate the model
    • 
+      
  • Key Idea
    • 
+      
  • Mathematics of CNNs
    • 
+      
  • Convolution Examples: Polynomial multiplication
    • 
+      
  • Efficient Polynomial Multiplication
    • 
+      
  • A more efficient way of coding the above Convolution
    • 
+      
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • 
+      
  • Principle of Superposition
    • 
+      
  • Simple Code Example
    • 
+      
  • Wrapping up Fourier transforms
    • 
+      
  • Finding the Coefficients
    • 
+      
  • Final words on Fourier Transforms
    • 
+      
  • Two-dimensional Objects
    • 
+      
  • Cross-Correlation
    • 
+      
  • More on Dimensionalities
    • 
+      
  • Further Dimensionality Remarks
    • 
+      
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • 
+      
  • Setting it up
    • 
+      
  • The MNIST dataset again
    • 
+      
  • Strong correlations
    • 
+      
  • Layers of a CNN
    • 
+      
  • Systematic reduction
    • 
+      
  • Prerequisites: Collect and pre-process data
    • 
+      
  • Importing Keras and Tensorflow
    • 
+      
  • Running with Keras
    • 
+      
  • Final part
    • 
+      
  • Final visualization
    • 
+      
  • The CIFAR01 data set
    • 
+      
  • Verifying the data set
    • 
+      
  • Set up  the model
    • 
+      
  • Add Dense layers on top
    • 
+      
  • Compile and train the model
    • 
+      
  • Finally, evaluate the model
    • 
 
         
       
@@ -402,39 +404,34 @@ MathJax.Hub.Config({
 
     
     
     
     
 
 
-
-
-
    Running with Keras 
    
+
 
+
    Importing Keras and Tensorflow 
    
 
    
 
 
-
    def create_convolutional_neural_network_keras(input_shape, receptive_field,
-                                              n_filters, n_neurons_connected, n_categories,
-                                              eta, lmbd):
-    model = Sequential()
-    model.add(layers.Conv2D(n_filters, (receptive_field, receptive_field), input_shape=input_shape, padding='same',
-              activation='relu', kernel_regularizer=regularizers.l2(lmbd)))
-    model.add(layers.MaxPooling2D(pool_size=(2, 2)))
-    model.add(layers.Flatten())
-    model.add(layers.Dense(n_neurons_connected, activation='relu', kernel_regularizer=regularizers.l2(lmbd)))
-    model.add(layers.Dense(n_categories, activation='softmax', kernel_regularizer=regularizers.l2(lmbd)))
-    
-    sgd = optimizers.SGD(lr=eta)
-    model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy'])
-    
-    return model
+
    from tensorflow.keras import datasets, layers, models
+from tensorflow.keras.layers import Input
+from tensorflow.keras.models import Sequential      #This allows appending layers to existing models
+from tensorflow.keras.layers import Dense           #This allows defining the characteristics of a particular layer
+from tensorflow.keras import optimizers             #This allows using whichever optimiser we want (sgd,adam,RMSprop)
+from tensorflow.keras import regularizers           #This allows using whichever regularizer we want (l1,l2,l1_l2)
+from tensorflow.keras.utils import to_categorical   #This allows using categorical cross entropy as the cost function
+#from tensorflow.keras import Conv2D
+#from tensorflow.keras import MaxPooling2D
+#from tensorflow.keras import Flatten
 
-epochs = 100
-batch_size = 100
-input_shape = X_train.shape[1:4]
-receptive_field = 3
-n_filters = 10
-n_neurons_connected = 50
-n_categories = 10
+from sklearn.model_selection import train_test_split
 
-eta_vals = np.logspace(-5, 1, 7)
-lmbd_vals = np.logspace(-5, 1, 7)
+# representation of labels
+labels = to_categorical(labels)
+
+# split into train and test data
+# one-liner from scikit-learn library
+train_size = 0.8
+test_size = 1 - train_size
+X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,
+                                                    test_size=test_size)
 
    
 
    
 
    
@@ -460,6 +457,7 @@ lmbd_vals = np.
   
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diff --git a/doc/pub/week42/html/._week42-bs083.html b/doc/pub/week42/html/._week42-bs083.html
index 21e2ec214..40c6de957 100644
--- a/doc/pub/week42/html/._week42-bs083.html
+++ b/doc/pub/week42/html/._week42-bs083.html
@@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source
                'layers-used-to-build-cnns'),
               ('Transforming images', 2, None, 'transforming-images'),
               ('CNNs in brief', 2, None, 'cnns-in-brief'),
+              ('Key Idea', 2, None, 'key-idea'),
               ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'),
               ('Convolution Examples: Polynomial multiplication',
                2,
@@ -358,37 +359,38 @@ MathJax.Hub.Config({
       
  • Layers used to build CNNs
    • 
       
  • Transforming images
    • 
       
  • CNNs in brief
    • 
-      
  • Mathematics of CNNs
    • 
-      
  • Convolution Examples: Polynomial multiplication
    • 
-      
  • Efficient Polynomial Multiplication
    • 
-      
  • A more efficient way of coding the above Convolution
    • 
-      
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • 
-      
  • Principle of Superposition
    • 
-      
  • Simple Code Example
    • 
-      
  • Wrapping up Fourier transforms
    • 
-      
  • Finding the Coefficients
    • 
-      
  • Final words on Fourier Transforms
    • 
-      
  • Two-dimensional Objects
    • 
-      
  • Cross-Correlation
    • 
-      
  • More on Dimensionalities
    • 
-      
  • Further Dimensionality Remarks
    • 
-      
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • 
-      
  • Setting it up
    • 
-      
  • The MNIST dataset again
    • 
-      
  • Strong correlations
    • 
-      
  • Layers of a CNN
    • 
-      
  • Systematic reduction
    • 
-      
  • Prerequisites: Collect and pre-process data
    • 
-      
  • Importing Keras and Tensorflow
    • 
-      
  • Running with Keras
    • 
-      
  • Final part
    • 
-      
  • Final visualization
    • 
-      
  • The CIFAR01 data set
    • 
-      
  • Verifying the data set
    • 
-      
  • Set up  the model
    • 
-      
  • Add Dense layers on top
    • 
-      
  • Compile and train the model
    • 
-      
  • Finally, evaluate the model
    • 
+      
  • Key Idea
    • 
+      
  • Mathematics of CNNs
    • 
+      
  • Convolution Examples: Polynomial multiplication
    • 
+      
  • Efficient Polynomial Multiplication
    • 
+      
  • A more efficient way of coding the above Convolution
    • 
+      
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • 
+      
  • Principle of Superposition
    • 
+      
  • Simple Code Example
    • 
+      
  • Wrapping up Fourier transforms
    • 
+      
  • Finding the Coefficients
    • 
+      
  • Final words on Fourier Transforms
    • 
+      
  • Two-dimensional Objects
    • 
+      
  • Cross-Correlation
    • 
+      
  • More on Dimensionalities
    • 
+      
  • Further Dimensionality Remarks
    • 
+      
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • 
+      
  • Setting it up
    • 
+      
  • The MNIST dataset again
    • 
+      
  • Strong correlations
    • 
+      
  • Layers of a CNN
    • 
+      
  • Systematic reduction
    • 
+      
  • Prerequisites: Collect and pre-process data
    • 
+      
  • Importing Keras and Tensorflow
    • 
+      
  • Running with Keras
    • 
+      
  • Final part
    • 
+      
  • Final visualization
    • 
+      
  • The CIFAR01 data set
    • 
+      
  • Verifying the data set
    • 
+      
  • Set up  the model
    • 
+      
  • Add Dense layers on top
    • 
+      
  • Compile and train the model
    • 
+      
  • Finally, evaluate the model
    • 
 
         
       
@@ -402,29 +404,39 @@ MathJax.Hub.Config({
 
     
     
     
     
 
 
-
+
 
-
    Final part 
    
+
    Running with Keras 
    
 
 
    
 
 
-
    CNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
-        
-for i, eta in enumerate(eta_vals):
-    for j, lmbd in enumerate(lmbd_vals):
-        CNN = create_convolutional_neural_network_keras(input_shape, receptive_field,
+
    def create_convolutional_neural_network_keras(input_shape, receptive_field,
                                               n_filters, n_neurons_connected, n_categories,
-                                              eta, lmbd)
-        CNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)
-        scores = CNN.evaluate(X_test, Y_test)
-        
-        CNN_keras[i][j] = CNN
-        
-        print("Learning rate = ", eta)
-        print("Lambda = ", lmbd)
-        print("Test accuracy: %.3f" % scores[1])
-        print()
+                                              eta, lmbd):
+    model = Sequential()
+    model.add(layers.Conv2D(n_filters, (receptive_field, receptive_field), input_shape=input_shape, padding='same',
+              activation='relu', kernel_regularizer=regularizers.l2(lmbd)))
+    model.add(layers.MaxPooling2D(pool_size=(2, 2)))
+    model.add(layers.Flatten())
+    model.add(layers.Dense(n_neurons_connected, activation='relu', kernel_regularizer=regularizers.l2(lmbd)))
+    model.add(layers.Dense(n_categories, activation='softmax', kernel_regularizer=regularizers.l2(lmbd)))
+    
+    sgd = optimizers.SGD(lr=eta)
+    model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy'])
+    
+    return model
+
+epochs = 100
+batch_size = 100
+input_shape = X_train.shape[1:4]
+receptive_field = 3
+n_filters = 10
+n_neurons_connected = 50
+n_categories = 10
+
+eta_vals = np.logspace(-5, 1, 7)
+lmbd_vals = np.logspace(-5, 1, 7)
 
    
 
    
 
    
@@ -449,6 +461,7 @@ MathJax.Hub.Config({
   
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diff --git a/doc/pub/week42/html/._week42-bs084.html b/doc/pub/week42/html/._week42-bs084.html
index 6ebb54a7b..70dbd7d04 100644
--- a/doc/pub/week42/html/._week42-bs084.html
+++ b/doc/pub/week42/html/._week42-bs084.html
@@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source
                'layers-used-to-build-cnns'),
               ('Transforming images', 2, None, 'transforming-images'),
               ('CNNs in brief', 2, None, 'cnns-in-brief'),
+              ('Key Idea', 2, None, 'key-idea'),
               ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'),
               ('Convolution Examples: Polynomial multiplication',
                2,
@@ -358,37 +359,38 @@ MathJax.Hub.Config({
       
  • Layers used to build CNNs
    • 
       
  • Transforming images
    • 
       
  • CNNs in brief
    • 
-      
  • Mathematics of CNNs
    • 
-      
  • Convolution Examples: Polynomial multiplication
    • 
-      
  • Efficient Polynomial Multiplication
    • 
-      
  • A more efficient way of coding the above Convolution
    • 
-      
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • 
-      
  • Principle of Superposition
    • 
-      
  • Simple Code Example
    • 
-      
  • Wrapping up Fourier transforms
    • 
-      
  • Finding the Coefficients
    • 
-      
  • Final words on Fourier Transforms
    • 
-      
  • Two-dimensional Objects
    • 
-      
  • Cross-Correlation
    • 
-      
  • More on Dimensionalities
    • 
-      
  • Further Dimensionality Remarks
    • 
-      
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • 
-      
  • Setting it up
    • 
-      
  • The MNIST dataset again
    • 
-      
  • Strong correlations
    • 
-      
  • Layers of a CNN
    • 
-      
  • Systematic reduction
    • 
-      
  • Prerequisites: Collect and pre-process data
    • 
-      
  • Importing Keras and Tensorflow
    • 
-      
  • Running with Keras
    • 
-      
  • Final part
    • 
-      
  • Final visualization
    • 
-      
  • The CIFAR01 data set
    • 
-      
  • Verifying the data set
    • 
-      
  • Set up  the model
    • 
-      
  • Add Dense layers on top
    • 
-      
  • Compile and train the model
    • 
-      
  • Finally, evaluate the model
    • 
+      
  • Key Idea
    • 
+      
  • Mathematics of CNNs
    • 
+      
  • Convolution Examples: Polynomial multiplication
    • 
+      
  • Efficient Polynomial Multiplication
    • 
+      
  • A more efficient way of coding the above Convolution
    • 
+      
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • 
+      
  • Principle of Superposition
    • 
+      
  • Simple Code Example
    • 
+      
  • Wrapping up Fourier transforms
    • 
+      
  • Finding the Coefficients
    • 
+      
  • Final words on Fourier Transforms
    • 
+      
  • Two-dimensional Objects
    • 
+      
  • Cross-Correlation
    • 
+      
  • More on Dimensionalities
    • 
+      
  • Further Dimensionality Remarks
    • 
+      
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • 
+      
  • Setting it up
    • 
+      
  • The MNIST dataset again
    • 
+      
  • Strong correlations
    • 
+      
  • Layers of a CNN
    • 
+      
  • Systematic reduction
    • 
+      
  • Prerequisites: Collect and pre-process data
    • 
+      
  • Importing Keras and Tensorflow
    • 
+      
  • Running with Keras
    • 
+      
  • Final part
    • 
+      
  • Final visualization
    • 
+      
  • The CIFAR01 data set
    • 
+      
  • Verifying the data set
    • 
+      
  • Set up  the model
    • 
+      
  • Add Dense layers on top
    • 
+      
  • Compile and train the model
    • 
+      
  • Finally, evaluate the model
    • 
 
         
       
@@ -404,41 +406,27 @@ MathJax.Hub.Config({
 
 
 
-
    Final visualization 
    
+
    Final part 
    
 
 
    
 
 
-
    # visual representation of grid search
-# uses seaborn heatmap, could probably do this in matplotlib
-import seaborn as sns
-
-sns.set()
-
-train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-
-for i in range(len(eta_vals)):
-    for j in range(len(lmbd_vals)):
-        CNN = CNN_keras[i][j]
-
-        train_accuracy[i][j] = CNN.evaluate(X_train, Y_train)[1]
-        test_accuracy[i][j] = CNN.evaluate(X_test, Y_test)[1]
-
+
    CNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
         
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Training Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
-
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Test Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
+for i, eta in enumerate(eta_vals):
+    for j, lmbd in enumerate(lmbd_vals):
+        CNN = create_convolutional_neural_network_keras(input_shape, receptive_field,
+                                              n_filters, n_neurons_connected, n_categories,
+                                              eta, lmbd)
+        CNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)
+        scores = CNN.evaluate(X_test, Y_test)
+        
+        CNN_keras[i][j] = CNN
+        
+        print("Learning rate = ", eta)
+        print("Lambda = ", lmbd)
+        print("Test accuracy: %.3f" % scores[1])
+        print()
 
    
 
    
 
    
@@ -462,6 +450,7 @@ plt.show()
   
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+  
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    • 
 
 
diff --git a/doc/pub/week42/html/._week42-bs085.html b/doc/pub/week42/html/._week42-bs085.html
index de3a7aae0..8c7408e29 100644
--- a/doc/pub/week42/html/._week42-bs085.html
+++ b/doc/pub/week42/html/._week42-bs085.html
@@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source
                'layers-used-to-build-cnns'),
               ('Transforming images', 2, None, 'transforming-images'),
               ('CNNs in brief', 2, None, 'cnns-in-brief'),
+              ('Key Idea', 2, None, 'key-idea'),
               ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'),
               ('Convolution Examples: Polynomial multiplication',
                2,
@@ -358,37 +359,38 @@ MathJax.Hub.Config({
       
  • Layers used to build CNNs
    • 
       
  • Transforming images
    • 
       
  • CNNs in brief
    • 
-      
  • Mathematics of CNNs
    • 
-      
  • Convolution Examples: Polynomial multiplication
    • 
-      
  • Efficient Polynomial Multiplication
    • 
-      
  • A more efficient way of coding the above Convolution
    • 
-      
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • 
-      
  • Principle of Superposition
    • 
-      
  • Simple Code Example
    • 
-      
  • Wrapping up Fourier transforms
    • 
-      
  • Finding the Coefficients
    • 
-      
  • Final words on Fourier Transforms
    • 
-      
  • Two-dimensional Objects
    • 
-      
  • Cross-Correlation
    • 
-      
  • More on Dimensionalities
    • 
-      
  • Further Dimensionality Remarks
    • 
-      
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • 
-      
  • Setting it up
    • 
-      
  • The MNIST dataset again
    • 
-      
  • Strong correlations
    • 
-      
  • Layers of a CNN
    • 
-      
  • Systematic reduction
    • 
-      
  • Prerequisites: Collect and pre-process data
    • 
-      
  • Importing Keras and Tensorflow
    • 
-      
  • Running with Keras
    • 
-      
  • Final part
    • 
-      
  • Final visualization
    • 
-      
  • The CIFAR01 data set
    • 
-      
  • Verifying the data set
    • 
-      
  • Set up  the model
    • 
-      
  • Add Dense layers on top
    • 
-      
  • Compile and train the model
    • 
-      
  • Finally, evaluate the model
    • 
+      
  • Key Idea
    • 
+      
  • Mathematics of CNNs
    • 
+      
  • Convolution Examples: Polynomial multiplication
    • 
+      
  • Efficient Polynomial Multiplication
    • 
+      
  • A more efficient way of coding the above Convolution
    • 
+      
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • 
+      
  • Principle of Superposition
    • 
+      
  • Simple Code Example
    • 
+      
  • Wrapping up Fourier transforms
    • 
+      
  • Finding the Coefficients
    • 
+      
  • Final words on Fourier Transforms
    • 
+      
  • Two-dimensional Objects
    • 
+      
  • Cross-Correlation
    • 
+      
  • More on Dimensionalities
    • 
+      
  • Further Dimensionality Remarks
    • 
+      
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • 
+      
  • Setting it up
    • 
+      
  • The MNIST dataset again
    • 
+      
  • Strong correlations
    • 
+      
  • Layers of a CNN
    • 
+      
  • Systematic reduction
    • 
+      
  • Prerequisites: Collect and pre-process data
    • 
+      
  • Importing Keras and Tensorflow
    • 
+      
  • Running with Keras
    • 
+      
  • Final part
    • 
+      
  • Final visualization
    • 
+      
  • The CIFAR01 data set
    • 
+      
  • Verifying the data set
    • 
+      
  • Set up  the model
    • 
+      
  • Add Dense layers on top
    • 
+      
  • Compile and train the model
    • 
+      
  • Finally, evaluate the model
    • 
 
         
       
@@ -404,27 +406,41 @@ MathJax.Hub.Config({
 
 
 
-
    The CIFAR01 data set 
    
-
-
    
-The CIFAR10 dataset contains 60,000 color images in 10 classes, with
-6,000 images in each class. The dataset is divided into 50,000
-training images and 10,000 testing images. The classes are mutually
-exclusive and there is no overlap between them.
+
    Final visualization 
    
 
 
    
 
 
-
    import tensorflow as tf
+
    # visual representation of grid search
+# uses seaborn heatmap, could probably do this in matplotlib
+import seaborn as sns
 
-from tensorflow.keras import datasets, layers, models
-import matplotlib.pyplot as plt
+sns.set()
 
-# We import the data set
-(train_images, train_labels), (test_images, test_labels) = datasets.cifar10.load_data()
+train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
 
-# Normalize pixel values to be between 0 and 1 by dividing by 255. 
-train_images, test_images = train_images / 255.0, test_images / 255.0
+for i in range(len(eta_vals)):
+    for j in range(len(lmbd_vals)):
+        CNN = CNN_keras[i][j]
+
+        train_accuracy[i][j] = CNN.evaluate(X_train, Y_train)[1]
+        test_accuracy[i][j] = CNN.evaluate(X_test, Y_test)[1]
+
+        
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Training Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Test Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
 
    
 
    
 
    
@@ -447,6 +463,7 @@ train_images, test_images = train_images 89
   
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diff --git a/doc/pub/week42/html/._week42-bs086.html b/doc/pub/week42/html/._week42-bs086.html
index f40123642..122a8e8fb 100644
--- a/doc/pub/week42/html/._week42-bs086.html
+++ b/doc/pub/week42/html/._week42-bs086.html
@@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source
                'layers-used-to-build-cnns'),
               ('Transforming images', 2, None, 'transforming-images'),
               ('CNNs in brief', 2, None, 'cnns-in-brief'),
+              ('Key Idea', 2, None, 'key-idea'),
               ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'),
               ('Convolution Examples: Polynomial multiplication',
                2,
@@ -358,37 +359,38 @@ MathJax.Hub.Config({
       
  • Layers used to build CNNs
    • 
       
  • Transforming images
    • 
       
  • CNNs in brief
    • 
-      
  • Mathematics of CNNs
    • 
-      
  • Convolution Examples: Polynomial multiplication
    • 
-      
  • Efficient Polynomial Multiplication
    • 
-      
  • A more efficient way of coding the above Convolution
    • 
-      
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • 
-      
  • Principle of Superposition
    • 
-      
  • Simple Code Example
    • 
-      
  • Wrapping up Fourier transforms
    • 
-      
  • Finding the Coefficients
    • 
-      
  • Final words on Fourier Transforms
    • 
-      
  • Two-dimensional Objects
    • 
-      
  • Cross-Correlation
    • 
-      
  • More on Dimensionalities
    • 
-      
  • Further Dimensionality Remarks
    • 
-      
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • 
-      
  • Setting it up
    • 
-      
  • The MNIST dataset again
    • 
-      
  • Strong correlations
    • 
-      
  • Layers of a CNN
    • 
-      
  • Systematic reduction
    • 
-      
  • Prerequisites: Collect and pre-process data
    • 
-      
  • Importing Keras and Tensorflow
    • 
-      
  • Running with Keras
    • 
-      
  • Final part
    • 
-      
  • Final visualization
    • 
-      
  • The CIFAR01 data set
    • 
-      
  • Verifying the data set
    • 
-      
  • Set up  the model
    • 
-      
  • Add Dense layers on top
    • 
-      
  • Compile and train the model
    • 
-      
  • Finally, evaluate the model
    • 
+      
  • Key Idea
    • 
+      
  • Mathematics of CNNs
    • 
+      
  • Convolution Examples: Polynomial multiplication
    • 
+      
  • Efficient Polynomial Multiplication
    • 
+      
  • A more efficient way of coding the above Convolution
    • 
+      
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • 
+      
  • Principle of Superposition
    • 
+      
  • Simple Code Example
    • 
+      
  • Wrapping up Fourier transforms
    • 
+      
  • Finding the Coefficients
    • 
+      
  • Final words on Fourier Transforms
    • 
+      
  • Two-dimensional Objects
    • 
+      
  • Cross-Correlation
    • 
+      
  • More on Dimensionalities
    • 
+      
  • Further Dimensionality Remarks
    • 
+      
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • 
+      
  • Setting it up
    • 
+      
  • The MNIST dataset again
    • 
+      
  • Strong correlations
    • 
+      
  • Layers of a CNN
    • 
+      
  • Systematic reduction
    • 
+      
  • Prerequisites: Collect and pre-process data
    • 
+      
  • Importing Keras and Tensorflow
    • 
+      
  • Running with Keras
    • 
+      
  • Final part
    • 
+      
  • Final visualization
    • 
+      
  • The CIFAR01 data set
    • 
+      
  • Verifying the data set
    • 
+      
  • Set up  the model
    • 
+      
  • Add Dense layers on top
    • 
+      
  • Compile and train the model
    • 
+      
  • Finally, evaluate the model
    • 
 
         
       
@@ -404,28 +406,27 @@ MathJax.Hub.Config({
 
 
 
-
    Verifying the data set 
    
+
    The CIFAR01 data set 
    
 
 
    
-To verify that the dataset looks correct, let's plot the first 25 images from the training set and display the class name below each image.
+The CIFAR10 dataset contains 60,000 color images in 10 classes, with
+6,000 images in each class. The dataset is divided into 50,000
+training images and 10,000 testing images. The classes are mutually
+exclusive and there is no overlap between them.
 
 
    
 
 
-
    class_names = ['airplane', 'automobile', 'bird', 'cat', 'deer',
-               'dog', 'frog', 'horse', 'ship', 'truck']
-​
-plt.figure(figsize=(10,10))
-for i in range(25):
-    plt.subplot(5,5,i+1)
-    plt.xticks([])
-    plt.yticks([])
-    plt.grid(False)
-    plt.imshow(train_images[i], cmap=plt.cm.binary)
-    # The CIFAR labels happen to be arrays, 
-    # which is why you need the extra index
-    plt.xlabel(class_names[train_labels[i][0]])
-plt.show()
+
    import tensorflow as tf
+
+from tensorflow.keras import datasets, layers, models
+import matplotlib.pyplot as plt
+
+# We import the data set
+(train_images, train_labels), (test_images, test_labels) = datasets.cifar10.load_data()
+
+# Normalize pixel values to be between 0 and 1 by dividing by 255. 
+train_images, test_images = train_images / 255.0, test_images / 255.0
 
    
 
    
 
    
@@ -447,6 +448,7 @@ plt.show()
   
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diff --git a/doc/pub/week42/html/._week42-bs087.html b/doc/pub/week42/html/._week42-bs087.html
index 15c11f9a5..a55fb61ea 100644
--- a/doc/pub/week42/html/._week42-bs087.html
+++ b/doc/pub/week42/html/._week42-bs087.html
@@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source
                'layers-used-to-build-cnns'),
               ('Transforming images', 2, None, 'transforming-images'),
               ('CNNs in brief', 2, None, 'cnns-in-brief'),
+              ('Key Idea', 2, None, 'key-idea'),
               ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'),
               ('Convolution Examples: Polynomial multiplication',
                2,
@@ -358,37 +359,38 @@ MathJax.Hub.Config({
       
  • Layers used to build CNNs
    • 
       
  • Transforming images
    • 
       
  • CNNs in brief
    • 
-      
  • Mathematics of CNNs
    • 
-      
  • Convolution Examples: Polynomial multiplication
    • 
-      
  • Efficient Polynomial Multiplication
    • 
-      
  • A more efficient way of coding the above Convolution
    • 
-      
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • 
-      
  • Principle of Superposition
    • 
-      
  • Simple Code Example
    • 
-      
  • Wrapping up Fourier transforms
    • 
-      
  • Finding the Coefficients
    • 
-      
  • Final words on Fourier Transforms
    • 
-      
  • Two-dimensional Objects
    • 
-      
  • Cross-Correlation
    • 
-      
  • More on Dimensionalities
    • 
-      
  • Further Dimensionality Remarks
    • 
-      
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • 
-      
  • Setting it up
    • 
-      
  • The MNIST dataset again
    • 
-      
  • Strong correlations
    • 
-      
  • Layers of a CNN
    • 
-      
  • Systematic reduction
    • 
-      
  • Prerequisites: Collect and pre-process data
    • 
-      
  • Importing Keras and Tensorflow
    • 
-      
  • Running with Keras
    • 
-      
  • Final part
    • 
-      
  • Final visualization
    • 
-      
  • The CIFAR01 data set
    • 
-      
  • Verifying the data set
    • 
-      
  • Set up  the model
    • 
-      
  • Add Dense layers on top
    • 
-      
  • Compile and train the model
    • 
-      
  • Finally, evaluate the model
    • 
+      
  • Key Idea
    • 
+      
  • Mathematics of CNNs
    • 
+      
  • Convolution Examples: Polynomial multiplication
    • 
+      
  • Efficient Polynomial Multiplication
    • 
+      
  • A more efficient way of coding the above Convolution
    • 
+      
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • 
+      
  • Principle of Superposition
    • 
+      
  • Simple Code Example
    • 
+      
  • Wrapping up Fourier transforms
    • 
+      
  • Finding the Coefficients
    • 
+      
  • Final words on Fourier Transforms
    • 
+      
  • Two-dimensional Objects
    • 
+      
  • Cross-Correlation
    • 
+      
  • More on Dimensionalities
    • 
+      
  • Further Dimensionality Remarks
    • 
+      
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • 
+      
  • Setting it up
    • 
+      
  • The MNIST dataset again
    • 
+      
  • Strong correlations
    • 
+      
  • Layers of a CNN
    • 
+      
  • Systematic reduction
    • 
+      
  • Prerequisites: Collect and pre-process data
    • 
+      
  • Importing Keras and Tensorflow
    • 
+      
  • Running with Keras
    • 
+      
  • Final part
    • 
+      
  • Final visualization
    • 
+      
  • The CIFAR01 data set
    • 
+      
  • Verifying the data set
    • 
+      
  • Set up  the model
    • 
+      
  • Add Dense layers on top
    • 
+      
  • Compile and train the model
    • 
+      
  • Finally, evaluate the model
    • 
 
         
       
@@ -404,31 +406,29 @@ MathJax.Hub.Config({
 
 
 
-
    Set up  the model 
    
+
    Verifying the data set 
    
 
 
    
-The 6 lines of code below define the convolutional base using a common pattern: a stack of Conv2D and MaxPooling2D layers.
-
-
    
-As input, a CNN takes tensors of shape (image_height, image_width, color_channels), ignoring the batch size. If you are new to these dimensions, color_channels refers to (R,G,B). In this example, you will configure our CNN to process inputs of shape (32, 32, 3), which is the format of CIFAR images. You can do this by passing the argument input_shape to our first layer.
+To verify that the dataset looks correct, let's plot the first 25 images from the training set and display the class name below each image.
 
 
    
 
 
-
    model = models.Sequential()
-model.add(layers.Conv2D(32, (3, 3), activation='relu', input_shape=(32, 32, 3)))
-model.add(layers.MaxPooling2D((2, 2)))
-model.add(layers.Conv2D(64, (3, 3), activation='relu'))
-model.add(layers.MaxPooling2D((2, 2)))
-model.add(layers.Conv2D(64, (3, 3), activation='relu'))
-
-# Let's display the architecture of our model so far.
-
-model.summary()
+
    class_names = ['airplane', 'automobile', 'bird', 'cat', 'deer',
+               'dog', 'frog', 'horse', 'ship', 'truck']
+​
+plt.figure(figsize=(10,10))
+for i in range(25):
+    plt.subplot(5,5,i+1)
+    plt.xticks([])
+    plt.yticks([])
+    plt.grid(False)
+    plt.imshow(train_images[i], cmap=plt.cm.binary)
+    # The CIFAR labels happen to be arrays, 
+    # which is why you need the extra index
+    plt.xlabel(class_names[train_labels[i][0]])
+plt.show()
 
    
-
    
-You can see that the output of every Conv2D and MaxPooling2D layer is a 3D tensor of shape (height, width, channels). The width and height dimensions tend to shrink as you go deeper in the network. The number of output channels for each Conv2D layer is controlled by the first argument (e.g., 32 or 64). Typically, as the width and height shrink, you can afford (computationally) to add more output channels in each Conv2D layer.
-
 
    
 
    
 
@@ -448,6 +448,7 @@ You can see that the output of every Conv2D and MaxPooling2D layer is a 3D tenso
   
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diff --git a/doc/pub/week42/html/._week42-bs088.html b/doc/pub/week42/html/._week42-bs088.html
index eb6e406ac..7d38e6705 100644
--- a/doc/pub/week42/html/._week42-bs088.html
+++ b/doc/pub/week42/html/._week42-bs088.html
@@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source
                'layers-used-to-build-cnns'),
               ('Transforming images', 2, None, 'transforming-images'),
               ('CNNs in brief', 2, None, 'cnns-in-brief'),
+              ('Key Idea', 2, None, 'key-idea'),
               ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'),
               ('Convolution Examples: Polynomial multiplication',
                2,
@@ -358,37 +359,38 @@ MathJax.Hub.Config({
       
  • Layers used to build CNNs
    • 
       
  • Transforming images
    • 
       
  • CNNs in brief
    • 
-      
  • Mathematics of CNNs
    • 
-      
  • Convolution Examples: Polynomial multiplication
    • 
-      
  • Efficient Polynomial Multiplication
    • 
-      
  • A more efficient way of coding the above Convolution
    • 
-      
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • 
-      
  • Principle of Superposition
    • 
-      
  • Simple Code Example
    • 
-      
  • Wrapping up Fourier transforms
    • 
-      
  • Finding the Coefficients
    • 
-      
  • Final words on Fourier Transforms
    • 
-      
  • Two-dimensional Objects
    • 
-      
  • Cross-Correlation
    • 
-      
  • More on Dimensionalities
    • 
-      
  • Further Dimensionality Remarks
    • 
-      
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • 
-      
  • Setting it up
    • 
-      
  • The MNIST dataset again
    • 
-      
  • Strong correlations
    • 
-      
  • Layers of a CNN
    • 
-      
  • Systematic reduction
    • 
-      
  • Prerequisites: Collect and pre-process data
    • 
-      
  • Importing Keras and Tensorflow
    • 
-      
  • Running with Keras
    • 
-      
  • Final part
    • 
-      
  • Final visualization
    • 
-      
  • The CIFAR01 data set
    • 
-      
  • Verifying the data set
    • 
-      
  • Set up  the model
    • 
-      
  • Add Dense layers on top
    • 
-      
  • Compile and train the model
    • 
-      
  • Finally, evaluate the model
    • 
+      
  • Key Idea
    • 
+      
  • Mathematics of CNNs
    • 
+      
  • Convolution Examples: Polynomial multiplication
    • 
+      
  • Efficient Polynomial Multiplication
    • 
+      
  • A more efficient way of coding the above Convolution
    • 
+      
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • 
+      
  • Principle of Superposition
    • 
+      
  • Simple Code Example
    • 
+      
  • Wrapping up Fourier transforms
    • 
+      
  • Finding the Coefficients
    • 
+      
  • Final words on Fourier Transforms
    • 
+      
  • Two-dimensional Objects
    • 
+      
  • Cross-Correlation
    • 
+      
  • More on Dimensionalities
    • 
+      
  • Further Dimensionality Remarks
    • 
+      
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • 
+      
  • Setting it up
    • 
+      
  • The MNIST dataset again
    • 
+      
  • Strong correlations
    • 
+      
  • Layers of a CNN
    • 
+      
  • Systematic reduction
    • 
+      
  • Prerequisites: Collect and pre-process data
    • 
+      
  • Importing Keras and Tensorflow
    • 
+      
  • Running with Keras
    • 
+      
  • Final part
    • 
+      
  • Final visualization
    • 
+      
  • The CIFAR01 data set
    • 
+      
  • Verifying the data set
    • 
+      
  • Set up  the model
    • 
+      
  • Add Dense layers on top
    • 
+      
  • Compile and train the model
    • 
+      
  • Finally, evaluate the model
    • 
 
         
       
@@ -404,29 +406,30 @@ MathJax.Hub.Config({
 
 
 
-
    Add Dense layers on top 
    
+
    Set up  the model 
    
 
 
    
-To complete our model, you will feed the last output tensor from the
-convolutional base (of shape (4, 4, 64)) into one or more Dense layers
-to perform classification. Dense layers take vectors as input (which
-are 1D), while the current output is a 3D tensor. First, you will
-flatten (or unroll) the 3D output to 1D, then add one or more Dense
-layers on top. CIFAR has 10 output classes, so you use a final Dense
-layer with 10 outputs and a softmax activation.
+The 6 lines of code below define the convolutional base using a common pattern: a stack of Conv2D and MaxPooling2D layers.
+
+
    
+As input, a CNN takes tensors of shape (image_height, image_width, color_channels), ignoring the batch size. If you are new to these dimensions, color_channels refers to (R,G,B). In this example, you will configure our CNN to process inputs of shape (32, 32, 3), which is the format of CIFAR images. You can do this by passing the argument input_shape to our first layer.
 
 
    
 
 
-
    model.add(layers.Flatten())
-model.add(layers.Dense(64, activation='relu'))
-model.add(layers.Dense(10))
-Here's the complete architecture of our model.
+
    model = models.Sequential()
+model.add(layers.Conv2D(32, (3, 3), activation='relu', input_shape=(32, 32, 3)))
+model.add(layers.MaxPooling2D((2, 2)))
+model.add(layers.Conv2D(64, (3, 3), activation='relu'))
+model.add(layers.MaxPooling2D((2, 2)))
+model.add(layers.Conv2D(64, (3, 3), activation='relu'))
+
+# Let's display the architecture of our model so far.
 
 model.summary()
 
    
 
    
-As you can see, our (4, 4, 64) outputs were flattened into vectors of shape (1024) before going through two Dense layers.
+You can see that the output of every Conv2D and MaxPooling2D layer is a 3D tensor of shape (height, width, channels). The width and height dimensions tend to shrink as you go deeper in the network. The number of output channels for each Conv2D layer is controlled by the first argument (e.g., 32 or 64). Typically, as the width and height shrink, you can afford (computationally) to add more output channels in each Conv2D layer.
 
 
    
 
    
@@ -446,6 +449,7 @@ As you can see, our (4, 4, 64) outputs were flattened into vectors of shape (102
   
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diff --git a/doc/pub/week42/html/._week42-bs089.html b/doc/pub/week42/html/._week42-bs089.html
index 0a7f5776f..9325f8a66 100644
--- a/doc/pub/week42/html/._week42-bs089.html
+++ b/doc/pub/week42/html/._week42-bs089.html
@@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source
                'layers-used-to-build-cnns'),
               ('Transforming images', 2, None, 'transforming-images'),
               ('CNNs in brief', 2, None, 'cnns-in-brief'),
+              ('Key Idea', 2, None, 'key-idea'),
               ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'),
               ('Convolution Examples: Polynomial multiplication',
                2,
@@ -358,37 +359,38 @@ MathJax.Hub.Config({
       
  • Layers used to build CNNs
    • 
       
  • Transforming images
    • 
       
  • CNNs in brief
    • 
-      
  • Mathematics of CNNs
    • 
-      
  • Convolution Examples: Polynomial multiplication
    • 
-      
  • Efficient Polynomial Multiplication
    • 
-      
  • A more efficient way of coding the above Convolution
    • 
-      
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • 
-      
  • Principle of Superposition
    • 
-      
  • Simple Code Example
    • 
-      
  • Wrapping up Fourier transforms
    • 
-      
  • Finding the Coefficients
    • 
-      
  • Final words on Fourier Transforms
    • 
-      
  • Two-dimensional Objects
    • 
-      
  • Cross-Correlation
    • 
-      
  • More on Dimensionalities
    • 
-      
  • Further Dimensionality Remarks
    • 
-      
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • 
-      
  • Setting it up
    • 
-      
  • The MNIST dataset again
    • 
-      
  • Strong correlations
    • 
-      
  • Layers of a CNN
    • 
-      
  • Systematic reduction
    • 
-      
  • Prerequisites: Collect and pre-process data
    • 
-      
  • Importing Keras and Tensorflow
    • 
-      
  • Running with Keras
    • 
-      
  • Final part
    • 
-      
  • Final visualization
    • 
-      
  • The CIFAR01 data set
    • 
-      
  • Verifying the data set
    • 
-      
  • Set up  the model
    • 
-      
  • Add Dense layers on top
    • 
-      
  • Compile and train the model
    • 
-      
  • Finally, evaluate the model
    • 
+      
  • Key Idea
    • 
+      
  • Mathematics of CNNs
    • 
+      
  • Convolution Examples: Polynomial multiplication
    • 
+      
  • Efficient Polynomial Multiplication
    • 
+      
  • A more efficient way of coding the above Convolution
    • 
+      
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • 
+      
  • Principle of Superposition
    • 
+      
  • Simple Code Example
    • 
+      
  • Wrapping up Fourier transforms
    • 
+      
  • Finding the Coefficients
    • 
+      
  • Final words on Fourier Transforms
    • 
+      
  • Two-dimensional Objects
    • 
+      
  • Cross-Correlation
    • 
+      
  • More on Dimensionalities
    • 
+      
  • Further Dimensionality Remarks
    • 
+      
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • 
+      
  • Setting it up
    • 
+      
  • The MNIST dataset again
    • 
+      
  • Strong correlations
    • 
+      
  • Layers of a CNN
    • 
+      
  • Systematic reduction
    • 
+      
  • Prerequisites: Collect and pre-process data
    • 
+      
  • Importing Keras and Tensorflow
    • 
+      
  • Running with Keras
    • 
+      
  • Final part
    • 
+      
  • Final visualization
    • 
+      
  • The CIFAR01 data set
    • 
+      
  • Verifying the data set
    • 
+      
  • Set up  the model
    • 
+      
  • Add Dense layers on top
    • 
+      
  • Compile and train the model
    • 
+      
  • Finally, evaluate the model
    • 
 
         
       
@@ -404,18 +406,30 @@ MathJax.Hub.Config({
 
 
 
-
    Compile and train the model 
    
+
    Add Dense layers on top 
    
+
+
    
+To complete our model, you will feed the last output tensor from the
+convolutional base (of shape (4, 4, 64)) into one or more Dense layers
+to perform classification. Dense layers take vectors as input (which
+are 1D), while the current output is a 3D tensor. First, you will
+flatten (or unroll) the 3D output to 1D, then add one or more Dense
+layers on top. CIFAR has 10 output classes, so you use a final Dense
+layer with 10 outputs and a softmax activation.
 
 
    
 
 
-
    model.compile(optimizer='adam',
-              loss=tf.keras.losses.SparseCategoricalCrossentropy(from_logits=True),
-              metrics=['accuracy'])
-​
-history = model.fit(train_images, train_labels, epochs=10, 
-                    validation_data=(test_images, test_labels))
+
    model.add(layers.Flatten())
+model.add(layers.Dense(64, activation='relu'))
+model.add(layers.Dense(10))
+Here's the complete architecture of our model.
+
+model.summary()
 
    
+
    
+As you can see, our (4, 4, 64) outputs were flattened into vectors of shape (1024) before going through two Dense layers.
+
 
    
 
    
 
@@ -433,6 +447,7 @@ history = model
   
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diff --git a/doc/pub/week42/html/._week42-bs090.html b/doc/pub/week42/html/._week42-bs090.html
index 9d4668ce0..2c42201c0 100644
--- a/doc/pub/week42/html/._week42-bs090.html
+++ b/doc/pub/week42/html/._week42-bs090.html
@@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source
                'layers-used-to-build-cnns'),
               ('Transforming images', 2, None, 'transforming-images'),
               ('CNNs in brief', 2, None, 'cnns-in-brief'),
+              ('Key Idea', 2, None, 'key-idea'),
               ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'),
               ('Convolution Examples: Polynomial multiplication',
                2,
@@ -358,37 +359,38 @@ MathJax.Hub.Config({
       
  • Layers used to build CNNs
    • 
       
  • Transforming images
    • 
       
  • CNNs in brief
    • 
-      
  • Mathematics of CNNs
    • 
-      
  • Convolution Examples: Polynomial multiplication
    • 
-      
  • Efficient Polynomial Multiplication
    • 
-      
  • A more efficient way of coding the above Convolution
    • 
-      
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • 
-      
  • Principle of Superposition
    • 
-      
  • Simple Code Example
    • 
-      
  • Wrapping up Fourier transforms
    • 
-      
  • Finding the Coefficients
    • 
-      
  • Final words on Fourier Transforms
    • 
-      
  • Two-dimensional Objects
    • 
-      
  • Cross-Correlation
    • 
-      
  • More on Dimensionalities
    • 
-      
  • Further Dimensionality Remarks
    • 
-      
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • 
-      
  • Setting it up
    • 
-      
  • The MNIST dataset again
    • 
-      
  • Strong correlations
    • 
-      
  • Layers of a CNN
    • 
-      
  • Systematic reduction
    • 
-      
  • Prerequisites: Collect and pre-process data
    • 
-      
  • Importing Keras and Tensorflow
    • 
-      
  • Running with Keras
    • 
-      
  • Final part
    • 
-      
  • Final visualization
    • 
-      
  • The CIFAR01 data set
    • 
-      
  • Verifying the data set
    • 
-      
  • Set up  the model
    • 
-      
  • Add Dense layers on top
    • 
-      
  • Compile and train the model
    • 
-      
  • Finally, evaluate the model
    • 
+      
  • Key Idea
    • 
+      
  • Mathematics of CNNs
    • 
+      
  • Convolution Examples: Polynomial multiplication
    • 
+      
  • Efficient Polynomial Multiplication
    • 
+      
  • A more efficient way of coding the above Convolution
    • 
+      
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • 
+      
  • Principle of Superposition
    • 
+      
  • Simple Code Example
    • 
+      
  • Wrapping up Fourier transforms
    • 
+      
  • Finding the Coefficients
    • 
+      
  • Final words on Fourier Transforms
    • 
+      
  • Two-dimensional Objects
    • 
+      
  • Cross-Correlation
    • 
+      
  • More on Dimensionalities
    • 
+      
  • Further Dimensionality Remarks
    • 
+      
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • 
+      
  • Setting it up
    • 
+      
  • The MNIST dataset again
    • 
+      
  • Strong correlations
    • 
+      
  • Layers of a CNN
    • 
+      
  • Systematic reduction
    • 
+      
  • Prerequisites: Collect and pre-process data
    • 
+      
  • Importing Keras and Tensorflow
    • 
+      
  • Running with Keras
    • 
+      
  • Final part
    • 
+      
  • Final visualization
    • 
+      
  • The CIFAR01 data set
    • 
+      
  • Verifying the data set
    • 
+      
  • Set up  the model
    • 
+      
  • Add Dense layers on top
    • 
+      
  • Compile and train the model
    • 
+      
  • Finally, evaluate the model
    • 
 
         
       
@@ -404,24 +406,19 @@ MathJax.Hub.Config({
 
 
 
-
    Finally, evaluate the model 
    
+
    Compile and train the model 
    
 
 
    
 
 
-
    plt.plot(history.history['accuracy'], label='accuracy')
-plt.plot(history.history['val_accuracy'], label = 'val_accuracy')
-plt.xlabel('Epoch')
-plt.ylabel('Accuracy')
-plt.ylim([0.5, 1])
-plt.legend(loc='lower right')
-
-test_loss, test_acc = model.evaluate(test_images,  test_labels, verbose=2)
-
-print(test_acc)
+
    model.compile(optimizer='adam',
+              loss=tf.keras.losses.SparseCategoricalCrossentropy(from_logits=True),
+              metrics=['accuracy'])
+​
+history = model.fit(train_images, train_labels, epochs=10, 
+                    validation_data=(test_images, test_labels))
 
    
 
    
-
 
    
 
 
      
@@ -437,6 +434,8 @@ test_loss, test_acc = model89
   
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diff --git a/doc/pub/week42/html/week42-bs.html b/doc/pub/week42/html/week42-bs.html
index 5522dc854..eaf81fcd0 100644
--- a/doc/pub/week42/html/week42-bs.html
+++ b/doc/pub/week42/html/week42-bs.html
@@ -190,6 +190,7 @@ Automatically generated HTML file from DocOnce source
                'layers-used-to-build-cnns'),
               ('Transforming images', 2, None, 'transforming-images'),
               ('CNNs in brief', 2, None, 'cnns-in-brief'),
+              ('Key Idea', 2, None, 'key-idea'),
               ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'),
               ('Convolution Examples: Polynomial multiplication',
                2,
@@ -358,37 +359,38 @@ MathJax.Hub.Config({
       
  • Layers used to build CNNs
    • 
       
  • Transforming images
    • 
       
  • CNNs in brief
    • 
-      
  • Mathematics of CNNs
    • 
-      
  • Convolution Examples: Polynomial multiplication
    • 
-      
  • Efficient Polynomial Multiplication
    • 
-      
  • A more efficient way of coding the above Convolution
    • 
-      
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • 
-      
  • Principle of Superposition
    • 
-      
  • Simple Code Example
    • 
-      
  • Wrapping up Fourier transforms
    • 
-      
  • Finding the Coefficients
    • 
-      
  • Final words on Fourier Transforms
    • 
-      
  • Two-dimensional Objects
    • 
-      
  • Cross-Correlation
    • 
-      
  • More on Dimensionalities
    • 
-      
  • Further Dimensionality Remarks
    • 
-      
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • 
-      
  • Setting it up
    • 
-      
  • The MNIST dataset again
    • 
-      
  • Strong correlations
    • 
-      
  • Layers of a CNN
    • 
-      
  • Systematic reduction
    • 
-      
  • Prerequisites: Collect and pre-process data
    • 
-      
  • Importing Keras and Tensorflow
    • 
-      
  • Running with Keras
    • 
-      
  • Final part
    • 
-      
  • Final visualization
    • 
-      
  • The CIFAR01 data set
    • 
-      
  • Verifying the data set
    • 
-      
  • Set up  the model
    • 
-      
  • Add Dense layers on top
    • 
-      
  • Compile and train the model
    • 
-      
  • Finally, evaluate the model
    • 
+      
  • Key Idea
    • 
+      
  • Mathematics of CNNs
    • 
+      
  • Convolution Examples: Polynomial multiplication
    • 
+      
  • Efficient Polynomial Multiplication
    • 
+      
  • A more efficient way of coding the above Convolution
    • 
+      
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • 
+      
  • Principle of Superposition
    • 
+      
  • Simple Code Example
    • 
+      
  • Wrapping up Fourier transforms
    • 
+      
  • Finding the Coefficients
    • 
+      
  • Final words on Fourier Transforms
    • 
+      
  • Two-dimensional Objects
    • 
+      
  • Cross-Correlation
    • 
+      
  • More on Dimensionalities
    • 
+      
  • Further Dimensionality Remarks
    • 
+      
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • 
+      
  • Setting it up
    • 
+      
  • The MNIST dataset again
    • 
+      
  • Strong correlations
    • 
+      
  • Layers of a CNN
    • 
+      
  • Systematic reduction
    • 
+      
  • Prerequisites: Collect and pre-process data
    • 
+      
  • Importing Keras and Tensorflow
    • 
+      
  • Running with Keras
    • 
+      
  • Final part
    • 
+      
  • Final visualization
    • 
+      
  • The CIFAR01 data set
    • 
+      
  • Verifying the data set
    • 
+      
  • Set up  the model
    • 
+      
  • Add Dense layers on top
    • 
+      
  • Compile and train the model
    • 
+      
  • Finally, evaluate the model
    • 
 
         
       
@@ -447,7 +449,7 @@ MathJax.Hub.Config({
   
  • 9
    • 
   
  • 10
    • 
   
  • ...
    • 
-  
  • 91
    • 
+  
  • 92
    • 
   
  • »
    • 
 
 
diff --git a/doc/pub/week42/html/week42-reveal.html b/doc/pub/week42/html/week42-reveal.html
index 5e3f382b4..959e2fb02 100644
--- a/doc/pub/week42/html/week42-reveal.html
+++ b/doc/pub/week42/html/week42-reveal.html
@@ -3191,6 +3191,24 @@ For more material on convolutional networks, we strongly recommend
 the course
 IN5400 – Machine Learning for Image Analysis
 and the slides of CS231 which is taught at Stanford University (consistently ranked as one of the top computer science programs in the world). Michael Nielsen's book is a must read, in particular chapter 6 which deals with CNNs.
+
+
    
+The textbook by Goodfellow et al, see chapter 9 contains an in depth discussion as well.
+
+
+
+
    
+
    Key Idea 
    
+
+
    
+A dense neural network is representd by an affine operation (like matrix-matrix multiplication) where all parameters are included.
+
+
    
+The key idea in CNNs for say imaging is that in images neighbor pixels tend to be related! So we connect
+only neighboring neurons in the input instead of connecting all with the first hidden layer.
+
+
    
+We say we perform a filtering (convolution is the mathematical operation).
 
    
 
 
diff --git a/doc/pub/week42/html/week42-solarized.html b/doc/pub/week42/html/week42-solarized.html
index 8a545de8c..da3cb0a00 100644
--- a/doc/pub/week42/html/week42-solarized.html
+++ b/doc/pub/week42/html/week42-solarized.html
@@ -210,6 +210,7 @@ div { text-align: justify; text-justify: inter-word; }
                'layers-used-to-build-cnns'),
               ('Transforming images', 2, None, 'transforming-images'),
               ('CNNs in brief', 2, None, 'cnns-in-brief'),
+              ('Key Idea', 2, None, 'key-idea'),
               ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'),
               ('Convolution Examples: Polynomial multiplication',
                2,
@@ -3230,6 +3231,24 @@ the course
 IN5400 – Machine Learning for Image Analysis
 and the slides of CS231 which is taught at Stanford University (consistently ranked as one of the top computer science programs in the world). Michael Nielsen's book is a must read, in particular chapter 6 which deals with CNNs.
 
+
    
+The textbook by Goodfellow et al, see chapter 9 contains an in depth discussion as well.
+
+
    
+









    
+
+
    Key Idea 
    
+
+
    
+A dense neural network is representd by an affine operation (like matrix-matrix multiplication) where all parameters are included.
+
+
    
+The key idea in CNNs for say imaging is that in images neighbor pixels tend to be related! So we connect
+only neighboring neurons in the input instead of connecting all with the first hidden layer.
+
+
    
+We say we perform a filtering (convolution is the mathematical operation).
+
 
    
 









    
 
diff --git a/doc/pub/week42/html/week42.html b/doc/pub/week42/html/week42.html
index bdce9a0ac..b7c1d0fd9 100644
--- a/doc/pub/week42/html/week42.html
+++ b/doc/pub/week42/html/week42.html
@@ -215,6 +215,7 @@ div { text-align: justify; text-justify: inter-word; }
                'layers-used-to-build-cnns'),
               ('Transforming images', 2, None, 'transforming-images'),
               ('CNNs in brief', 2, None, 'cnns-in-brief'),
+              ('Key Idea', 2, None, 'key-idea'),
               ('Mathematics of CNNs', 2, None, 'mathematics-of-cnns'),
               ('Convolution Examples: Polynomial multiplication',
                2,
@@ -3235,6 +3236,24 @@ the course
 IN5400 – Machine Learning for Image Analysis
 and the slides of CS231 which is taught at Stanford University (consistently ranked as one of the top computer science programs in the world). Michael Nielsen's book is a must read, in particular chapter 6 which deals with CNNs.
 
+
    
+The textbook by Goodfellow et al, see chapter 9 contains an in depth discussion as well.
+
+
    
+









    
+
+
    Key Idea 
    
+
+
    
+A dense neural network is representd by an affine operation (like matrix-matrix multiplication) where all parameters are included.
+
+
    
+The key idea in CNNs for say imaging is that in images neighbor pixels tend to be related! So we connect
+only neighboring neurons in the input instead of connecting all with the first hidden layer.
+
+
    
+We say we perform a filtering (convolution is the mathematical operation).
+
 
    
 









    
 
diff --git a/doc/pub/week42/ipynb/ipynb-week42-src.tar.gz b/doc/pub/week42/ipynb/ipynb-week42-src.tar.gz
index 750fb55ccbb742066abad697c6020c4e181f1645..ee3d5823cb721411c06fcc40eaee6b64ecf98bb7 100644
GIT binary patch
delta 20
bcmeBP%i6t`l})~zgTX$ik!>p*V`~@yLa+tm

delta 20
bcmeBP%i6t`l})~zgJGv(BimLs#?~+ZMe+ta

diff --git a/doc/pub/week42/ipynb/week42.ipynb b/doc/pub/week42/ipynb/week42.ipynb
index 21d701d87..86309df61 100644
--- a/doc/pub/week42/ipynb/week42.ipynb
+++ b/doc/pub/week42/ipynb/week42.ipynb
@@ -3298,6 +3298,17 @@
     "[IN5400 – Machine Learning for Image Analysis](https://www.uio.no/studier/emner/matnat/ifi/IN5400/index-eng.html)\n",
     "and the slides of [CS231](http://cs231n.github.io/convolutional-networks/) which is taught at Stanford University (consistently ranked as one of the top computer science programs in the world). [Michael Nielsen's book is a must read, in particular chapter 6 which deals with CNNs](http://neuralnetworksanddeeplearning.com/chap6.html).\n",
     "\n",
+    "The textbook by Goodfellow et al, see chapter 9 contains an in depth discussion as well.\n",
+    "\n",
+    "## Key Idea\n",
+    "\n",
+    "A dense neural network is representd by an affine operation (like matrix-matrix multiplication) where all parameters are included.\n",
+    "\n",
+    "The key idea in CNNs for say imaging is that in images neighbor pixels tend to be related! So we connect\n",
+    "only neighboring neurons in the input instead of connecting all with the first hidden layer.\n",
+    "\n",
+    "We say we perform a filtering (convolution is the mathematical operation). \n",
+    "\n",
     "\n",
     "## Mathematics of CNNs\n",
     "\n",
diff --git a/doc/src/week42/week42.do.txt b/doc/src/week42/week42.do.txt
index f6fc309d1..49d873545 100644
--- a/doc/src/week42/week42.do.txt
+++ b/doc/src/week42/week42.do.txt
@@ -2608,6 +2608,18 @@ the course
 "IN5400 – Machine Learning for Image Analysis":"https://www.uio.no/studier/emner/matnat/ifi/IN5400/index-eng.html"
 and the slides of "CS231":"http://cs231n.github.io/convolutional-networks/" which is taught at Stanford University (consistently ranked as one of the top computer science programs in the world). "Michael Nielsen's book is a must read, in particular chapter 6 which deals with CNNs":"http://neuralnetworksanddeeplearning.com/chap6.html".
 
+The textbook by Goodfellow et al, see chapter 9 contains an in depth discussion as well.
+
+!split
+===== Key Idea =====
+
+A dense neural network is representd by an affine operation (like matrix-matrix multiplication) where all parameters are included.
+
+The key idea in CNNs for say imaging is that in images neighbor pixels tend to be related! So we connect
+only neighboring neurons in the input instead of connecting all with the first hidden layer.
+
+We say we perform a filtering (convolution is the mathematical operation). 
+
 
 !split
 ===== Mathematics of CNNs =====