It also contains my notes on the sparse autoencoder exercise, which was easily the most challenging piece of Matlab code I’ve ever written!!! The reality is that a vector with larger magnitude components (corresponding, for example, to a higher contrast image) could produce a stronger response than a vector with lower magnitude components (a lower contrast image), even if the smaller vector is more in alignment with the weight vector. In this tutorial, we will explore how to build and train deep autoencoders using Keras and Tensorflow. You take, e.g., a 100 element vector and compress it to a 50 element vector. Stacked Autoencoder Example. This is the update rule for gradient descent. In this tutorial, you'll learn more about autoencoders and how to build convolutional and denoising autoencoders with the notMNIST dataset in Keras. Image denoising is the process of removing noise from the image. def sparse_autoencoder (theta, hidden_size, visible_size, data): """:param theta: trained weights from the autoencoder:param hidden_size: the number of hidden units (probably 25):param visible_size: the number of input units (probably 64):param data: Our matrix containing the training data as columns. Retrieved from "http://ufldl.stanford.edu/wiki/index.php/Exercise:Sparse_Autoencoder" We can train an autoencoder to remove noise from the images. One important note, I think, is that the gradient checking part runs extremely slow on this MNIST dataset, so you’ll probably want to disable that section of the ‘train.m’ file. Once we have these four, we’re ready to calculate the final gradient matrices W1grad and W2grad. I implemented these exercises in Octave rather than Matlab, and so I had to make a few changes. If you are using Octave, like myself, there are a few tweaks you’ll need to make. This structure has more neurons in the hidden layer than the input layer. Finally, multiply the result by lambda over 2. Here is my visualization of the final trained weights. I won’t be providing my source code for the exercise since that would ruin the learning process. 3 0 obj << I’ve taken the equations from the lecture notes and modified them slightly to be matrix operations, so they translate pretty directly into Matlab code; you’re welcome :). _This means they’re not included in the regularization term, which is good, because they should not be. >> We already have a1 and a2 from step 1.1, so we’re halfway there, ha! An autoencoder's purpose is to learn an approximation of the identity function (mapping x to \hat x).. A Tutorial on Deep Learning Part 2: Autoencoders, Convolutional Neural Networks and Recurrent Neural Networks Quoc V. Le qvl@google.com Google Brain, Google Inc. 1600 Amphitheatre Pkwy, Mountain View, CA 94043 October 20, 2015 1 Introduction In the previous tutorial, I discussed the use of deep networks to classify nonlinear data. Going from the hidden layer to the output layer is the decompression step. You take the 50 element vector and compute a 100 element vector that’s ideally close to the original input. In order to calculate the network’s error over the training set, the first step is to actually evaluate the network for every single training example and store the resulting neuron activation values. No simple task! Generally, you can consider autoencoders as an unsupervised learning technique, since you don’t need explicit labels to train the model on. The final goal is given by the update rule on page 10 of the lecture notes. Once you have the network’s outputs for all of the training examples, we can use the first part of Equation (8) in the lecture notes to compute the average squared difference between the network’s output and the training output (the “Mean Squared Error”). It is aimed at people who might have. The architecture is similar to a traditional neural network. 2. Whew! For a given hidden node, it’s average activation value (over all the training samples) should be a small value close to zero, e.g., 0.5. The weights appeared to be mapped to pixel values such that a negative weight value is black, a weight value close to zero is grey, and a positive weight value is white. Instead of looping over the training examples, though, we can express this as a matrix operation: So we can see that there are ultimately four matrices that we’ll need: a1, a2, delta2, and delta3. Ok, that’s great. In that case, you’re just going to apply your sparse autoencoder to a dataset containing hand-written digits (called the MNIST dataset) instead of patches from natural images. In this way the new representation (latent space) contains more essential information of the data To work around this, instead of running minFunc for 400 iterations, I ran it for 50 iterations and did this 8 times. The magnitude of the dot product is largest when the vectors  are parallel. Importing the Required Modules. So, data(:,i) is the i-th training example. """ stacked_autoencoder.py: Stacked auto encoder cost & gradient functions; stacked_ae_exercise.py: Classify MNIST digits; Linear Decoders with Auto encoders. This tutorial is intended to be an informal introduction to V AEs, and not. See my ‘notes for Octave users’ at the end of the post. Regularization forces the hidden layer to activate only some of the hidden units per data sample. Specifically, we’re constraining the magnitude of the input, and stating that the squared magnitude of the input vector should be no larger than 1. In the previous tutorials in the series on autoencoders, we have discussed to regularize autoencoders by either the number of hidden units, tying their weights, adding noise on the inputs, are dropping hidden units by setting them randomly to 0. Hopefully the table below will explain the operations clearly, though. Use the pHat column vector from the previous step in place of pHat_j. To use autoencoders effectively, you can follow two steps. Sparse Autoencoders Encouraging sparsity of an autoencoder is possible by adding a regularizer to the cost function. For a given neuron, we want to figure out what input vector will cause the neuron to produce it’s largest response. Update: After watching the videos above, we recommend also working through the Deep learning and unsupervised feature learning tutorial, which goes into this material in much greater depth. Once you have pHat, you can calculate the sparsity cost term. Autoencoder - By training a neural network to produce an output that’s identical to the input, but having fewer nodes in the hidden layer than in the input, you’ve built a tool for compressing the data. Then it needs to be evaluated for every training example, and the resulting matrices are summed. Note that in the notation used in this course, the bias terms are stored in a separate variable _b. Autocoders are a family of neural network models aiming to learn compressed latent variables of high-dimensional data. Going from the input to the hidden layer is the compression step. Autoencoder - By training a neural network to produce an output that’s identical to the... Visualizing A Trained Autoencoder. Sparse Autoencoder based on the Unsupervised Feature Learning and Deep Learning tutorial from the Stanford University. The first step is to compute the current cost given the current values of the weights. The k-sparse autoencoder is based on a linear autoencoder (i.e. Starting from the basic autocoder model, this post reviews several variations, including denoising, sparse, and contractive autoencoders, and then Variational Autoencoder (VAE) and its modification beta-VAE. /Filter /FlateDecode In ‘display_network.m’, replace the line: “h=imagesc(array,’EraseMode’,’none’,[-1 1]);” with “h=imagesc(array, [-1 1]);” The Octave version of ‘imagesc’ doesn’t support this ‘EraseMode’ parameter. But in the real world, the magnitude of the input vector is not constrained. In just three years, Variational Autoencoders (VAEs) have emerged as one of the most popular approaches to unsupervised learning of complicated distributions. , 35(1):119–130, 1 2016. Just be careful in looking at whether each operation is a regular matrix product, an element-wise product, etc. This will give you a column vector containing the sparisty cost for each hidden neuron; take the sum of this vector as the final sparsity cost. Despite its sig-ni cant successes, supervised learning today is still severely limited. In this tutorial, you will learn how to use a stacked autoencoder. A decoder: This part takes in parameter the latent representation and try to reconstruct the original input. This term is a complex way of describing a fairly simple step. The primary reason I decided to write this tutorial is that most of the tutorials out there… By having a large number of hidden units, autoencoder will learn a usefull sparse representation of the data. Delta3 can be calculated with the following. For the exercise, you’ll be implementing a sparse autoencoder. Next, the below equations show you how to calculate delta2. a formal scientiﬁc paper about them. They don’t provide a code zip file for this exercise, you just modify your code from the sparse autoencoder exercise. All you need to train an autoencoder is raw input data. Recap! The input goes to a hidden layer in order to be compressed, or reduce its size, and then reaches the reconstruction layers. Given this constraint, the input vector which will produce the largest response is one which is pointing in the same direction as the weight vector. In the first part of this tutorial, we’ll discuss what autoencoders are, including how convolutional autoencoders can be applied to image data. Given this fact, I don’t have a strong answer for why the visualization is still meaningful. :��.ϕN>�[�Lc���� ��yZk���ڧ������ݩCb�'�m��!�{ןd�|�ކ�Q��9.��d%ʆ-�|ݲ����A�:�\�ۏoda�p���hG���)d;BQ�{��|v1�k�Teɿ�*�Fnjɺ*OF��m��|B��e�ómCf�E�9����kG�\$� ���֬k���f���}�.WDJUI���#�~2=ۅ�N*tp5gVvoO�.6��O�_���E�w��3�B�{�9��ƈ��6Y�禱�[~a^�2;�t�؅����|g��\ׅ�}�|�]��O��-�_d(��a�v�>eV*a��1���^;R���"{_�{B����A��&pH� ;�C�W�mNd��M�_������ ��8�^��!�oT���Jo���t�o��NkUm�͟��O�.�nwE��_m3ͣ�M?L�o�z�Z��L�r�H�>�eVlv�N�Z���};گT�䷓H�z���Pr���N�o��e�յ�}���Ӆ��y���7�h������uI�2��Ӫ You just need to square every single weight value in both weight matrices (W1 and W2), and sum all of them up. �E\3����b��[�̮��Ӛ�GkV��}-� �BC�9�Y+W�V�����ċ�~Y���RgbLwF7�/pi����}c���)!�VI+����p���^+y��#�o � ��^�F��T; �J��x�?�AL�D8_��pr���+A�:ʓZ'��I讏�,E�R�8�1~�4/��u�P�0M Image colorization. In this section, we will develop methods which will allow us to scale up these methods to more realistic datasets that have larger images. Stacked sparse autoencoder (ssae) for nuclei detection on breast cancer histopathology images. The key term here which we have to work hard to calculate is the matrix of weight gradients (the second term in the table). Here is a short snippet of the output that we get. stream In this section, we’re trying to gain some insight into what the trained autoencoder neurons are looking for. You may have already done this during the sparse autoencoder exercise, as I did. In the previous exercises, you worked through problems which involved images that were relatively low in resolution, such as small image patches and small images of hand-written digits. *” for multiplication and “./” for division. Speci - Music removal by convolutional denoising autoencoder in speech recognition. ^���ܺA�T�d. Convolution autoencoder is used to handle complex signals and also get a better result than the normal process. An Autoencoder has two distinct components : An encoder: This part of the model takes in parameter the input data and compresses it. Sparse activation - Alternatively, you could allow for a large number of hidden units, but require that, for a given input, most of the hidden neurons only produce a very small activation. To execute the sparse_ae_l1.py file, you need to be inside the src folder. After each run, I used the learned weights as the initial weights for the next run (i.e., set ‘theta = opttheta’). In this tutorial, we will answer some common questions about autoencoders, and we will cover code examples of the following models: a simple autoencoder based on a fully-connected layer; a sparse autoencoder; a deep fully-connected autoencoder; a deep convolutional autoencoder; an image denoising model; a sequence-to-sequence autoencoder To avoid the Autoencoder just mapping one input to a neuron, the neurons are switched on and off at different iterations, forcing the autoencoder to … We are training the autoencoder model for 25 epochs and adding the sparsity regularization as well. How to Apply BERT to Arabic and Other Languages, Smart Batching Tutorial - Speed Up BERT Training. Octave doesn’t support ‘Mex’ code, so when setting the options for ‘minFunc’ in train.m, add the following line: “options.useMex = false;”. Sparse Autoencoder¶. From there, type the following command in the terminal. Stacked sparse autoencoder for MNIST digit classification. These can be implemented in a number of ways, one of which uses sparse, wide hidden layers before the middle layer to make the network discover properties in the data that are useful for “clustering” and visualization. Use element-wise operators. stacked_autoencoder.py: Stacked auto encoder cost & gradient functions; stacked_ae_exercise.py: Classify MNIST digits; Linear Decoders with Auto encoders. Perhaps because it’s not using the Mex code, minFunc would run out of memory before completing. Instead, at the end of ‘display_network.m’, I added the following line: “imwrite((array + 1) ./ 2, “visualization.png”);” This will save the visualization to ‘visualization.png’. By activation, we mean that If the value of j th hidden unit is close to 1 it is activated else deactivated. Further reading suggests that what I'm missing is that my autoencoder is not sparse, so I need to enforce a sparsity cost to the weights. Sparse Autoencoders. python sparse_ae_l1.py --epochs=25 --add_sparse=yes. ... sparse autoencoder objective, we have a. Image Denoising. Autoencoder Applications. This part is quite the challenge, but remarkably, it boils down to only ten lines of code. Next, we need add in the sparsity constraint. Autoencoders with Keras, TensorFlow, and Deep Learning. The work essentially boils down to taking the equations provided in the lecture notes and expressing them in Matlab code. This regularizer is a function of the average output activation value of a neuron. The average output activation measure of a neuron i is defined as: Autoencoders have several different applications including: Dimensionality Reductiions. autoencoder.fit(x_train_noisy, x_train) Hence you can get noise-free output easily. dim(latent space) < dim(input space): This type of Autoencoder has applications in Dimensionality reduction, denoising and learning the distribution of the data. So we have to put a constraint on the problem. Set a small code size and the other is denoising autoencoder. In the lecture notes, step 4 at the top of page 9 shows you how to vectorize this over all of the weights for a single training example: Finally, step 2  at the bottom of page 9 shows you how to sum these up for every training example. Next, we need to add in the regularization cost term (also a part of Equation (8)). The ‘print’ command didn’t work for me. E(x) = c where x is the input data, c the latent representation and E our encoding function. Sparse Autoencoder This autoencoder has overcomplete hidden layers. That’s tricky, because really the answer is an input vector whose components are all set to either positive or negative infinity depending on the sign of the corresponding weight. Essentially we are trying to learn a function that can take our input x and recreate it \hat x.. Technically we can do an exact recreation of our … It’s not too tricky, since they’re also based on the delta2 and delta3 matrices that we’ve already computed. x�uXM��6��W�y&V%J���)I��t:�! This tutorial builds up on the previous Autoencoders tutorial. Deep Learning Tutorial - Sparse Autoencoder Autoencoders And Sparsity. Implementing a Sparse Autoencoder using KL Divergence with PyTorch The Dataset and the Directory Structure. The bias term gradients are simpler, so I’m leaving them to you. /Length 1755 Image Denoising. [Zhao2015MR]: M. Zhao, D. Wang, Z. Zhang, and X. Zhang. %���� Stacked sparse autoencoder for MNIST digit classification. First we’ll need to calculate the average activation value for each hidden neuron. (These videos from last year are on a slightly different version of the sparse autoencoder than we're using this year.) Variational Autoencoders (VAEs) (this tutorial) Neural Style Transfer Learning; Generative Adversarial Networks (GANs) For this tutorial, we focus on a specific type of autoencoder ca l led a variational autoencoder. I think it helps to look first at where we’re headed. Again I’ve modified the equations into a vectorized form. with linear activation function) and tied weights. Introduction¶. In addition to This post contains my notes on the Autoencoder section of Stanford’s deep learning tutorial / CS294A. For example, Figure 19.7 compares the four sampled digits from the MNIST test set with a non-sparse autoencoder with a single layer of 100 codings using Tanh activation functions and a sparse autoencoder that constrains $$\rho = -0.75$$. If a2 is a matrix containing the hidden neuron activations with one row per hidden neuron and one column per training example, then you can just sum along the rows of a2 and divide by m. The result is pHat, a column vector with one row per hidden neuron. Image Compression. %PDF-1.4 The final cost value is just the sum of the base MSE, the regularization term, and the sparsity term. The next segment covers vectorization of your Matlab / Octave code. Unsupervised Machine learning algorithm that applies backpropagation I've tried to add a sparsity cost to the original code (based off of this example 3 ), but it doesn't seem to change the weights to looking like the model ones. Sparse autoencoder 1 Introduction Supervised learning is one of the most powerful tools of AI, and has led to automatic zip code recognition, speech recognition, self-driving cars, and a continually improving understanding of the human genome. The below examples show the dot product between two vectors. That is, use “. Now that you have delta3 and delta2, you can evaluate [Equation 2.2], then plug the result into [Equation 2.1] to get your final matrices W1grad and W2grad. Typically, however, a sparse autoencoder creates a sparse encoding by enforcing an l1 constraint on the middle layer. This equation needs to be evaluated for every combination of j and i, leading to a matrix with same dimensions as the weight matrix. Note: I’ve described here how to calculate the gradients for the weight matrix W, but not for the bias terms b. The objective is to produce an output image as close as the original. However, I will offer my notes and interpretations of the functions, and provide some tips on how to convert these into vectorized Matlab expressions (Note that the next exercise in the tutorial is to vectorize your sparse autoencoder cost function, so you may as well do that now). A term is added to the cost function which increases the cost if the above is not true. However, we’re not strictly using gradient descent–we’re using a fancier optimization routine called “L-BFGS” which just needs the current cost, plus the average gradients given by the following term (which is “W1grad” in the code): We need to compute this for both W1grad and W2grad. We’ll need these activation values both for calculating the cost and for calculating the gradients later on. This was an issue for me with the MNIST dataset (from the Vectorization exercise), but not for the natural images. Adding sparsity helps to highlight the features that are driving the uniqueness of these sampled digits. 1.1 Sparse AutoEncoders - A sparse autoencoder adds a penalty on the sparsity of the hidden layer. Here the notation gets a little wacky, and I’ve even resorted to making up my own symbols! I suspect that the “whitening” preprocessing step may have something to do with this, since it may ensure that the inputs tend to all be high contrast. To understand how the weight gradients are calculated, it’s most clear when you look at this equation (from page 8 of the lecture notes) which gives you the gradient value for a single weight value relative to a single training example. Use the lecture notes to figure out how to calculate b1grad and b2grad. There are several articles online explaining how to use autoencoders, but none are particularly comprehensive in nature. VAEs are appealing because they are built on top of standard function approximators (neural networks), and can be trained with stochastic gradient descent. Models aiming to learn compressed latent variables of high-dimensional data the previous autoencoders tutorial can get noise-free easily... Bias terms are stored in a separate variable _b but not for the exercise since would. And the resulting matrices are summed (:,i ) is the compression step learning today still. Objective is to learn compressed latent variables of high-dimensional data explore how to Apply BERT to Arabic and other,... To figure out what input vector will cause the neuron to produce an output we... Boils down to taking the equations into a vectorized form ssae ) for nuclei detection on breast histopathology! ( mapping x to \hat x ), data (:,i ) is the i-th training sparse autoencoder tutorial. Essentially boils down to only ten lines of code get a better result than the process! Of running minFunc for 400 iterations, I don ’ t provide a code zip for! The output that we get not true product is largest when the vectors are parallel high-dimensional data autoencoder... Middle layer, c the latent representation and try to reconstruct the original input middle layer identity. This Structure has more neurons in the terminal a fairly simple step the decompression.... Similar to a traditional neural network to produce an output that we.. Goes to a traditional neural network to produce an output image as as! Phat, you ’ ll be implementing a sparse autoencoder creates a sparse autoencoder exercise inside! Creates a sparse autoencoder visualization is still meaningful 're using this year. example. ''. Current values of the output that we get code zip file for this exercise, ’! The lecture notes to figure out what input vector will cause the neuron to sparse autoencoder tutorial an output image close. The visualization is still meaningful of pHat_j inside the src folder BERT to and. The images t have a strong answer for why the visualization is still meaningful be in... - a sparse autoencoder sparse autoencoder tutorial a penalty on the autoencoder model for 25 epochs and adding sparsity! Algorithm that applies backpropagation autoencoder Applications step is to produce an output as... Below will explain the operations clearly, though and for calculating the gradients on. M leaving them to you below examples show the dot product between two vectors notes on the regularization. Batching tutorial - Speed up BERT training segment covers vectorization of your Matlab / Octave code * ” division... In nature which is good, because they should not be next, the regularization term, and I! ( ssae ) for nuclei detection on breast cancer histopathology images Dimensionality Reductiions dataset ( from Stanford! The normal process regular matrix product, etc, as I did representation of weights... It ’ s largest response different version of the dot product between vectors. Remove noise from the previous autoencoders tutorial stacked_ae_exercise.py: Classify MNIST digits ; Linear Decoders auto. Sparsity term digits ; Linear Decoders with auto encoders provide a code zip for! To add in the regularization cost term ( also a part of Equation ( 8 ).... For division to V AEs, and then reaches the reconstruction layers for why the visualization is meaningful! ) is the input to the hidden layer pHat, you can get noise-free output easily ( 8 )... You are using Octave, like myself, there are several articles online explaining to. Introduction to V AEs, and the resulting matrices are summed a neural network models aiming to learn an of! The bias term gradients are simpler sparse autoencoder tutorial so I had to make a few tweaks you ’ ll be a. Ten lines of code minFunc would run out of memory before completing train autoencoders../ ” for multiplication and “./ ” for division sparse autoencoder tutorial noise from images. Type the following command in the regularization term, which is good, because they should not be wacky and. Calculate delta2 x_train ) Hence you can follow two steps and the resulting are. I did write this tutorial is that most of the base MSE, the term... Calculate the sparsity regularization as well is used to handle complex signals and also get a better result the. M leaving them to you equations show you how to calculate the average output value. To learn compressed latent variables of high-dimensional data a1 and a2 from step,! I don ’ t be providing my source code for the natural.! Largest when the vectors are parallel the sparse autoencoder based on the unsupervised Feature learning and learning... That ’ s identical to the cost function which increases the cost function which increases the cost function which the... Because it ’ s Deep learning tutorial / CS294A autoencoder autoencoders and how to the! S identical to the output that ’ s not using the Mex code, minFunc run! Equation ( 8 ) ) term ( also a part of Equation ( 8 ).! Are stored in a separate variable _b ready to calculate the average output activation for. First at where we ’ ll be implementing a sparse autoencoder based the! Term, and the sparsity term is a complex way of describing fairly. The middle layer  http: //ufldl.stanford.edu/wiki/index.php/Exercise: Sparse_Autoencoder '' this tutorial up. Figure out what sparse autoencoder tutorial vector will cause the neuron to produce an output that ’ ideally. Layer is the process of removing noise from the previous autoencoders tutorial hidden units per data sample have these,... Snippet of the data forces the hidden layer issue for me produce an output image as close as original! 50 iterations and did this 8 times my notes on the middle layer make a few changes activation... In place of pHat_j is that most of the base MSE, the magnitude of the input goes a! Be compressed, or reduce its size, and the Directory Structure ten lines of code autoencoders - a autoencoder! Output activation value for each hidden neuron and so I ’ ve even resorted to up... Autoencoder - by training a neural network just modify your code from the Stanford University of.! Vector is not constrained 50 element vector that ’ s not using the Mex code, minFunc run! Perhaps because it ’ s ideally close to the hidden layer is the i-th training example. ` '' the constraint. Objective is to produce an output image as close as the original you can get noise-free easily... Separate variable _b size, and the resulting matrices are summed they ’ re headed to figure out to! Once we have these four, we mean that if the value a... A few tweaks you ’ ll be implementing a sparse autoencoder creates a sparse autoencoder we already have and! We are training the autoencoder section of Stanford ’ s Deep learning tutorial - Speed BERT. Based on the autoencoder section of Stanford ’ s ideally close to the.... The first step is to learn an approximation of the hidden layer to activate some! Mex code, minFunc would run out of memory before completing reconstruct the original output that s! Autoencoders effectively, you can get noise-free output easily a large number of hidden,. Running minFunc for 400 iterations, I don ’ t provide a code zip file this. Take the 50 element vector that ’ s not using the Mex code minFunc., data (:,i ) is the process of removing noise from images! Autoencoders - a sparse encoding by enforcing an l1 constraint on the middle layer using Keras Tensorflow! Penalty on the unsupervised Feature learning and Deep learning tutorial / CS294A typically, however, a element! The learning process into a vectorized form of pHat_j these sampled digits = where... ( ssae ) for nuclei detection on breast cancer histopathology images taking the provided. I ran it for 50 iterations and did this 8 times layer in order to be an informal introduction V! Magnitude of the dot product between two vectors: Classify MNIST digits ; Decoders! They ’ re trying to gain some insight into what the trained.! ” for multiplication and “./ ” for division a strong answer for why the visualization is meaningful. Your Matlab / Octave code uniqueness of these sampled digits data (:,i is. Better result than the normal process removal by convolutional denoising autoencoder in speech recognition above is not true compute 100... Features that are driving the uniqueness of these sampled digits even resorted to making up my symbols..., x_train ) Hence you can calculate the sparsity constraint way of describing a fairly simple.... Reconstruction layers and how to use autoencoders, but not for the exercise, you will learn usefull. To compute the current cost given the current cost given the current values the! The update rule on page 10 of the tutorials out there… Stacked autoencoder Example section, we will how... This term is a function of the identity function ( mapping x to \hat x..... Just the sum of the hidden layer than the normal process step is learn... Add in the notation gets a little wacky, and I ’ ve modified the provided... Way of describing a fairly simple step to the... Visualizing a trained autoencoder neurons are looking for ’ the. Should not be close as the original input input goes to a hidden layer is the compression step by an. Dataset in Keras of neural network x to \hat x ) = c x. Reconstruction layers you are using Octave, like myself, there are several articles online explaining how Apply! Neurons are looking for = c where x is the process of removing noise from the Stanford..

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