Exercise:Convolution and Pooling
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Exercise:Convolution and Pooling
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== Convolution and Pooling == In this problem set, you will use the features you learned on 8x8 patches sampled from images from the STL10 dataset in [[Exercise:Learning color features with Sparse Autoencoders | the earlier exercise on linear decoders]] for classifying 64x64 STL10 images by applying [[Feature extraction using convolution | convolution]] and [[Pooling | pooling]]. In the file <tt>[http://ufldl.stanford.edu/wiki/resources/cnn_exercise.zip cnn_exercise.zip]</tt> we have provided some starter code. You should write your code at the places indicated "YOUR CODE HERE" in the files. For this exercise, you will need to modify'''<tt>cnnConvolve.m</tt>''' and '''<tt>cnnPool.m</tt>'''. === Dependencies === The following additional files are required for this exercise: * STL10 dataset You will also need: * <tt>STL10Features.mat</tt> - saved features from [[Exercise:Learning color features with Sparse Autoencoders]] * <tt>softmaxTrain.m</tt> (and related functions) from [[Exercise:Softmax Regression]] ''If you have not completed the exercises listed above, we strongly suggest you complete them first.'' === Step 1: Load learned features === In this step, we will load the color features you learned in [[Exercise:Learning color features with Sparse Autoencoders]]. To verify that the features are correct, the loaded features will be visualized, and you should get something like the following: [[File:CNN_Features_Good.png|480px]] === Step 2: Convolution and pooling === Now that you have learned features for small patches, you will convolved these learned features with the large images, and pool these convolved features for use in a classifier later. ==== Step 2a: Convolution ==== Implement convolution, as described in [[feature extraction using convolution]], in the function <tt>cnnConvolve</tt> in <tt>cnnConvolve.m</tt>. Implementing convolution is somewhat involved, so we will guide you through the process below. First of all, what we want to compute is <math>\sigma(Wx_{(r,c)} + b)</math> for all ''valid'' <math>(r, c)</math> (''valid'' meaning that the entire 8x8 patch is contained within the image; as opposed to a ''full'' convolution which allows the patch to extend outside the image, with the area outside the image assumed to be 0) , where <math>W</math> and <math>b</math> are the learned weights and biases from the input layer to the hidden layer, and <math>x_{(r,c)}</math> is the 8x8 patch with the upper left corner at <math>(r, c)</math>. To accomplish this, what we could do is loop over all such patches and compute <math>\sigma(Wx_{(r,c)} + b)</math> for each of them. In theory, this is correct. However, in practice, the convolution is usually done in three small steps to take advantage of MATLAB's optimized convolution functions. Observe that the convolution above can be broken down into the following three small steps. First, compute <math>Wx_{(r,c)}</math> for all <math>(r, c)</math>. Next, add b to all the computed values. Finally, apply the sigmoid function to the resultant values. This doesn't seem to buy you anything, since the first step still requires a loop. However, you can replace the loop in the first step with one of MATLAB's optimized convolution functions, <tt>conv2</tt>, speeding up the process slightly. However, there are two complications in using <tt>conv2</tt>. First, <tt>conv2</tt> performs a 2-D convolution, but you have 5 "dimensions" - image number, feature number, row of image, column of image, and channel of image - that you want to convolve over. Because of this, you will have to convolve each feature and image channel separately for each image, using the row and column of the image as the 2 dimensions you convolve over. This means that you will need three outer loops over the image number <tt>imageNum</tt>, feature number <tt>featureNum</tt>, and the channel number of the image <tt>channel</tt>, with the 2-D convolution of the weight matrix for the <tt>featureNum</tt>-th feature and <tt>channel</tt>-th channel with the image matrix for the <tt>imageNum</tt>-th image going inside. Second, because of the mathematical definition of convolution, the feature matrix must be "flipped" before passing it to <tt>conv2</tt>. This is explained in greater detail in the implementation tip section following the code. Concretely, the code to do the convolution using <tt>conv2</tt> will look something like the following: <syntaxhighlight lang="matlab"> convolvedFeatures = zeros(hiddenSize, numImages, imageDim - patchDim + 1, imageDim - patchDim + 1); for imageNum = 1:numImages for featureNum = 1:hiddenSize % Obtain the feature matrix for this feature Wt = W(featureNum, :); Wt = reshape(Wt, patchDim, patchDim, 3); % Get convolution of image with feature matrix for each channel convolvedTemp = zeros(imageDim - patchDim + 1, imageDim - patchDim + 1, 3); for channel = 1:3 % Flip the feature matrix because of the definition of convolution, as explained later Wt(:, :, channel) = flipud(fliplr(squeeze(Wt(:, :, channel)))); convolvedTemp(:, :, channel) = conv2(squeeze(images(:, :, channel, imageNum)), squeeze(Wt(:, :, channel)), 'valid'); end % The convolved feature is the sum of the convolved values for all channels convolvedFeatures(featureNum, imageNum, :, :) = sum(convolvedTemp, 3); end end </syntaxhighlight> The following implementation tip explains the "flipping" of feature matrices when using MATLAB's convolution functions: <div style="border:1px solid black; padding: 5px"> '''Implementation tip:''' Using <tt>conv2</tt> and <tt>convn</tt> Because the mathematical definition of convolution involves "flipping" the matrix to convolve with (reversing its rows and its columns), to use MATLAB's convolution functions, you must first "flip" the weight matrix so that when MATLAB "flips" it according to the mathematical definition the entries will be at the correct place. For example, suppose you wanted to convolve two matrices <tt>image</tt> (a large image) and <tt>W</tt> (the feature) using <tt>conv2(image, W)</tt>, and W is a 3x3 matrix as below: <math> W = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{pmatrix} </math> If you use <tt>conv2(image, W)</tt>, MATLAB will first "flip" <tt>W</tt>, reversing its rows and columns, before convolving <tt>W</tt> with <tt>image</tt>, as below: <math> \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{pmatrix} \xrightarrow{flip} \begin{pmatrix} 9 & 8 & 7 \\ 6 & 5 & 4 \\ 3 & 2 & 1 \\ \end{pmatrix} </math> If the original layout of <tt>W</tt> was correct, after flipping, it would be incorrect. For the layout to be correct after flipping, you will have to flip <tt>W</tt> before passing it into <tt>conv2</tt>, so that after MATLAB flips <tt>W</tt> in <tt>conv2</tt>, the layout will be correct. For <tt>conv2</tt>, this means reversing the rows and columns, which can be done with <tt>flipud</tt> and <tt>fliplr</tt>, as we did in the example code above. This is also true for the general convolution function <tt>convn</tt>, in which case MATLAB reverses every dimension. In general, you can flip the matrix <tt>W</tt> using the following code snippet, which works for <tt>W</tt> of any dimension <syntaxhighlight lang="matlab"> % Flip W for use in conv2 / convn temp = W(:); temp = flipud(temp); temp = reshape(temp, size(W)); </syntaxhighlight> </div> To each of <tt>convolvedFeatures</tt>, you should then add <tt>b</tt>, the corresponding bias for the <tt>featureNum</tt>-th feature. If you had done no preprocessing of the patches, you could then apply the sigmoid function to obtain the convolved features. However, because you preprocessed the patches before learning features on them, you must also apply the same preprocessing steps to the convolved patches to get the correct feature activations. In particular, you did the following to the patches: <ol> <li> divide by 255 to normalize them into the range <math>[0, 1]</math> <li> subtract the mean patch, <tt>meanPatch</tt> to zero the mean of the patches <li> ZCA whiten using the whitening matrix <tt>ZCAWhite</tt>. </ol> These same three steps must also be applied to the convolved patches. Taking the preprocessing steps into account, the feature activations that you should compute is <math>\sigma(W(T(x-\bar{x})) + b)</math>, where <math>T</math> is the whitening matrix and <math>\bar{x}</math> is the mean patch. Expanding this, you obtain <math>\sigma(WTx - WT\bar{x} + b)</math>, which suggests that you should convolve the images with <math>WT</math> rather than <math>W</math> as earlier, and you should add <math>(b - WT\bar{x})</math>, rather than just <math>b</math> to <tt>convolvedFeatures</tt>, before finally applying the sigmoid function. ==== Step 2b: Checking ==== We have provided some code for you to check that you have done the convolution correctly. The code randomly checks the convolved values for a number of (feature, row, column) tuples by computing the feature activations for the selected features and patches directly using the sparse autoencoder. ==== Step 2c: Pooling ==== Implement [[pooling]] in the function <tt>cnnPool</tt> in <tt>cnnPool.m</tt>. === Step 3: Use pooled features for classification === Once you have implemented pooling, you will use the pooled features to train a softmax classifier to map the pooled features to the class labels. The code in this section uses <tt>softmaxTrain</tt> from the softmax exercise to train a softmax classifier on the pooled features for 500 iterations, which should take around 5 minutes. === Step 4: Test classifier === Now that you have a trained softmax classifier, you can see how well it performs on the test set. This section contains code that will load the test set (which is a smaller part of the STL10 dataset, specifically, 3200 rescaled 64x64 images from 4 different classes) and obtain the pooled, convolved features for the images using the functions <tt>cnnConvolve</tt> and <tt>cnnPool</tt> which you wrote earlier, as well as the preprocessing matrices <tt>ZCAWhite</tt> and <tt>meanImage</tt> which were computed earlier in preprocessing the training images. These pooled features will then be run through the softmax classifier, and the accuracy of the predictions will be computed. You should expect to get an accuracy of around 77-78%.
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