# The mechanics of Convolutional Neural Networks in Tensorflow and Keras

Convolutional Neural Networks (CNNs), have been very popular in the last decade or so. CNNs have been used in multiple applications like image recognition, image classification, facial recognition, neural style transfer etc. CNN’s have been extremely successful in handling these kind of problems. How do they work? What makes them so successful? What is the principle behind CNN’s ?

Note: this post is based on two Coursera courses I did, namely namely Deep Learning specialisation by Prof Andrew Ng and Tensorflow Specialisation by  Laurence Moroney.

In this post I show you how CNN’s work. To understand how CNNs work, we need to understand the concept behind machine learning algorithms. If you take a simple machine learning algorithm in which you are trying to do multi-class classification using softmax or binary classification with the sigmoid function, for a set of for a set of input features against a target variable we need to create an objective function of the input features versus the target variable. Then we need to minimise this objective function, while performing gradient descent, such that the cost  is the lowest. This will give the set of weights for the different variables in the objective function.

The central problem in ML algorithms is to do feature selection, i.e.  we need to find the set of features that actually influence the target.  There are various methods for doing features selection – best fit, forward fit, backward fit, ridge and lasso regression. All these methods try to pick out the predictors that influence the output most, by making the weights of the other features close to zero. Please look at my post – Practical Machine Learning in R and Python – Part 3, where I show you the different methods for doing features selection.

In image classification or Image recognition we need to find the important features in the image. How do we do that? Many years back, have played around with OpenCV.  While working with OpenCV I came across are numerous filters like the Sobel ,the Laplacian, Canny, Gaussian filter et cetera which can be used to identify key features of the image. For example the Canny filter feature can be used for edge detection, Gaussian for smoothing, Sobel for determining the derivative and we have other filters for detecting vertical or horizontal edges. Take a look at my post Computer Vision: Ramblings on derivatives, histograms and contours So for handling images we need to apply these filters to pick  out the key features of the image namely the edges and other features. So rather than using the entire image’s pixels against the target class we can pick out the features from the image and use that as predictors of the target output.

Note: that in Convolutional Neural Network, fixed filter values like the those shown above  are not used directly. Rather the filter values are learned through back propagation and gradient descent as shown below.

In CNNs the filter values are considered to be weights which are then learned and updated in each forward/backward propagation cycle much like the way a fully connected Deep Learning Network learns the weights of the network.

Here is a short derivation of the most important parts of how a CNNs work

The convolution of a filter F with the input X can be represented as.

Convolving we get

This the forward propagation as it passes through a non-linear function like Relu

To go through back propagation we need to compute the $\partial L$  at every node of Convolutional Neural network

The loss with respect to the output is $\partial L/\partial O$. $\partial O/\partial X$ & $\partial O/\partial F$ are the local derivatives

We need these local derivatives because we can learn the filter values using gradient descent

where $\alpha$ is the learning rate. Also $\partial L/\partial X$ is the loss which is back propagated to the previous layers. You can see the detailed derivation of back propagation in my post Deep Learning from first principles in Python, R and Octave – Part 3 in a L-layer, multi-unit Deep Learning network.

In the fully connected layers the weights associated with each connection is computed in every cycle of forward and backward propagation using gradient descent. Similarly, the filter values are also computed and updated in each forward and backward propagation cycle. This is done so as to minimize the loss at the output layer.

By using the chain rule and simplifying the back propagation for the Convolutional layers we get these 2 equations. The first equation is used to learn the filter values and the second is used pass the loss to layers before

(for the detailed derivation see Convolutions and Backpropagations

An important aspect of performing convolutions is to reduce the size of  the flattened image that is passed into the fully connected DL network. Successively convolving with 2D filters and doing a max pooling helps to reduce the size of the features that we can use for learning the images. Convolutions also enable a sparsity of connections  as you can see in the diagram below. In the LeNet-5 Convolution Neural Network of Yann Le Cunn, successive convolutions reduce the image size from 28 x 28=784 to 120 flattened values.

Here is an interesting Deep Learning problem. Convolutions help in picking out important features of images and help in image classification/ detection. What would be its equivalent if we wanted to identify the Carnatic ragam of a song? A Carnatic ragam is roughly similar to Western scales (major, natural, melodic, blues) with all its modes Lydian, Aeolion, Phyrgian etc. Except in the case of the ragams, it is more nuanced, complex and involved. Personally, I can rarely identify a ragam on which a carnatic song is based (I am tone deaf when it comes to identifying ragams). I have come to understand that each Carnatic ragam has its own character, which is made up of several melodic phrases which are unique to that flavor of a ragam. What operation like convolution would be needed so that we can pick out these unique phrases in a Carnatic ragam? Of course, we would need to use it in Recurrent Neural Networks with LSTMs as a song is a time sequence of notes to identify sequences. Nevertheless, if there was some operation with which we can pick up the distinct, unique phrases from a song and then run it through a classifier, maybe we would be able to identify the ragam of the song.

Below I implement 3 simple CNN using the Dogs vs Cats Dataset from Kaggle. The first CNN uses regular Convolutions a Fully connected network to classify the images. The second approach uses Image Augmentation. For some reason, I did not get a better performance with Image Augumentation. Thirdly I use the pre-trained Inception v3 network.

# 1. Basic Convolutional Neural Network in Tensorflow & Keras

You can view the Colab notebook here – Cats_vs_dogs_1.ipynb

Here some important parts of the notebook

## Create CNN Model

• Use 3 Convolution + Max pooling layers with 32,64 and 128 filters respectively
• Flatten the data
• Have 2 Fully connected layers with 128, 512 neurons with relu activation
• Use sigmoid for binary classification
In [0]:
model = tf.keras.models.Sequential([
tf.keras.layers.Conv2D(32,(3,3),activation='relu',input_shape=(150,150,3)),
tf.keras.layers.MaxPooling2D(2,2),
tf.keras.layers.Conv2D(64,(3,3),activation='relu'),
tf.keras.layers.MaxPooling2D(2,2),
tf.keras.layers.Conv2D(128,(3,3),activation='relu'),
tf.keras.layers.MaxPooling2D(2,2),
tf.keras.layers.Flatten(),
tf.keras.layers.Dense(128,activation='relu'),
tf.keras.layers.Dense(512,activation='relu'),
tf.keras.layers.Dense(1,activation='sigmoid')
])


## Print model summary

In [13]:
model.summary()

Model: "sequential"
_________________________________________________________________
Layer (type)                 Output Shape              Param #
=================================================================
conv2d (Conv2D)              (None, 148, 148, 32)      896
_________________________________________________________________
max_pooling2d (MaxPooling2D) (None, 74, 74, 32)        0
_________________________________________________________________
conv2d_1 (Conv2D)            (None, 72, 72, 64)        18496
_________________________________________________________________
max_pooling2d_1 (MaxPooling2 (None, 36, 36, 64)        0
_________________________________________________________________
conv2d_2 (Conv2D)            (None, 34, 34, 128)       73856
_________________________________________________________________
max_pooling2d_2 (MaxPooling2 (None, 17, 17, 128)       0
_________________________________________________________________
flatten (Flatten)            (None, 36992)             0
_________________________________________________________________
dense (Dense)                (None, 128)               4735104
_________________________________________________________________
dense_1 (Dense)              (None, 512)               66048
_________________________________________________________________
dense_2 (Dense)              (None, 1)                 513
=================================================================
Total params: 4,894,913
Trainable params: 4,894,913
Non-trainable params: 0
_________________________________________________________________


## Use the Adam Optimizer with binary cross entropy

model.compile(optimizer='adam',
loss='binary_crossentropy',
metrics=['accuracy'])


## Perform Gradient Descent

• Do Gradient Descent for 15 epochs
history=model.fit(train_generator,
validation_data=validation_generator,
steps_per_epoch=100,
epochs=15,
validation_steps=50,
verbose=2)

Epoch 1/15
100/100 - 13s - loss: 0.6821 - accuracy: 0.5425 - val_loss: 0.6484 - val_accuracy: 0.6131
Epoch 2/15
100/100 - 13s - loss: 0.6227 - accuracy: 0.6456 - val_loss: 0.6161 - val_accuracy: 0.6394
Epoch 3/15
100/100 - 13s - loss: 0.5975 - accuracy: 0.6719 - val_loss: 0.5558 - val_accuracy: 0.7206
Epoch 4/15
100/100 - 13s - loss: 0.5480 - accuracy: 0.7241 - val_loss: 0.5431 - val_accuracy: 0.7138
Epoch 5/15
100/100 - 13s - loss: 0.5182 - accuracy: 0.7447 - val_loss: 0.4839 - val_accuracy: 0.7606
Epoch 6/15
100/100 - 13s - loss: 0.4773 - accuracy: 0.7781 - val_loss: 0.5029 - val_accuracy: 0.7506
Epoch 7/15
100/100 - 13s - loss: 0.4466 - accuracy: 0.7972 - val_loss: 0.4573 - val_accuracy: 0.7912
Epoch 8/15
100/100 - 13s - loss: 0.4395 - accuracy: 0.7997 - val_loss: 0.4252 - val_accuracy: 0.8119
Epoch 9/15
100/100 - 13s - loss: 0.4314 - accuracy: 0.8019 - val_loss: 0.4931 - val_accuracy: 0.7481
Epoch 10/15
100/100 - 13s - loss: 0.4309 - accuracy: 0.7969 - val_loss: 0.4203 - val_accuracy: 0.8109
Epoch 11/15
100/100 - 13s - loss: 0.4329 - accuracy: 0.7916 - val_loss: 0.4189 - val_accuracy: 0.8069
Epoch 12/15
100/100 - 13s - loss: 0.4248 - accuracy: 0.8050 - val_loss: 0.4476 - val_accuracy: 0.7925
Epoch 13/15
100/100 - 13s - loss: 0.3868 - accuracy: 0.8306 - val_loss: 0.3900 - val_accuracy: 0.8236
Epoch 14/15
100/100 - 13s - loss: 0.3710 - accuracy: 0.8328 - val_loss: 0.4520 - val_accuracy: 0.7900
Epoch 15/15
100/100 - 13s - loss: 0.3654 - accuracy: 0.8353 - val_loss: 0.3999 - val_accuracy: 0.8100

## Plot results

• Plot training and validation accuracy

• Plot training and validation loss

#-----------------------------------------------------------
# Retrieve a list of list results on training and test data
# sets for each training epoch
#-----------------------------------------------------------
acc      = history.history[     'accuracy' ]
val_acc  = history.history[ 'val_accuracy' ]
loss     = history.history[    'loss' ]
val_loss = history.history['val_loss' ]

epochs   = range(len(acc)) # Get number of epochs

#------------------------------------------------
# Plot training and validation accuracy per epoch
#------------------------------------------------
plt.plot  ( epochs,     acc,label="training accuracy" )
plt.plot  ( epochs, val_acc, label='validation acuracy' )
plt.title ('Training and validation accuracy')
plt.legend()

plt.figure()

#------------------------------------------------
# Plot training and validation loss per epoch
#------------------------------------------------
plt.plot  ( epochs,     loss , label="training loss")
plt.plot  ( epochs, val_loss,label="validation loss" )
plt.title ('Training and validation loss'   )
plt.legend()



# 2. CNN with Image Augmentation

You can check the Cats_vs_Dogs_2.ipynb

Including the important parts of this implementation below

## Use Image Augumentation

Use Image Augumentation to improve performance

• Use the same model parameters as before
• Perform the following image augmentation
• width, height shift
• shear and zoom

Note: Adding rotation made the performance worse

import tensorflow as tf
from tensorflow import keras
from tensorflow.keras.optimizers import RMSprop
from tensorflow.keras.preprocessing.image import ImageDataGenerator
model = tf.keras.models.Sequential([
tf.keras.layers.Conv2D(32,(3,3),activation='relu',input_shape=(150,150,3)),
tf.keras.layers.MaxPooling2D(2,2),
tf.keras.layers.Conv2D(64,(3,3),activation='relu'),
tf.keras.layers.MaxPooling2D(2,2),
tf.keras.layers.Conv2D(128,(3,3),activation='relu'),
tf.keras.layers.MaxPooling2D(2,2),
tf.keras.layers.Flatten(),
tf.keras.layers.Dense(128,activation='relu'),
tf.keras.layers.Dense(512,activation='relu'),
tf.keras.layers.Dense(1,activation='sigmoid')
])

train_datagen = ImageDataGenerator(
rescale=1./255,
#rotation_range=90,
width_shift_range=0.2,
height_shift_range=0.2,
shear_range=0.2,
zoom_range=0.2)
#horizontal_flip=True,
#fill_mode='nearest')

validation_datagen = ImageDataGenerator(rescale=1./255)
#
train_generator = train_datagen.flow_from_directory(train_dir,
batch_size=32,
class_mode='binary',
target_size=(150, 150))
# --------------------
# Flow validation images in batches of 20 using test_datagen generator
# --------------------
validation_generator =  validation_datagen.flow_from_directory(validation_dir,
batch_size=32,
class_mode  = 'binary',
target_size = (150, 150))

# Use Adam Optmizer
model.compile(optimizer='adam',
loss='binary_crossentropy',
metrics=['accuracy'])

Found 20000 images belonging to 2 classes.
Found 5000 images belonging to 2 classes.


## Perform Gradient Descent

history=model.fit(train_generator,
validation_data=validation_generator,
steps_per_epoch=100,
epochs=15,
validation_steps=50,
verbose=2)

Epoch 1/15
100/100 - 27s - loss: 0.5716 - accuracy: 0.6922 - val_loss: 0.4843 - val_accuracy: 0.7744
Epoch 2/15
100/100 - 27s - loss: 0.5575 - accuracy: 0.7084 - val_loss: 0.4683 - val_accuracy: 0.7750
Epoch 3/15
100/100 - 26s - loss: 0.5452 - accuracy: 0.7228 - val_loss: 0.4856 - val_accuracy: 0.7665
Epoch 4/15
100/100 - 27s - loss: 0.5294 - accuracy: 0.7347 - val_loss: 0.4654 - val_accuracy: 0.7812
Epoch 5/15
100/100 - 27s - loss: 0.5352 - accuracy: 0.7350 - val_loss: 0.4557 - val_accuracy: 0.7981
Epoch 6/15
100/100 - 26s - loss: 0.5136 - accuracy: 0.7453 - val_loss: 0.4964 - val_accuracy: 0.7621
Epoch 7/15
100/100 - 27s - loss: 0.5249 - accuracy: 0.7334 - val_loss: 0.4959 - val_accuracy: 0.7556
Epoch 8/15
100/100 - 26s - loss: 0.5035 - accuracy: 0.7497 - val_loss: 0.4555 - val_accuracy: 0.7969
Epoch 9/15
100/100 - 26s - loss: 0.5024 - accuracy: 0.7487 - val_loss: 0.4675 - val_accuracy: 0.7728
Epoch 10/15
100/100 - 27s - loss: 0.5015 - accuracy: 0.7500 - val_loss: 0.4276 - val_accuracy: 0.8075
Epoch 11/15
100/100 - 26s - loss: 0.5002 - accuracy: 0.7581 - val_loss: 0.4193 - val_accuracy: 0.8131
Epoch 12/15
100/100 - 27s - loss: 0.4733 - accuracy: 0.7706 - val_loss: 0.5209 - val_accuracy: 0.7398
Epoch 13/15
100/100 - 27s - loss: 0.4999 - accuracy: 0.7538 - val_loss: 0.4109 - val_accuracy: 0.8075
Epoch 14/15
100/100 - 27s - loss: 0.4550 - accuracy: 0.7859 - val_loss: 0.3770 - val_accuracy: 0.8288
Epoch 15/15
100/100 - 26s - loss: 0.4688 - accuracy: 0.7688 - val_loss: 0.4764 - val_accuracy: 0.7786


## Plot results

• Plot training and validation accuracy
• Plot training and validation loss
In [15]:
import matplotlib.pyplot as plt
#-----------------------------------------------------------
# Retrieve a list of list results on training and test data
# sets for each training epoch
#-----------------------------------------------------------
acc      = history.history[     'accuracy' ]
val_acc  = history.history[ 'val_accuracy' ]
loss     = history.history[    'loss' ]
val_loss = history.history['val_loss' ]

epochs   = range(len(acc)) # Get number of epochs

#------------------------------------------------
# Plot training and validation accuracy per epoch
#------------------------------------------------
plt.plot  ( epochs,     acc,label="training accuracy" )
plt.plot  ( epochs, val_acc, label='validation acuracy' )
plt.title ('Training and validation accuracy')
plt.legend()

plt.figure()

#------------------------------------------------
# Plot training and validation loss per epoch
#------------------------------------------------
plt.plot  ( epochs,     loss , label="training loss")
plt.plot  ( epochs, val_loss,label="validation loss" )
plt.title ('Training and validation loss'   )
plt.legend()



# Implementation using Inception Network V3

The implementation is in the Colab notebook Cats_vs_Dog_3.ipynb

This is implemented as below

## Use Inception V3

import os

from tensorflow.keras import layers
from tensorflow.keras import Model

from tensorflow.keras.applications.inception_v3 import InceptionV3
pre_trained_model = InceptionV3(input_shape = (150, 150, 3),
include_top = False,
weights = 'imagenet')

for layer in pre_trained_model.layers:
layer.trainable = False

# pre_trained_model.summary()

last_layer = pre_trained_model.get_layer('mixed7')
print('last layer output shape: ', last_layer.output_shape)
last_output = last_layer.output

Downloading data from https://storage.googleapis.com/tensorflow/keras-applications/inception_v3/inception_v3_weights_tf_dim_ordering_tf_kernels_notop.h5
87916544/87910968 [==============================] - 1s 0us/step
last layer output shape:  (None, 7, 7, 768)


## Use Layer 7 of Inception Network

• Use Image Augumentation
• Use Adam Optimizer
In [0]:
import tensorflow as tf
from tensorflow import keras
from tensorflow.keras.optimizers import RMSprop
from tensorflow.keras.preprocessing.image import ImageDataGenerator
# Flatten the output layer to 1 dimension
x = layers.Flatten()(last_output)
# Add a fully connected layer with 1,024 hidden units and ReLU activation
x = layers.Dense(1024, activation='relu')(x)
# Add a dropout rate of 0.2
x = layers.Dropout(0.2)(x)
# Add a final sigmoid layer for classification
x = layers.Dense  (1, activation='sigmoid')(x)

model = Model( pre_trained_model.input, x)
#train_datagen = ImageDataGenerator( rescale = 1.0/255. )
#validation_datagen = ImageDataGenerator( rescale = 1.0/255. )

train_datagen = ImageDataGenerator(
rescale=1./255,
#rotation_range=90,
width_shift_range=0.2,
height_shift_range=0.2,
shear_range=0.2,
zoom_range=0.2)
#horizontal_flip=True,
#fill_mode='nearest')

validation_datagen = ImageDataGenerator(rescale=1./255)
#
train_generator = train_datagen.flow_from_directory(train_dir,
batch_size=32,
class_mode='binary',
target_size=(150, 150))
# --------------------
# Flow validation images in batches of 20 using test_datagen generator
# --------------------
validation_generator =  validation_datagen.flow_from_directory(validation_dir,
batch_size=32,
class_mode  = 'binary',
target_size = (150, 150))

model.compile(optimizer='adam',
loss='binary_crossentropy',
metrics=['accuracy'])

Found 20000 images belonging to 2 classes.
Found 5000 images belonging to 2 classes.


## Fit model

history=model.fit(train_generator,
validation_data=validation_generator,
steps_per_epoch=100,
epochs=15,
validation_steps=50,
verbose=2)

Epoch 1/15
100/100 - 31s - loss: 0.5961 - accuracy: 0.8909 - val_loss: 0.1919 - val_accuracy: 0.9456
Epoch 2/15
100/100 - 30s - loss: 0.2002 - accuracy: 0.9259 - val_loss: 0.1025 - val_accuracy: 0.9550
Epoch 3/15
100/100 - 30s - loss: 0.1618 - accuracy: 0.9366 - val_loss: 0.0920 - val_accuracy: 0.9581
Epoch 4/15
100/100 - 29s - loss: 0.1442 - accuracy: 0.9381 - val_loss: 0.0960 - val_accuracy: 0.9600
Epoch 5/15
100/100 - 30s - loss: 0.1402 - accuracy: 0.9381 - val_loss: 0.0703 - val_accuracy: 0.9794
Epoch 6/15
100/100 - 30s - loss: 0.1437 - accuracy: 0.9413 - val_loss: 0.1090 - val_accuracy: 0.9531
Epoch 7/15
100/100 - 30s - loss: 0.1325 - accuracy: 0.9428 - val_loss: 0.0756 - val_accuracy: 0.9670
Epoch 8/15
100/100 - 29s - loss: 0.1341 - accuracy: 0.9491 - val_loss: 0.0625 - val_accuracy: 0.9737
Epoch 9/15
100/100 - 29s - loss: 0.1186 - accuracy: 0.9513 - val_loss: 0.0934 - val_accuracy: 0.9581
Epoch 10/15
100/100 - 29s - loss: 0.1171 - accuracy: 0.9513 - val_loss: 0.0642 - val_accuracy: 0.9727
Epoch 11/15
100/100 - 29s - loss: 0.1018 - accuracy: 0.9591 - val_loss: 0.0930 - val_accuracy: 0.9606
Epoch 12/15
100/100 - 29s - loss: 0.1190 - accuracy: 0.9541 - val_loss: 0.0737 - val_accuracy: 0.9719
Epoch 13/15
100/100 - 29s - loss: 0.1223 - accuracy: 0.9494 - val_loss: 0.0740 - val_accuracy: 0.9695
Epoch 14/15
100/100 - 29s - loss: 0.1158 - accuracy: 0.9516 - val_loss: 0.0659 - val_accuracy: 0.9744
Epoch 15/15
100/100 - 29s - loss: 0.1168 - accuracy: 0.9591 - val_loss: 0.0788 - val_accuracy: 0.9669


## Plot results

• Plot training and validation accuracy
• Plot training and validation loss
In [14]:
import matplotlib.pyplot as plt
#-----------------------------------------------------------
# Retrieve a list of list results on training and test data
# sets for each training epoch
#-----------------------------------------------------------
acc      = history.history[     'accuracy' ]
val_acc  = history.history[ 'val_accuracy' ]
loss     = history.history[    'loss' ]
val_loss = history.history['val_loss' ]

epochs   = range(len(acc)) # Get number of epochs

#------------------------------------------------
# Plot training and validation accuracy per epoch
#------------------------------------------------
plt.plot  ( epochs,     acc,label="training accuracy" )
plt.plot  ( epochs, val_acc, label='validation acuracy' )
plt.title ('Training and validation accuracy')
plt.legend()

plt.figure()

#------------------------------------------------
# Plot training and validation loss per epoch
#------------------------------------------------
plt.plot  ( epochs,     loss , label="training loss")
plt.plot  ( epochs, val_loss,label="validation loss" )
plt.title ('Training and validation loss'   )
plt.legend()

I intend to do some interesting stuff with Convolutional Neural Networks.

Watch this space!

To see all posts click Index of posts