406 lines
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HTML
406 lines
25 KiB
HTML
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._week44-bs001.html#plan-for-week-44" style="font-size: 80%;"><b>Plan for week 44</b></a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs002.html#material-for-lecture-thursday-november-2" style="font-size: 80%;"><b>Material for Lecture Thursday November 2</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs003.html#convolutional-neural-networks-recognizing-images" style="font-size: 80%;"><b>Convolutional Neural Networks (recognizing images)</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs004.html#what-is-the-difference" style="font-size: 80%;"><b>What is the Difference</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs005.html#neural-networks-vs-cnns" style="font-size: 80%;"><b>Neural Networks vs CNNs</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs006.html#why-cnns-for-images-sound-files-medical-images-from-ct-scans-etc" style="font-size: 80%;"><b>Why CNNS for images, sound files, medical images from CT scans etc?</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs007.html#regular-nns-don-t-scale-well-to-full-images" style="font-size: 80%;"><b>Regular NNs don’t scale well to full images</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs008.html#3d-volumes-of-neurons" style="font-size: 80%;"><b>3D volumes of neurons</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs009.html#layers-used-to-build-cnns" style="font-size: 80%;"><b>Layers used to build CNNs</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs010.html#transforming-images" style="font-size: 80%;"><b>Transforming images</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs011.html#cnns-in-brief" style="font-size: 80%;"><b>CNNs in brief</b></a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs012.html#key-idea" style="font-size: 80%;"><b>Key Idea</b></a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs013.html#mathematics-of-cnns" style="font-size: 80%;"><b>Mathematics of CNNs</b></a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs014.html#convolution-examples-polynomial-multiplication" style="font-size: 80%;"><b>Convolution Examples: Polynomial multiplication</b></a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs015.html#efficient-polynomial-multiplication" style="font-size: 80%;"><b>Efficient Polynomial Multiplication</b></a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs016.html#a-more-efficient-way-of-coding-the-above-convolution" style="font-size: 80%;"><b>A more efficient way of coding the above Convolution</b></a></li>
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<!-- navigation toc: --> <li><a href="#convolution-examples-principle-of-superposition-and-periodic-forces-fourier-transforms" style="font-size: 80%;"><b>Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)</b></a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs018.html#simple-code-example" style="font-size: 80%;"><b>Simple Code Example</b></a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs019.html#wrapping-up-fourier-transforms" style="font-size: 80%;"><b>Wrapping up Fourier transforms</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs020.html#finding-the-coefficients" style="font-size: 80%;"><b>Finding the Coefficients</b></a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs021.html#final-words-on-fourier-transforms" style="font-size: 80%;"><b>Final words on Fourier Transforms</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs021.html#fourier-transforms-and-convolution" style="font-size: 80%;"> Fourier transforms and convolution</a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs021.html#two-dimensional-objects" style="font-size: 80%;"><b>Two-dimensional Objects</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs022.html#more-on-dimensionalities" style="font-size: 80%;"><b>More on Dimensionalities</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs023.html#further-dimensionality-remarks" style="font-size: 80%;"><b>Further Dimensionality Remarks</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs024.html#cnns-in-more-detail" style="font-size: 80%;"><b>CNNs in more detail</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs025.html#pooling" style="font-size: 80%;"><b>Pooling</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs026.html#no-zero-padding-unit-strides" style="font-size: 80%;"><b>No zero padding, unit strides</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs027.html#zero-padding-unit-strides" style="font-size: 80%;"><b>Zero padding, unit strides</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs028.html#half-same-padding" style="font-size: 80%;"><b>Half (same) padding</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs029.html#full-padding" style="font-size: 80%;"><b>Full padding</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs030.html#pooling-arithmetic" style="font-size: 80%;"><b>Pooling arithmetic</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs031.html#cnns-in-more-detail-building-convolutional-neural-networks-in-tensorflow-and-keras" style="font-size: 80%;"><b>CNNs in more detail, building convolutional neural networks in Tensorflow and Keras</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs032.html#setting-it-up" style="font-size: 80%;"><b>Setting it up</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs033.html#the-mnist-dataset-again" style="font-size: 80%;"><b>The MNIST dataset again</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs034.html#strong-correlations" style="font-size: 80%;"><b>Strong correlations</b></a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs035.html#layers-of-a-cnn" style="font-size: 80%;"><b>Layers of a CNN</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs036.html#systematic-reduction" style="font-size: 80%;"><b>Systematic reduction</b></a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs037.html#prerequisites-collect-and-pre-process-data" style="font-size: 80%;"><b>Prerequisites: Collect and pre-process data</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs038.html#importing-keras-and-tensorflow" style="font-size: 80%;"><b>Importing Keras and Tensorflow</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs039.html#running-with-keras" style="font-size: 80%;"><b>Running with Keras</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs040.html#final-part" style="font-size: 80%;"><b>Final part</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs041.html#final-visualization" style="font-size: 80%;"><b>Final visualization</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs042.html#the-cifar01-data-set" style="font-size: 80%;"><b>The CIFAR01 data set</b></a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs043.html#verifying-the-data-set" style="font-size: 80%;"><b>Verifying the data set</b></a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs044.html#set-up-the-model" style="font-size: 80%;"><b>Set up the model</b></a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs045.html#add-dense-layers-on-top" style="font-size: 80%;"><b>Add Dense layers on top</b></a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs046.html#compile-and-train-the-model" style="font-size: 80%;"><b>Compile and train the model</b></a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs047.html#finally-evaluate-the-model" style="font-size: 80%;"><b>Finally, evaluate the model</b></a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#building-our-own-cnn-code" style="font-size: 80%;"><b>Building our own CNN code</b></a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#list-of-contents" style="font-size: 80%;"> List of contents:</a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#schedulers" style="font-size: 80%;"> Schedulers</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#usage-of-schedulers" style="font-size: 80%;"> Usage of schedulers</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#cost-functions" style="font-size: 80%;"> Cost functions</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#usage-of-cost-functions" style="font-size: 80%;"> Usage of cost functions</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#usage-of-activation-functions" style="font-size: 80%;"> Usage of activation functions</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#convolution" style="font-size: 80%;"> Convolution</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#layers" style="font-size: 80%;"> Layers</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#convolution2dlayer-convolution-in-a-hidden-layer" style="font-size: 80%;"> Convolution2DLayer: convolution in a hidden layer</a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#backpropagation-in-the-convolutional-layer" style="font-size: 80%;"> Backpropagation in the convolutional layer</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#demonstration" style="font-size: 80%;"> Demonstration</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#pooling-layer" style="font-size: 80%;"> Pooling Layer</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#flattening-layer" style="font-size: 80%;"> Flattening Layer</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#fully-connected-layers" style="font-size: 80%;"> Fully Connected Layers</a></li>
|
||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#optimized-convolution2dlayer" style="font-size: 80%;"> Optimized Convolution2DLayer</a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs048.html#the-convolutional-neural-network-cnn" style="font-size: 80%;"> The Convolutional Neural Network (CNN)</a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs048.html#usage-of-cnn-code" style="font-size: 80%;"> Usage of CNN code</a></li>
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<!-- navigation toc: --> <li><a href="._week44-bs048.html#additional-remarks" style="font-size: 80%;"> Additional Remarks</a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#remarks-on-the-speed" style="font-size: 80%;"> Remarks on the speed</a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#convolution-using-separable-kernels" style="font-size: 80%;"> Convolution using separable kernels</a></li>
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||
<!-- navigation toc: --> <li><a href="._week44-bs048.html#convolution-in-the-fourier-domain" style="font-size: 80%;"> Convolution in the Fourier domain</a></li>
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<div class="container">
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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||
<a name="part0017"></a>
|
||
<!-- !split -->
|
||
<h2 id="convolution-examples-principle-of-superposition-and-periodic-forces-fourier-transforms" class="anchor">Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms) </h2>
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||
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||
<p>For problems with so-called harmonic oscillations, given by for example the following differential equation</p>
|
||
$$
|
||
m\frac{d^2x}{dt^2}+\eta\frac{dx}{dt}+x(t)=F(t),
|
||
$$
|
||
|
||
<p>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.
|
||
</p>
|
||
|
||
<p>If one has several driving forces, \( F(t)=\sum_n F_n(t) \), one can find
|
||
the particular solution \( x_{pn}(t) \) to the above differential equation for each \( F_n \). The particular
|
||
solution for the entire driving force is then given by a series like
|
||
</p>
|
||
|
||
$$
|
||
\begin{equation}
|
||
x_p(t)=\sum_nx_{pn}(t).
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||
\tag{1}
|
||
\end{equation}
|
||
$$
|
||
|
||
<p>This is known as the principle of superposition. It only applies when
|
||
the homogenous equation is linear.
|
||
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.
|
||
</p>
|
||
|
||
<p>Driving forces are often periodic, even when they are not
|
||
sinusoidal. Periodicity implies that for some time \( t \) our function repeats itself periodically after a period \( \tau \), that is
|
||
</p>
|
||
|
||
$$
|
||
\begin{eqnarray}
|
||
F(t+\tau)=F(t).
|
||
\end{eqnarray}
|
||
$$
|
||
|
||
<p>One example of a non-sinusoidal periodic force is a square wave. Many
|
||
components in electric circuits are non-linear, for example diodes. This
|
||
makes many wave forms non-sinusoidal even when the circuits are being
|
||
driven by purely sinusoidal sources.
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||
</p>
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<p>
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