Here we present a flexible and readable python code for a CNN implemented with NumPy. We will present the code, showcase how to use the codebase and fit a CNN that yields a 99% accuracy on the 28x28 MNIST dataset within reasonable time.
The CNN is compatible with all schedulers, cost functions and activation functions discussed in constructing our neural network codes.
The CNN code consists of different types of Layer classes, including Convolution2DLayer, Pooling2DLayer, FlattenLayer, FullyConnectedLayer and OutputLayer, which can be added to the CNN object using the interface of the CNN class. This allows you to easily construct your own CNN, as well as allowing you to get used to an interface similar to that of TensorFlow which is used for real world applications.
Another important feature of this code is that it throws errors if unreasonable decisions are made (for example using a kernel that is larger than the image, not using a FlattenLayer, etc), and provides the user with an informative error message.
The code below shows object oriented implementations of the Constant, Momentum, Adagrad, AdagradMomentum, RMS prop and Adam schedulers. All of the classes belong to the shared abstract Scheduler class, and share the update_change() and reset() methods allowing for any of the schedulers to be seamlessly used during the training stage, as will later be shown in the fit() method of the neural network. Update_change() only has one parameter, the gradient (\( \delta^{l}_{j}a^{l-1}_k \)), and returns the change which will be subtracted from the weights. The reset() function takes no parameters, and resets the desired variables. For Constant and Momentum, reset does nothing.
import autograd.numpy as np
class Scheduler:
"""
Abstract class for Schedulers
"""
def __init__(self, eta):
self.eta = eta
# should be overwritten
def update_change(self, gradient):
raise NotImplementedError
# overwritten if needed
def reset(self):
pass
class Constant(Scheduler):
def __init__(self, eta):
super().__init__(eta)
def update_change(self, gradient):
return self.eta * gradient
def reset(self):
pass
class Momentum(Scheduler):
def __init__(self, eta: float, momentum: float):
super().__init__(eta)
self.momentum = momentum
self.change = 0
def update_change(self, gradient):
self.change = self.momentum * self.change + self.eta * gradient
return self.change
def reset(self):
pass
class Adagrad(Scheduler):
def __init__(self, eta):
super().__init__(eta)
self.G_t = None
def update_change(self, gradient):
delta = 1e-8 # avoid division ny zero
if self.G_t is None:
self.G_t = np.zeros((gradient.shape[0], gradient.shape[0]))
self.G_t += gradient @ gradient.T
G_t_inverse = 1 / (
delta + np.sqrt(np.reshape(np.diagonal(self.G_t), (self.G_t.shape[0], 1)))
)
return self.eta * gradient * G_t_inverse
def reset(self):
self.G_t = None
class AdagradMomentum(Scheduler):
def __init__(self, eta, momentum):
super().__init__(eta)
self.G_t = None
self.momentum = momentum
self.change = 0
def update_change(self, gradient):
delta = 1e-8 # avoid division ny zero
if self.G_t is None:
self.G_t = np.zeros((gradient.shape[0], gradient.shape[0]))
self.G_t += gradient @ gradient.T
G_t_inverse = 1 / (
delta + np.sqrt(np.reshape(np.diagonal(self.G_t), (self.G_t.shape[0], 1)))
)
self.change = self.change * self.momentum + self.eta * gradient * G_t_inverse
return self.change
def reset(self):
self.G_t = None
class RMS_prop(Scheduler):
def __init__(self, eta, rho):
super().__init__(eta)
self.rho = rho
self.second = 0.0
def update_change(self, gradient):
delta = 1e-8 # avoid division ny zero
self.second = self.rho * self.second + (1 - self.rho) * gradient * gradient
return self.eta * gradient / (np.sqrt(self.second + delta))
def reset(self):
self.second = 0.0
class Adam(Scheduler):
def __init__(self, eta, rho, rho2):
super().__init__(eta)
self.rho = rho
self.rho2 = rho2
self.moment = 0
self.second = 0
self.n_epochs = 1
def update_change(self, gradient):
delta = 1e-8 # avoid division ny zero
self.moment = self.rho * self.moment + (1 - self.rho) * gradient
self.second = self.rho2 * self.second + (1 - self.rho2) * gradient * gradient
moment_corrected = self.moment / (1 - self.rho**self.n_epochs)
second_corrected = self.second / (1 - self.rho2**self.n_epochs)
return self.eta * moment_corrected / (np.sqrt(second_corrected + delta))
def reset(self):
self.n_epochs += 1
self.moment = 0
self.second = 0
To initalize a scheduler, simply create the object and pass in the necessary parameters such as the learning rate and the momentum as shown below. As the Scheduler class is an abstract class it should not called directly, and will raise an error upon usage.
momentum_scheduler = Momentum(eta=1e-3, momentum=0.9)
adam_scheduler = Adam(eta=1e-3, rho=0.9, rho2=0.999)
Here is a small example for how a segment of code using schedulers could look. Switching out the schedulers is simple.
weights = np.ones((3,3))
print(f"Before scheduler:\n{weights=}")
epochs = 10
for e in range(epochs):
gradient = np.random.rand(3, 3)
change = adam_scheduler.update_change(gradient)
weights = weights - change
adam_scheduler.reset()
print(f"\nAfter scheduler:\n{weights=}")
In this section we will quickly look at cost functions that can be used when creating the neural network. Every cost function takes the target vector as its parameter, and returns a function valued only at X such that it may easily be differentiated.
def CostOLS(target):
"""
Return OLS function valued only at X, so
that it may be easily differentiated
"""
def func(X):
return (1.0 / target.shape[0]) * np.sum((target - X) ** 2)
return func
def CostLogReg(target):
"""
Return Logistic Regression cost function
valued only at X, so that it may be easily differentiated
"""
def func(X):
return -(1.0 / target.shape[0]) * np.sum(
(target * np.log(X + 10e-10)) + ((1 - target) * np.log(1 - X + 10e-10))
)
return func
def CostCrossEntropy(target):
"""
Return cross entropy cost function valued only at X, so
that it may be easily differentiated
"""
def func(X):
return -(1.0 / target.size) * np.sum(target * np.log(X + 10e-10))
return func
Below we will provide a short example of how these cost function may be used to obtain results if you wish to test them out on your own using AutoGrad's automatic differentiation.
from autograd import grad
target = np.array([[1, 2, 3]]).T
a = np.array([[4, 5, 6]]).T
cost_func = CostCrossEntropy
cost_func_derivative = grad(cost_func(target))
valued_at_a = cost_func_derivative(a)
print(f"Derivative of cost function {cost_func.__name__} valued at a:\n{valued_at_a}")
Finally, before we look at the layers that make up the neural network, we will look at the activation functions which can be specified between the hidden layers and as the output function. Each function can be valued for any given vector or matrix X, and can be differentiated via derivate().
import autograd.numpy as np
from autograd import elementwise_grad
def identity(X):
return X
def sigmoid(X):
try:
return 1.0 / (1 + np.exp(-X))
except FloatingPointError:
return np.where(X > np.zeros(X.shape), np.ones(X.shape), np.zeros(X.shape))
def softmax(X):
X = X - np.max(X, axis=-1, keepdims=True)
delta = 10e-10
return np.exp(X) / (np.sum(np.exp(X), axis=-1, keepdims=True) + delta)
def RELU(X):
return np.where(X > np.zeros(X.shape), X, np.zeros(X.shape))
def LRELU(X):
delta = 10e-4
return np.where(X > np.zeros(X.shape), X, delta * X)
def derivate(func):
if func.__name__ == "RELU":
def func(X):
return np.where(X > 0, 1, 0)
return func
elif func.__name__ == "LRELU":
def func(X):
delta = 10e-4
return np.where(X > 0, 1, delta)
return func
else:
return elementwise_grad(func)
Below we present a short demonstration of how to use an activation function. The derivative of the activation function will be important when calculating the output delta term during backpropagation. Note that derivate() can also be used for cost functions for a more generalized approach.
z = np.array([[4, 5, 6]]).T
print(f"Input to activation function:\n{z}")
act_func = sigmoid
a = act_func(z)
print(f"\nOutput from {act_func.__name__} activation function:\n{a}")
act_func_derivative = derivate(act_func)
valued_at_z = act_func_derivative(a)
print(f"\nDerivative of {act_func.__name__} activation function valued at z:\n{valued_at_z}")
In order to construct a convolutional neural network (CNN), it is crucial to comprehend the fundamental principles of convolution and how it aids in extracting information from images. Convolution, at its core, is merely a mathematical operation between two functions that yields another function. It is represented by an integral between two functions, which is typically expressed as:
$$ (f \ast g)(t):=\int_{-\infty}^{\infty} f(\tau) g(t-\tau) d \tau. $$Here, f and g are the two functions on which we want to perform an operation. The outcome of the convolution operation is represented by \( (f \ast g) \), and it is derived by sliding the function g over f and computing the integral of their product at each position. If both functions are continuous, convolution takes the form shown above. However, if we discretize both f and g, the convolution operation will take the form of a sum between the elements of f and g:
$$ (f \ast g)[n]=\sum_{m=0}^{n-1} f[m] g[n-m]. $$The key idea we utilize to extract the information contained in an image is to slide an \( m \times n \) matrix g over an \( m \times n \) matrix f. In our case, f represents the image, while g represents the kernel, oftentimes called a filter. However, since our convolution will be a two-dimensional variant, we need to extend our mathematical formula with an additional summation:
$$ (f \ast g)[i, j]\sum_{m=0}^{M-1}\sum_{n=0}^{N-1} f[m,n] g[i-m, j-n]. $$It is imperative to note that the size of the kernel g is significantly smaller than the size of the input image f, thereby reducing the amount of computation necessary for feature extraction. Furthermore, the kernel is usually a trainable parameter in a convolutional neural network, allowing the network to learn appropriate kernels for specific tasks.
To give you an example of how 2D convolution works in practice, suppose we have an image f of dimension \( 6 \times 6 \)
$$ f = \begin{bmatrix} 4 & 1 & 2 & 9 & 8 & 6 \\ 9 & 5 & 9 & 5 & 8 & 5 \\ 1 & 5 & 9 & 7 & 6 & 4 \\ 2 & 9 & 8 & 3 & 7 & 1 \\ 8 & 1 & 6 & 4 & 2 & 2 \\ 1 & 0 & 5 & 7 & 8 & 2 \\ \end{bmatrix} $$and a \( 3 \times 3 \) kernel g called a low-pass filter. Note that the kernel is usually rotated by 180 degrees during convolution, however this has no effect on this kernel.
$$ g = \frac{1}{9} \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{bmatrix} $$In order to filter the image, we have to extract a \( 3 \times 3 \) element from the upper left corner of f, and perform element-wise multiplication of the extracted image pixels with the elements of the kernel g:
$$ \begin{bmatrix} 4 & 1 & 2 \\ 9 & 5 & 9 \\ 1 & 5 & 9 \\ \end{bmatrix} \cdot \begin{bmatrix} \frac{1}{9} & \frac{1}{9} & \frac{1}{9} \\ \frac{1}{9} & \frac{1}{9} & \frac{1}{9} \\ \frac{1}{9} & \frac{1}{9} & \frac{1}{9} \\ \end{bmatrix} = \begin{bmatrix} \frac{4}{9} & \frac{1}{9} & \frac{2}{9} \\ \frac{9}{9} & \frac{5}{9} & \frac{9}{9} \\ \frac{1}{9} & \frac{5}{9} & \frac{9}{9} \\ \end {bmatrix} = \textbf{A} $$Then, following the multiplication, we summarize all the elements of the resulting matrix A:
$$ (f \ast g)[0, 0]= \sum_{i=0}^{2} \sum_{j=0}^{2} a_{i,j} = 5 $$Which corresponds to the first element of the filtered image \( (f \ast g) \).
Here we use a stride of 1, a parameter denoted s which describes how many indexes we move the kernel g to the right before repeating the calculations above for the next \( 3 \times 3 \) element of the image f. It is usually presumed that *s*=1, however, larger values for s can be used to reduce the dimentionality of the filtered image such that the convolution operation is more computationally efficient. In the context of a convolutional neural network, this will become very useful.
The full result of the convolution is:
$$ (f \ast g) = \begin{bmatrix} 5 & 5.78 & 7 & 6.44 \\ 6.33 & 6.67 & 6.89 & 5.11 \\ 5.44 & 5.78 & 5.78 & 4 \\ 4.44 & 4.78 & 5.56 & 4 \\ \end{bmatrix} $$The result is markedly smaller in shape than the original image. This occurs when using convolution without first padding the image with additional columns and rows, allowing us to keep the original image shape after sliding the kernel over the image. How many rows and columns we wish to pad the image with depends strictly on the shape of the kernel, as we wish to pad the image with r additional rows and c additional columns.
$$ r =\lfloor \frac{kernel\ height}{2} \rfloor \cdot 2 \\ c =\lfloor \frac{kernel\ width}{2} \rfloor \cdot 2 $$Note the notation \( \lfloor \frac{kernel width}{2} \rfloor \) means that we floor the result of the division, meaning we round down to a whole number in case \( \frac{kernel width}{2} \) results in a floating point number.
Using those simple equations, we find out by how much we have to extend the dimensions of the original image. Before proceeding, however, we might ask what we shall fill the additional rows and columns with? One of the most common approaches to padding is zero-padding, which as the name suggest, involves filling the rows and columns with zeros. This is the approach that we will be using for this demonstration. If we apply this padding to out original \( 6 \times 6 \) image, the result will be an \( 8 \times 8 \) image as the kernel has a width and height of 3. Note that the original image is encapsuled by the zero-padded rows and columns:
$$ \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 4 & 1 & 2 & 9 & 8 & 6 & 0 \\ 0 & 9 & 5 & 9 & 5 & 8 & 5 & 0 \\ 0 & 1 & 5 & 9 & 7 & 6 & 4 & 0 \\ 0 & 2 & 9 & 8 & 3 & 7 & 1 & 0 \\ 0 & 8 & 1 & 6 & 4 & 2 & 2 & 0 \\ 0 & 1 & 0 & 5 & 7 & 8 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \] \end{bmatrix} $$Below we have provided code that demonstrates padding and convolution. As you will see when we run the code, the size of the image will remain unchanged when using padding.
import numpy as np
def padding(image, kernel):
# calculate r and c
r = (kernel.shape[0] // 2) * 2
c = (kernel.shape[1] // 2) * 2
# padded image dimensions
padded_height = image.shape[0] + r
padded_width = image.shape[1] + c
# for more readable code
k_half_height = kernel.shape[0] // 2
k_half_width = kernel.shape[1] // 2
# zero matrix with padded dimensions
padded_img = np.zeros((padded_height, padded_width))
# place image into zero matrix
padded_img[k_half_height : padded_height - k_half_height,
k_half_width : padded_width - k_half_width] = image[:, :]
return padded_img
def convolve(original_image, padded_image, kernel, stride=1):
# rotate kernel by 180 degrees
kernel = np.rot90(np.rot90(kernel))
# note that kernel height // 2 is written as 'm'
# and kernel width // 2 as 'n' in the mathematical notation
m = kernel.shape[0] // 2
n = kernel.shape[1] // 2
r = (kernel.shape[0] // 2) * 2
c = (kernel.shape[1] // 2) * 2
# initialize output array
convolved_image = np.zeros(original_image.shape)
image_height = original_image.shape[0]
image_width = original_image.shape[1]
# the convolution
for i in range(m, image_height + m, stride):
for j in range(n, image_width + n, stride):
convolved_image[i-m, j-n] = np.sum(
padded_image[i : i + m, j : j + n]
* kernel
)
return convolved_image
def convolve(image, kernel, stride=1):
for i in range(2):
kernel = np.rot90(kernel)
k_half_height = kernel.shape[0] // 2
k_half_width = kernel.shape[0] // 2
conv_image = np.zeros(image.shape)
pad_image = padding(image, kernel)
for i in range(k_half_height, conv_image.shape[0] + k_half_height, stride):
for j in range(k_half_width, conv_image.shape[1] + k_half_width, stride):
conv_image[i - k_half_height, j - k_half_width] = np.sum(
pad_image[
i - k_half_height : i + k_half_height + 1, j - k_half_width : j + k_half_width + 1
]
* kernel
)
return conv_image
Fun fact: When filtering images, you will see that convolution involves rotating the kernel by 180 degrees. However, this is not the case when applying convolution in a CNN, where the same operation not rotated by 180 degrees is called cross-correlation.
original_image = np.array([[4, 1, 2, 9, 8, 6],
[9, 5, 9, 5, 8, 5],
[1, 5, 9, 7, 6, 4],
[2, 9, 8, 3, 7, 1],
[8, 1, 6, 4, 2, 2],
[1, 0, 5, 7, 8, 2]])
kernel = (1/9)*np.ones((3,3))
print(f"{original_image.shape=}")
# note that convolve() performs padding
convolved_image = convolve(original_image, kernel, stride=1)
print(f"{convolved_image.shape=}")
As you can see, the resulting image is of the same size as the original image. To round of our demonstration of convolution, we will present the results of convolution using commonly used kernels. In a CNN, the values of the kernels are randomly initialized, and then learned during training. These kernels will extract information regarding the picture, such as for example the edge detection filter demonstrated below extracts the edges present in the picture. Of course, there is no guarantee that the CNN will learn an edge detection filter, but this should provide some intuiton as to how the CNN is able to use kernels to make better predictions than a regular feed forward neural network.
# Now an example using a real image and first a gaussian low-pass filter and then a sobel filter
import numpy as np
import imageio.v3 as imageio
import matplotlib.pyplot as plt
import time
def generate_gauss_mask(sigma, K=1):
side = np.ceil(1 + 8 * sigma)
y, x = np.mgrid[-side // 2 + 1 : (side // 2) + 1, -side // 2 + 1 : (side // 2) + 1]
ker_coef = K / (2 * np.pi * sigma**2)
g = np.exp(-((x**2 + y**2) / (2.0 * sigma**2)))
return g, ker_coef
img_path = "data/IMG-2167.JPG"
image_of_cute_dog = imageio.imread(img_path, mode='L')
plt.imshow(image_of_cute_dog, cmap="gray", vmin=0, vmax=255, aspect="auto")
plt.title("Original image")
plt.show()
gauss, kernel = generate_gauss_mask(sigma=6)
gauss_kernel = gauss*kernel
filtered_image = convolve(image_of_cute_dog, gauss_kernel)
plt.imshow(filtered_image, cmap="gray", vmin=0, vmax=255, aspect="auto")
plt.title("Result of convolution with gauss kernel (blurring filter)")
plt.show()
sobel_kernel = np.array([[1, 2, 1],
[0, 0, 0],
[-1, -2, -1]])
filtered_image = convolve(image_of_cute_dog, sobel_kernel)
plt.imshow(filtered_image, cmap="gray", vmin=0, vmax=255, aspect="auto")
plt.title("Result of convolution with sobel kernel (edge detection filter)")
plt.show()
The code below initialises global variables for readability and describes the abstract class Layers. This is not important in order to understand the CNN, but is benefitial for organizing the code neatly.
import math
import autograd.numpy as np
from copy import deepcopy, copy
from autograd import grad
from typing import Callable
# global variables for index readability
input_index = 0
node_index = 1
bias_index = 1
input_channel_index = 1
feature_maps_index = 1
height_index = 2
width_index = 3
kernel_feature_maps_index = 1
kernel_input_channels_index = 0
class Layer:
def __init__(self, seed):
self.seed = seed
def _feedforward(self):
raise NotImplementedError
def _backpropagate(self):
raise NotImplementedError
def _reset_weights(self, previous_nodes):
raise NotImplementedError
After establishing the foundational understanding of applying convolution to spatial data, let us delve into the intricate workings of a convolutional layer in a Convolutional Neural Network (CNN). The primary function of convolution, as previously discussed, is to extract pertinent information from images while simultaneously decreasing the scale of our data. To initiate the image processing, we shall begin by partitioning the images into color channels (unless the image is grayscale), comprising three primary colors: red, green, and blue. We will subsequently utilize trainable kernels to construct a higher-dimensional encoding of each channel called feature maps. Successive layers will receive these feature maps as inputs, generating further encodings, albeit with reduced dimensions. The term trainable kernels denotes the initialization of pre-defined kernel-shaped weights, which we will then train via backpropagation, similar to how weights are trained in a Feedforward Neural Network.
To ensure seamless integration between our implementation of the convolutional layer and popular machine learning frameworks like Tensorflow (Keras) and PyTorch, we have adopted a design pattern that mirrors the construction of models using these APIs. This involves implementing our convolutional layer as a Python class or object, which allows for a more modular and flexible approach to building neural networks. By structuring our code in this way, users can easily incorporate our implementation into their existing machine learning pipelines without having to make significant changes to their codebase. Additionally, this design pattern promotes code reusability and makes it easier to maintain and update our convolutional layer implementation over time.
Note that the Convolution2DLayer takes in an activation function as a parameter, as it also performs non-linearity.
class Convolution2DLayer(Layer):
def __init__(
self,
input_channels,
feature_maps,
kernel_height,
kernel_width,
v_stride,
h_stride,
pad,
act_func: Callable,
seed=None,
reset_weights_independently=True,
):
super().__init__(seed)
self.input_channels = input_channels
self.feature_maps = feature_maps
self.kernel_height = kernel_height
self.kernel_width = kernel_width
self.v_stride = v_stride
self.h_stride = h_stride
self.pad = pad
self.act_func = act_func
# such that the layer can be used on its own
# outside of the CNN module
if reset_weights_independently == True:
self._reset_weights_independently()
def _feedforward(self, X_batch):
# note that the shape of X_batch = [inputs, input_maps, img_height, img_width]
# pad the input batch
X_batch_padded = self._padding(X_batch)
# calculate height_index and width_index after stride
strided_height = int(np.ceil(X_batch.shape[height_index] / self.v_stride))
strided_width = int(np.ceil(X_batch.shape[width_index] / self.h_stride))
# create output array
output = np.ndarray(
(
X_batch.shape[input_index],
self.feature_maps,
strided_height,
strided_width,
)
)
# save input and output for backpropagation
self.X_batch_feedforward = X_batch
self.output_shape = output.shape
# checking for errors, no need to look here :)
self._check_for_errors()
# convolve input with kernel
for img in range(X_batch.shape[input_index]):
for chin in range(self.input_channels):
for fmap in range(self.feature_maps):
out_h = 0
for h in range(0, X_batch.shape[height_index], self.v_stride):
out_w = 0
for w in range(0, X_batch.shape[width_index], self.h_stride):
output[img, fmap, out_h, out_w] = np.sum(
X_batch_padded[
img,
chin,
h : h + self.kernel_height,
w : w + self.kernel_width,
]
* self.kernel[chin, fmap, :, :]
)
out_w += 1
out_h += 1
# Pay attention to the fact that we're not rotating the kernel by 180 degrees when filtering the image in
# the convolutional layer, as convolution in terms of Machine Learning is a procedure known as cross-correlation
# in image processing and signal processing
# return a
return self.act_func(output / (self.kernel_height))
def _backpropagate(self, delta_term_next):
# intiate matrices
delta_term = np.zeros((self.X_batch_feedforward.shape))
gradient_kernel = np.zeros((self.kernel.shape))
# pad input for convolution
X_batch_padded = self._padding(self.X_batch_feedforward)
# Since an activation function is used at the output of the convolution layer, its derivative
# has to be accounted for in the backpropagation -> as if ReLU was a layer on its own.
act_derivative = derivate(self.act_func)
delta_term_next = act_derivative(delta_term_next)
# fill in 0's for values removed by vertical stride in feedforward
if self.v_stride > 1:
v_ind = 1
for i in range(delta_term_next.shape[height_index]):
for j in range(self.v_stride - 1):
delta_term_next = np.insert(
delta_term_next, v_ind, 0, axis=height_index
)
v_ind += self.v_stride
# fill in 0's for values removed by horizontal stride in feedforward
if self.h_stride > 1:
h_ind = 1
for i in range(delta_term_next.shape[width_index]):
for k in range(self.h_stride - 1):
delta_term_next = np.insert(
delta_term_next, h_ind, 0, axis=width_index
)
h_ind += self.h_stride
# crops out 0-rows and 0-columns
delta_term_next = delta_term_next[
:,
:,
: self.X_batch_feedforward.shape[height_index],
: self.X_batch_feedforward.shape[width_index],
]
# the gradient received from the next layer also needs to be padded
delta_term_next = self._padding(delta_term_next)
# calculate delta term by convolving next delta term with kernel
for img in range(self.X_batch_feedforward.shape[input_index]):
for chin in range(self.input_channels):
for fmap in range(self.feature_maps):
for h in range(self.X_batch_feedforward.shape[height_index]):
for w in range(self.X_batch_feedforward.shape[width_index]):
delta_term[img, chin, h, w] = np.sum(
delta_term_next[
img,
fmap,
h : h + self.kernel_height,
w : w + self.kernel_width,
]
* np.rot90(np.rot90(self.kernel[chin, fmap, :, :]))
)
# calculate gradient for kernel for weight update
# also via convolution
for chin in range(self.input_channels):
for fmap in range(self.feature_maps):
for k_x in range(self.kernel_height):
for k_y in range(self.kernel_width):
gradient_kernel[chin, fmap, k_x, k_y] = np.sum(
X_batch_padded[
img,
chin,
h : h + self.kernel_height,
w : w + self.kernel_width,
]
* delta_term_next[
img,
fmap,
h : h + self.kernel_height,
w : w + self.kernel_width,
]
)
# all kernels are updated with weight gradient of kernel
self.kernel -= gradient_kernel
# return delta term
return delta_term
def _padding(self, X_batch, batch_type="image"):
# same padding for images
if self.pad == "same" and batch_type == "image":
padded_height = X_batch.shape[height_index] + (self.kernel_height // 2) * 2
padded_width = X_batch.shape[width_index] + (self.kernel_width // 2) * 2
half_kernel_height = self.kernel_height // 2
half_kernel_width = self.kernel_width // 2
# initialize padded array
X_batch_padded = np.ndarray(
(
X_batch.shape[input_index],
X_batch.shape[feature_maps_index],
padded_height,
padded_width,
)
)
# zero pad all images in X_batch
for img in range(X_batch.shape[input_index]):
padded_img = np.zeros(
(X_batch.shape[feature_maps_index], padded_height, padded_width)
)
padded_img[
:,
half_kernel_height : padded_height - half_kernel_height,
half_kernel_width : padded_width - half_kernel_width,
] = X_batch[img, :, :, :]
X_batch_padded[img, :, :, :] = padded_img[:, :, :]
return X_batch_padded
# same padding for gradients
elif self.pad == "same" and batch_type == "grad":
padded_height = X_batch.shape[height_index] + (self.kernel_height // 2) * 2
padded_width = X_batch.shape[width_index] + (self.kernel_width // 2) * 2
half_kernel_height = self.kernel_height // 2
half_kernel_width = self.kernel_width // 2
# initialize padded array
delta_term_padded = np.zeros(
(
X_batch.shape[input_index],
X_batch.shape[feature_maps_index],
padded_height,
padded_width,
)
)
# zero pad delta term
delta_term_padded[
:, :, : X_batch.shape[height_index], : X_batch.shape[width_index]
] = X_batch[:, :, :, :]
return delta_term_padded
else:
return X_batch
def _reset_weights_independently(self):
# sets seed to remove randomness inbetween runs
if self.seed is not None:
np.random.seed(self.seed)
# initializes kernel matrix
self.kernel = np.ndarray(
(
self.input_channels,
self.feature_maps,
self.kernel_height,
self.kernel_width,
)
)
# randomly initializes weights
for chin in range(self.kernel.shape[kernel_input_channels_index]):
for fmap in range(self.kernel.shape[kernel_feature_maps_index]):
self.kernel[chin, fmap, :, :] = np.random.rand(
self.kernel_height, self.kernel_width
)
def _reset_weights(self, previous_nodes):
# sets weights
self._reset_weights_independently()
# returns shape of output used for subsequent layer's weight initiation
strided_height = int(
np.ceil(previous_nodes.shape[height_index] / self.v_stride)
)
strided_width = int(np.ceil(previous_nodes.shape[width_index] / self.h_stride))
next_nodes = np.ones(
(
previous_nodes.shape[input_index],
self.feature_maps,
strided_height,
strided_width,
)
)
return next_nodes / self.kernel_height
def _check_for_errors(self):
if self.X_batch_feedforward.shape[input_channel_index] != self.input_channels:
raise AssertionError(
f"ERROR: Number of input channels in data ({self.X_batch_feedforward.shape[input_channel_index]}) is not equal to input channels in Convolution2DLayerOPT ({self.input_channels})! Please change the number of input channels of the Convolution2DLayer such that they are equal"
)
As you may have noticed, we have not yet explained how the backpropagation algorithm works in a convolutional layer. However, having covered all other major details about convolutional layers, we are now prepared to do so. It should come as no surprise that the calculation of delta terms at each convolutional layer takes the form of convolution. After the gradient has been propagated backwards through the flattening layer, where it was reshaped into an appropriate form, calculating the update value for the kernel is simply a matter of convolving the output gradient with the input of the layer for which we are updating the weights. For more detail, this article serves as an excellent resource, see https://pavisj.medium.com/convolutions-and-backpropagations-46026a8f5d2c
We can use the convolutional layer above to perform a simple convolution on an image of the now familiar cute dog.
import numpy as np
import imageio.v3 as imageio
import matplotlib.pyplot as plt
def plot_convolution_result(X, layer):
plt.imshow(X[0, 0, :, :], vmin=0, vmax=255, cmap="gray")
plt.title("Original image")
plt.colorbar()
plt.show()
conv_result = layer._feedforward(X)
plt.title("Result of convolutional layer")
plt.imshow(conv_result[0, 0, :, :], vmin=0, vmax=255, cmap="gray")
plt.colorbar()
plt.show()
# create layer
layer = Convolution2DLayer(
input_channels=3,
feature_maps=1,
kernel_height=4,
kernel_width=4,
v_stride=2,
h_stride=2,
pad="same",
act_func=identity,
seed=2023,
)
# read in image path, make data correct format
img_path = img_path = "data/IMG-2167.JPG"
image_of_cute_dog = imageio.imread(img_path)
image_shape = image_of_cute_dog.shape
image_of_cute_dog = image_of_cute_dog.reshape(1, image_shape[0], image_shape[1], image_shape[2])
image_of_cute_dog = image_of_cute_dog.transpose(0, 3, 1, 2)
# plot the result of the convolution
plot_convolution_result(image_of_cute_dog, layer)
We cobserve that the result has half the pixels on each axis due to the fact that we've used a horizontal and vertical stride of 2. The result of this convolution is not very insightfull, as the kernel has completely random values for the first feedforward pass. However, as we perform multiple forward and backward passes, the results of the convolution should provide identifying features of the image it uses for classification.
Note that image data usually comes in many different shapes and sizes, but for our CNN we require the input data be formatted as \[Number of inputs, input channels, input height, input width\]. Occasionally, the data you come accross use will be formatted like this, but on many occasions reshaping and transposing the dimensions is sadly necessary.
The pooling layer is another widely used type of layer in convolutional neural networks that enables data downsampling to a more manageable size. Despite recent technological advancements that allow for convolution without excessive size reduction of the data, the pooling layer still remains a fundamental component of convolutional neural networks. It can be used before, after, or in between convolutional layers, although finding the optimal placement of layers and network depth requires experimentation to achieve the best performance for a given problem. The code we provide allows you to perform two types of pooling known as max pooling and average pooling.
class Pooling2DLayer(Layer):
def __init__(
self,
kernel_height,
kernel_width,
v_stride,
h_stride,
pooling="max",
seed=None,
):
super().__init__(seed)
self.kernel_height = kernel_height
self.kernel_width = kernel_width
self.v_stride = v_stride
self.h_stride = h_stride
self.pooling = pooling
def _feedforward(self, X_batch):
# Saving the input for use in the backwardpass
self.X_batch_feedforward = X_batch
# check if user is silly
self._check_for_errors()
# Computing the size of the feature maps based on kernel size and the stride parameter
strided_height = (
X_batch.shape[height_index] - self.kernel_height
) // self.v_stride + 1
if X_batch.shape[height_index] == X_batch.shape[width_index]:
strided_width = strided_height
else:
strided_width = (
X_batch.shape[width_index] - self.kernel_width
) // self.h_stride + 1
# initialize output array
output = np.ndarray(
(
X_batch.shape[input_index],
X_batch.shape[feature_maps_index],
strided_height,
strided_width,
)
)
# select pooling action, either max or average pooling
if self.pooling == "max":
self.pooling_action = np.max
elif self.pooling == "average":
self.pooling_action = np.mean
# pool based on kernel size and stride
for img in range(output.shape[input_index]):
for fmap in range(output.shape[feature_maps_index]):
for h in range(strided_height):
for w in range(strided_width):
output[img, fmap, h, w] = self.pooling_action(
X_batch[
img,
fmap,
(h * self.v_stride) : (h * self.v_stride)
+ self.kernel_height,
(w * self.h_stride) : (w * self.h_stride)
+ self.kernel_width,
]
)
# output for feedforward in next layer
return output
def _backpropagate(self, delta_term_next):
# initiate delta term array
delta_term = np.zeros((self.X_batch_feedforward.shape))
for img in range(delta_term_next.shape[input_index]):
for fmap in range(delta_term_next.shape[feature_maps_index]):
for h in range(0, delta_term_next.shape[height_index], self.v_stride):
for w in range(
0, delta_term_next.shape[width_index], self.h_stride
):
# max pooling
if self.pooling == "max":
# get window
window = self.X_batch_feedforward[
img,
fmap,
h : h + self.kernel_height,
w : w + self.kernel_width,
]
# find max values indices in window
max_h, max_w = np.unravel_index(
window.argmax(), window.shape
)
# set values in new, upsampled delta term
delta_term[
img,
fmap,
(h + max_h),
(w + max_w),
] += delta_term_next[img, fmap, h, w]
# average pooling
if self.pooling == "average":
delta_term[
img,
fmap,
h : h + self.kernel_height,
w : w + self.kernel_width,
] = (
delta_term_next[img, fmap, h, w]
/ self.kernel_height
/ self.kernel_width
)
# returns input to backpropagation in previous layer
return delta_term
def _reset_weights(self, previous_nodes):
# calculate strided height, strided width
strided_height = (
previous_nodes.shape[height_index] - self.kernel_height
) // self.v_stride + 1
if previous_nodes.shape[height_index] == previous_nodes.shape[width_index]:
strided_width = strided_height
else:
strided_width = (
previous_nodes.shape[width_index] - self.kernel_width
) // self.h_stride + 1
# initiate output array
output = np.ones(
(
previous_nodes.shape[input_index],
previous_nodes.shape[feature_maps_index],
strided_height,
strided_width,
)
)
# returns output with shape used for reset weights in next layer
return output
def _check_for_errors(self):
# check if input is smaller than kernel size -> error
assert (
self.X_batch_feedforward.shape[width_index] >= self.kernel_width
), f"ERROR: Pooling kernel width_index ({self.kernel_width}) larger than data width_index ({self.X_batch_feedforward.input.shape[2]}), please lower the kernel width_index of the Pooling2DLayer"
assert (
self.X_batch_feedforward.shape[height_index] >= self.kernel_height
), f"ERROR: Pooling kernel height_index ({self.kernel_height}) larger than data height_index ({self.X_batch_feedforward.input.shape[3]}), please lower the kernel height_index of the Pooling2DLayer"
Before we can begin building our first CNN model, we need to introduce the flattening layer. As its name suggests, the flattening layer transforms the data into a one-dimensional vector that can be fed into the feedforward layers of our network. This layer plays a crucial role in preparing the data for further processing in the network. Additionally, the flattening layer is responsible for reshaping the gradient to the proper shape during backpropagation. This ensures that the kernels are correctly updated, allowing for effective learning in the network.
class FlattenLayer(Layer):
def __init__(self, act_func=LRELU, seed=None):
super().__init__(seed)
self.act_func = act_func
def _feedforward(self, X_batch):
# save input for backpropagation
self.X_batch_feedforward_shape = X_batch.shape
# Remember, the data has the following shape: (I, FM, H, W, ) in the convolutional layers
# whilst the data has the shape (I, FM * H * W) in the fully connected layers
# I = Inputs, FM = Feature Maps, H = Height and W = Width.
X_batch = X_batch.reshape(
X_batch.shape[input_index],
X_batch.shape[feature_maps_index]
* X_batch.shape[height_index]
* X_batch.shape[width_index],
)
# add bias to a
self.z_matrix = X_batch
bias = np.ones((X_batch.shape[input_index], 1)) * 0.01
self.a_matrix = np.hstack([bias, X_batch])
# return a, the input to feedforward in next layer
return self.a_matrix
def _backpropagate(self, weights_next, delta_term_next):
activation_derivative = derivate(self.act_func)
# calculate delta term
delta_term = (
weights_next[bias_index:, :] @ delta_term_next.T
).T * activation_derivative(self.z_matrix)
# FlattenLayer does not update weights
# reshapes delta layer to convolutional layer data format [Input, Feature_Maps, Height, Width]
return delta_term.reshape(self.X_batch_feedforward_shape)
def _reset_weights(self, previous_nodes):
# note that the previous nodes to the FlattenLayer are from the convolutional layers
previous_nodes = previous_nodes.reshape(
previous_nodes.shape[input_index],
previous_nodes.shape[feature_maps_index]
* previous_nodes.shape[height_index]
* previous_nodes.shape[width_index],
)
# return shape used in reset_weights in next layer
return previous_nodes.shape[node_index]
def get_prev_a(self):
return self.a_matrix
Finally, the result from the flatten layer will pass to a series of fully connected layers, which function as a normal feed forward neural network. The fully connected layers are split into two classes; FullyConnectedLayer which acts as a hidden layer, and OutputLayer, which acts as the single output layer at the end of the CNN. If one wishes to use this codebase to construct a normal feed forward neural network, it must start with a FlattenLayer due to techincal details regarding weight intitialization. However many FullyConnectedLayers can be added to the CNN, and in each layer the amount of nodes, which activation function and scheduler to use can be specified. In practice, the scheduler will be specified in the CNN object initialization, and inherited if no other scheduler is specified.
class FullyConnectedLayer(Layer):
# FullyConnectedLayer per default uses LRELU and Adam scheduler
# with an eta of 0.0001, rho of 0.9 and rho2 of 0.999
def __init__(
self,
nodes: int,
act_func: Callable = LRELU,
scheduler: Scheduler = Adam(eta=1e-4, rho=0.9, rho2=0.999),
seed: int = None,
):
super().__init__(seed)
self.nodes = nodes
self.act_func = act_func
self.scheduler_weight = copy(scheduler)
self.scheduler_bias = copy(scheduler)
# initiate matrices for later
self.weights = None
self.a_matrix = None
self.z_matrix = None
def _feedforward(self, X_batch):
# calculate z
self.z_matrix = X_batch @ self.weights
# calculate a, add bias
bias = np.ones((X_batch.shape[input_index], 1)) * 0.01
self.a_matrix = self.act_func(self.z_matrix)
self.a_matrix = np.hstack([bias, self.a_matrix])
# return a, the input for feedforward in next layer
return self.a_matrix
def _backpropagate(self, weights_next, delta_term_next, a_previous, lam):
# take the derivative of the activation function
activation_derivative = derivate(self.act_func)
# calculate the delta term
delta_term = (
weights_next[bias_index:, :] @ delta_term_next.T
).T * activation_derivative(self.z_matrix)
# intitiate matrix to store gradient
# note that we exclude the bias term, which we will calculate later
gradient_weights = np.zeros(
(
a_previous.shape[input_index],
a_previous.shape[node_index] - bias_index,
delta_term.shape[node_index],
)
)
# calculate gradient = delta term * previous a
for i in range(len(delta_term)):
gradient_weights[i, :, :] = np.outer(
a_previous[i, bias_index:], delta_term[i, :]
)
# sum the gradient, divide by input_index
gradient_weights = np.mean(gradient_weights, axis=input_index)
# for the bias gradient we do not multiply by previous a
gradient_bias = np.mean(delta_term, axis=input_index).reshape(
1, delta_term.shape[node_index]
)
# regularization term
gradient_weights += self.weights[bias_index:, :] * lam
# send gradients into scheduler
# returns update matrix which will be used to update the weights and bias
update_matrix = np.vstack(
[
self.scheduler_bias.update_change(gradient_bias),
self.scheduler_weight.update_change(gradient_weights),
]
)
# update weights
self.weights -= update_matrix
# return weights and delta term, input for backpropagation in previous layer
return self.weights, delta_term
def _reset_weights(self, previous_nodes):
# sets seed to remove randomness inbetween runs
if self.seed is not None:
np.random.seed(self.seed)
# add bias, initiate random weights
bias = 1
self.weights = np.random.randn(previous_nodes + bias, self.nodes)
# returns number of nodes, used for reset_weights in next layer
return self.nodes
def _reset_scheduler(self):
# resets scheduler per epoch
self.scheduler_weight.reset()
self.scheduler_bias.reset()
def get_prev_a(self):
# returns a matrix, used in backpropagation
return self.a_matrix
class OutputLayer(FullyConnectedLayer):
def __init__(
self,
nodes: int,
output_func: Callable = LRELU,
cost_func: Callable = CostCrossEntropy,
scheduler: Scheduler = Adam(eta=1e-4, rho=0.9, rho2=0.999),
seed: int = None,
):
super().__init__(nodes, output_func, copy(scheduler), seed)
self.cost_func = cost_func
# initiate matrices for later
self.weights = None
self.a_matrix = None
self.z_matrix = None
# decides if the output layer performs binary or multi-class classification
self._set_pred_format()
def _feedforward(self, X_batch: np.ndarray):
# calculate a, z
# note that bias is not added as this would create an extra output class
self.z_matrix = X_batch @ self.weights
self.a_matrix = self.act_func(self.z_matrix)
# returns prediction
return self.a_matrix
def _backpropagate(self, target, a_previous, lam):
# note that in the OutputLayer the activation function is the output function
activation_derivative = derivate(self.act_func)
# calculate output delta terms
# for multi-class or binary classification
if self.pred_format == "Multi-class":
delta_term = self.a_matrix - target
else:
cost_func_derivative = grad(self.cost_func(target))
delta_term = activation_derivative(self.z_matrix) * cost_func_derivative(
self.a_matrix
)
# intiate matrix that stores gradient
gradient_weights = np.zeros(
(
a_previous.shape[input_index],
a_previous.shape[node_index] - bias_index,
delta_term.shape[node_index],
)
)
# calculate gradient = delta term * previous a
for i in range(len(delta_term)):
gradient_weights[i, :, :] = np.outer(
a_previous[i, bias_index:], delta_term[i, :]
)
# sum the gradient, divide by input_index
gradient_weights = np.mean(gradient_weights, axis=input_index)
# for the bias gradient we do not multiply by previous a
gradient_bias = np.mean(delta_term, axis=input_index).reshape(
1, delta_term.shape[node_index]
)
# regularization term
gradient_weights += self.weights[bias_index:, :] * lam
# send gradients into scheduler
# returns update matrix which will be used to update the weights and bias
update_matrix = np.vstack(
[
self.scheduler_bias.update_change(gradient_bias),
self.scheduler_weight.update_change(gradient_weights),
]
)
# update weights
self.weights -= update_matrix
# return weights and delta term, input for backpropagation in previous layer
return self.weights, delta_term
def _reset_weights(self, previous_nodes):
# sets seed to remove randomness inbetween runs
if self.seed is not None:
np.random.seed(self.seed)
# add bias, initiate random weights
bias = 1
self.weights = np.random.rand(previous_nodes + bias, self.nodes)
# returns number of nodes, used for reset_weights in next layer
return self.nodes
def _reset_scheduler(self):
# resets scheduler per epoch
self.scheduler_weight.reset()
self.scheduler_bias.reset()
def _set_pred_format(self):
# sets prediction format to either regression, binary or multi-class classification
if self.act_func.__name__ is None or self.act_func.__name__ == "identity":
self.pred_format = "Regression"
elif self.act_func.__name__ == "sigmoid" or self.act_func.__name__ == "tanh":
self.pred_format = "Binary"
else:
self.pred_format = "Multi-class"
def get_pred_format(self):
# returns format of prediction
return self.pred_format
For our CNN, we have also implemented an optimized version of the Convolution2DLayer, Convolution2DLayerOPT, which runs much faster. See VII. Remarks for discussion. This layer will per default be used by the CNN due to its computational advantages, but is much less readable. We've documented it such that specially interested students can understand the principles behind it, but it is not recommended to read. In short, we reshape and transpose parts of the image such that the convolutional operation can be swapped out for a simple matrix multiplication.
class Convolution2DLayerOPT(Convolution2DLayer):
"""
Am optimized version of the convolution layer above which
utilizes an approach of extracting windows of size equivalent
in size to the filter. The convoution is then performed on those
windows instead of a full feature map.
"""
def __init__(
self,
input_channels,
feature_maps,
kernel_height,
kernel_width,
v_stride,
h_stride,
pad,
act_func: Callable,
seed=None,
reset_weights_independently=True,
):
super().__init__(
input_channels,
feature_maps,
kernel_height,
kernel_width,
v_stride,
h_stride,
pad,
act_func,
seed,
)
# true if layer is used outside of CNN
if reset_weights_independently == True:
self._reset_weights_independently()
def _feedforward(self, X_batch):
# The optimized _feedforward method is difficult to understand but computationally more efficient
# for a more "by the book" approach, please look at the _feedforward method of Convolution2DLayer
# save the input for backpropagation
self.X_batch_feedforward = X_batch
# check that there are the correct amount of input channels
self._check_for_errors()
# calculate new shape after stride
strided_height = int(np.ceil(X_batch.shape[height_index] / self.v_stride))
strided_width = int(np.ceil(X_batch.shape[width_index] / self.h_stride))
# get windows of the image for more computationally efficient convolution
# the idea is that we want to align the dimensions that we wish to matrix
# multiply, then use a simple matrix multiplication instead of convolution.
# then, we reshape the size back to its intended shape
windows = self._extract_windows(X_batch)
windows = windows.transpose(1, 0, 2, 3, 4).reshape(
X_batch.shape[input_index],
strided_height * strided_width,
-1,
)
# reshape the kernel for more computationally efficient convolution
kernel = self.kernel
kernel = kernel.transpose(0, 2, 3, 1).reshape(
kernel.shape[kernel_input_channels_index]
* kernel.shape[height_index]
* kernel.shape[width_index],
-1,
)
# use simple matrix calculation to obtain output
output = (
(windows @ kernel)
.reshape(
X_batch.shape[input_index],
strided_height,
strided_width,
-1,
)
.transpose(0, 3, 1, 2)
)
# The output is reshaped and rearranged to appropriate shape
return self.act_func(
output / (self.kernel_height * X_batch.shape[feature_maps_index])
)
def _backpropagate(self, delta_term_next):
# The optimized _backpropagate method is difficult to understand but computationally more efficient
# for a more "by the book" approach, please look at the _backpropagate method of Convolution2DLayer
act_derivative = derivate(self.act_func)
delta_term_next = act_derivative(delta_term_next)
# calculate strided dimensions
strided_height = int(
np.ceil(self.X_batch_feedforward.shape[height_index] / self.v_stride)
)
strided_width = int(
np.ceil(self.X_batch_feedforward.shape[width_index] / self.h_stride)
)
# copy kernel
kernel = self.kernel
# get windows, reshape for matrix multiplication
windows = self._extract_windows(self.X_batch_feedforward, "image").reshape(
self.X_batch_feedforward.shape[input_index]
* strided_height
* strided_width,
-1,
)
# initialize output gradient, reshape and transpose into correct shape
# for matrix multiplication
output_grad_tr = delta_term_next.transpose(0, 2, 3, 1).reshape(
self.X_batch_feedforward.shape[input_index]
* strided_height
* strided_width,
-1,
)
# calculate gradient kernel via simple matrix multiplication and reshaping
gradient_kernel = (
(windows.T @ output_grad_tr)
.reshape(
kernel.shape[kernel_input_channels_index],
kernel.shape[height_index],
kernel.shape[width_index],
kernel.shape[kernel_feature_maps_index],
)
.transpose(0, 3, 1, 2)
)
# for computing the input gradient
windows_out, upsampled_height, upsampled_width = self._extract_windows(
delta_term_next, "grad"
)
# calculate new window dimensions
new_windows_first_dim = (
self.X_batch_feedforward.shape[input_index]
* upsampled_height
* upsampled_width
)
# ceil allows for various asymmetric kernels
new_windows_sec_dim = int(np.ceil(windows_out.size / new_windows_first_dim))
# reshape for matrix multiplication
windows_out = windows_out.transpose(1, 0, 2, 3, 4).reshape(
new_windows_first_dim, new_windows_sec_dim
)
# reshape for matrix multiplication
kernel_reshaped = kernel.reshape(self.input_channels, -1)
# calculating input gradient for next convolutional layer
input_grad = (windows_out @ kernel_reshaped.T).reshape(
self.X_batch_feedforward.shape[input_index],
upsampled_height,
upsampled_width,
kernel.shape[kernel_input_channels_index],
)
input_grad = input_grad.transpose(0, 3, 1, 2)
# Update the weights in the kernel
self.kernel -= gradient_kernel
# Output the gradient to propagate backwards
return input_grad
def _extract_windows(self, X_batch, batch_type="image"):
"""
Receives as input the X_batch with shape (inputs, feature_maps, image_height, image_width)
and extract windows of size kernel_height * kernel_width for every image and every feature_map.
It then returns an np.ndarray of shape (image_height * image_width, inputs, feature_maps, kernel_height, kernel_width)
which will be used either to filter the images in feedforward or to calculate the gradient.
"""
# initialize list of windows
windows = []
if batch_type == "image":
# pad the images
X_batch_padded = self._padding(X_batch, batch_type="image")
img_height, img_width = X_batch_padded.shape[2:]
# For each location in the image...
for h in range(
0,
X_batch.shape[height_index],
self.v_stride,
):
for w in range(
0,
X_batch.shape[width_index],
self.h_stride,
):
# ...obtain an image patch of the original size (strided)
# get window
window = X_batch_padded[
:,
:,
h : h + self.kernel_height,
w : w + self.kernel_width,
]
# append to list of windows
windows.append(window)
# return numpy array instead of list
return np.stack(windows)
# In order to be able to perform backprogagation by the method of window extraction,
# here is a modified approach to extracting the windows which allow for the necessary
# upsampling of the gradient in case the on of the stride parameters is larger than one.
if batch_type == "grad":
# In the case of one of the stride parameters being odd, we have to take some
# extra care in calculating the upsampled size of X_batch. We solve this
# by simply flooring the result of dividing stride by 2.
if self.v_stride < 2 or self.v_stride % 2 == 0:
v_stride = 0
else:
v_stride = int(np.floor(self.v_stride / 2))
if self.h_stride < 2 or self.h_stride % 2 == 0:
h_stride = 0
else:
h_stride = int(np.floor(self.h_stride / 2))
upsampled_height = (X_batch.shape[height_index] * self.v_stride) - v_stride
upsampled_width = (X_batch.shape[width_index] * self.h_stride) - h_stride
# When upsampling, we need to insert rows and columns filled with zeros
# into each feature map. How many of those we have to insert is purely
# dependant on the value of stride parameter in the vertical and horizontal
# direction.
if self.v_stride > 1:
v_ind = 1
for i in range(X_batch.shape[height_index]):
for j in range(self.v_stride - 1):
X_batch = np.insert(X_batch, v_ind, 0, axis=height_index)
v_ind += self.v_stride
if self.h_stride > 1:
h_ind = 1
for i in range(X_batch.shape[width_index]):
for k in range(self.h_stride - 1):
X_batch = np.insert(X_batch, h_ind, 0, axis=width_index)
h_ind += self.h_stride
# Since the insertion of zero-filled rows and columns isn't perfect, we have
# to assure that the resulting feature maps will have the expected upsampled height
# and width by cutting them og at desired dimensions.
X_batch = X_batch[:, :, :upsampled_height, :upsampled_width]
X_batch_padded = self._padding(X_batch, batch_type="grad")
# initialize list of windows
windows = []
# For each location in the image...
for h in range(
0,
X_batch.shape[height_index],
self.v_stride,
):
for w in range(
0,
X_batch.shape[width_index],
self.h_stride,
):
# ...obtain an image patch of the original size (strided)
# get window
window = X_batch_padded[
:, :, h : h + self.kernel_height, w : w + self.kernel_width
]
# append window to list
windows.append(window)
# return numpy array, unsampled dimensions
return np.stack(windows), upsampled_height, upsampled_width
def _check_for_errors(self):
# compares input channels of data to input channels of Convolution2DLayer
if self.X_batch_feedforward.shape[input_channel_index] != self.input_channels:
raise AssertionError(
f"ERROR: Number of input channels in data ({self.X_batch_feedforward.shape[input_channel_index]}) is not equal to input channels in Convolution2DLayerOPT ({self.input_channels})! Please change the number of input channels of the Convolution2DLayer such that they are equal"
)
Finally, we present the code for the CNN. The CNN class organizes all the layers, and allows for training on image data.
import math
import autograd.numpy as np
import sys
import warnings
from autograd import grad, elementwise_grad
from random import random, seed
from copy import deepcopy
from typing import Tuple, Callable
from sklearn.utils import resample
warnings.simplefilter("error")
class CNN:
def __init__(
self,
cost_func: Callable = CostCrossEntropy,
scheduler: Scheduler = Adam(eta=1e-4, rho=0.9, rho2=0.999),
seed: int = None,
):
"""
Description:
------------
Instantiates CNN object
Parameters:
------------
I output_func (costFunctions) cost function for feed forward neural network part of CNN,
such as "CostLogReg", "CostOLS" or "CostCrossEntropy"
II scheduler (Scheduler) optional parameter, default set to Adam. Can also be set to other
schedulers such as AdaGrad, Momentum, RMS_prop and Constant. Note that schedulers have
to be instantiated first with proper parameters (for example eta, rho and rho2 for Adam)
III seed (int) used for seeding all random operations
"""
self.layers = list()
self.cost_func = cost_func
self.scheduler = scheduler
self.schedulers_weight = list()
self.schedulers_bias = list()
self.seed = seed
self.pred_format = None
def add_FullyConnectedLayer(
self, nodes: int, act_func=LRELU, scheduler=None
) -> None:
"""
Description:
------------
Add a FullyConnectedLayer to the CNN, i.e. a hidden layer in the feed forward neural
network part of the CNN. Often called a Dense layer in literature
Parameters:
------------
I nodes (int) number of nodes in FullyConnectedLayer
II act_func (activationFunctions) activation function of FullyConnectedLayer,
such as "sigmoid", "RELU", "LRELU", "softmax" or "identity"
III scheduler (Scheduler) optional parameter, default set to Adam. Can also be set to other
schedulers such as AdaGrad, Momentum, RMS_prop and Constant
"""
assert self.layers, "FullyConnectedLayer should follow FlattenLayer in CNN"
if scheduler is None:
scheduler = self.scheduler
layer = FullyConnectedLayer(nodes, act_func, scheduler, self.seed)
self.layers.append(layer)
def add_OutputLayer(self, nodes: int, output_func=sigmoid, scheduler=None) -> None:
"""
Description:
------------
Add an OutputLayer to the CNN, i.e. a the final layer in the feed forward neural
network part of the CNN
Parameters:
------------
I nodes (int) number of nodes in OutputLayer. Set nodes=1 for binary classification and
nodes = number of classes for multi-class classification
II output_func (activationFunctions) activation function for the output layer, such as
"identity" for regression, "sigmoid" for binary classification and "softmax" for multi-class
classification
III scheduler (Scheduler) optional parameter, default set to Adam. Can also be set to other
schedulers such as AdaGrad, Momentum, RMS_prop and Constant
"""
assert self.layers, "OutputLayer should follow FullyConnectedLayer in CNN"
if scheduler is None:
scheduler = self.scheduler
output_layer = OutputLayer(
nodes, output_func, self.cost_func, scheduler, self.seed
)
self.layers.append(output_layer)
self.pred_format = output_layer.get_pred_format()
def add_FlattenLayer(self, act_func=LRELU) -> None:
"""
Description:
------------
Add a FlattenLayer to the CNN, which flattens the image data such that it is formatted to
be used in the feed forward neural network part of the CNN
"""
self.layers.append(FlattenLayer(act_func=act_func, seed=self.seed))
def add_Convolution2DLayer(
self,
input_channels=1,
feature_maps=1,
kernel_height=3,
kernel_width=3,
v_stride=1,
h_stride=1,
pad="same",
act_func=LRELU,
optimized=True,
) -> None:
"""
Description:
------------
Add a Convolution2DLayer to the CNN, i.e. a convolutional layer with a 2 dimensional kernel. Should be
the first layer added to the CNN
Parameters:
------------
I input_channels (int) specifies amount of input channels. For monochrome images, use input_channels
= 1, and input_channels = 3 for colored images, where each channel represents one of R, G and B
II feature_maps (int) amount of feature maps in CNN
III kernel_height (int) height of the kernel, also called 'convolutional filter' in literature
IV kernel_width (int) width of the kernel, also called 'convolutional filter' in literature
V v_stride (int) value of vertical stride for dimentionality reduction
VI h_stride (int) value of horizontal stride for dimentionality reduction
VII pad (str) default = "same" ensures output size is the same as input size (given stride=1)
VIII act_func (activationFunctions) default = "LRELU", nonlinear activation function
IX optimized (bool) default = True, uses Convolution2DLayerOPT if True which is much faster when
compared to Convolution2DLayer, which is a more straightforward, understandable implementation
"""
if optimized:
conv_layer = Convolution2DLayerOPT(
input_channels,
feature_maps,
kernel_height,
kernel_width,
v_stride,
h_stride,
pad,
act_func,
self.seed,
reset_weights_independently=False,
)
else:
conv_layer = Convolution2DLayer(
input_channels,
feature_maps,
kernel_height,
kernel_width,
v_stride,
h_stride,
pad,
act_func,
self.seed,
reset_weights_independently=False,
)
self.layers.append(conv_layer)
def add_PoolingLayer(
self, kernel_height=2, kernel_width=2, v_stride=1, h_stride=1, pooling="max"
) -> None:
"""
Description:
------------
Add a Pooling2DLayer to the CNN, i.e. a pooling layer that reduces the dimentionality of
the image data. It is not necessary to use a Pooling2DLayer when creating a CNN, but it
can be used to speed up the training
Parameters:
------------
I kernel_height (int) height of the kernel used for pooling
II kernel_width (int) width of the kernel used for pooling
III v_stride (int) value of vertical stride for dimentionality reduction
IV h_stride (int) value of horizontal stride for dimentionality reduction
V pooling (str) either "max" or "average", describes type of pooling performed
"""
pooling_layer = Pooling2DLayer(
kernel_height, kernel_width, v_stride, h_stride, pooling, self.seed
)
self.layers.append(pooling_layer)
def fit(
self,
X: np.ndarray,
t: np.ndarray,
epochs: int = 100,
lam: float = 0,
batches: int = 1,
X_val: np.ndarray = None,
t_val: np.ndarray = None,
) -> dict:
"""
Description:
------------
Fits the CNN to input X for a given amount of epochs. Performs feedforward and backpropagation passes,
can utilize batches, regulariziation and validation if desired.
Parameters:
------------
X (numpy array) with input data in format [images, input channels,
image height, image_width]
t (numpy array) target labels for input data
epochs (int) amount of epochs
lam (float) regulariziation term lambda
batches (int) amount of batches input data splits into
X_val (numpy array) validation data
t_val (numpy array) target labels for validation data
Returns:
------------
scores (dict) a dictionary with "train_error", "train_acc", "val_error", val_acc" keys
that contain numpy arrays with float values of all accuracies/errors over all epochs.
Can be used to create plots. Also used to update the progress bar during training
"""
# setup
if self.seed is not None:
np.random.seed(self.seed)
# initialize weights
self._initialize_weights(X)
# create arrays for score metrics
scores = self._initialize_scores(epochs)
assert batches <= t.shape[0]
batch_size = X.shape[0] // batches
try:
for epoch in range(epochs):
for batch in range(batches):
# minibatch gradient descent
# If the for loop has reached the last batch, take all thats left
if batch == batches - 1:
X_batch = X[batch * batch_size :, :, :, :]
t_batch = t[batch * batch_size :, :]
else:
X_batch = X[
batch * batch_size : (batch + 1) * batch_size, :, :, :
]
t_batch = t[batch * batch_size : (batch + 1) * batch_size, :]
self._feedforward(X_batch)
self._backpropagate(t_batch, lam)
# reset schedulers for each epoch (some schedulers pass in this call)
for layer in self.layers:
if isinstance(layer, FullyConnectedLayer):
layer._reset_scheduler()
# computing performance metrics
scores = self._compute_scores(scores, epoch, X, t, X_val, t_val)
# printing progress bar
print_length = self._progress_bar(
epoch,
epochs,
scores,
)
# allows for stopping training at any point and seeing the result
except KeyboardInterrupt:
pass
# visualization of training progression (similiar to tensorflow progression bar)
sys.stdout.write("\r" + " " * print_length)
sys.stdout.flush()
self._progress_bar(
epochs,
epochs,
scores,
)
sys.stdout.write("")
return scores
def _feedforward(self, X_batch) -> np.ndarray:
"""
Description:
------------
Performs the feedforward pass for all layers in the CNN. Called from fit()
"""
a = X_batch
for layer in self.layers:
a = layer._feedforward(a)
return a
def _backpropagate(self, t_batch, lam) -> None:
"""
Description:
------------
Performs backpropagation for all layers in the CNN. Called from fit()
"""
assert len(self.layers) >= 2
reversed_layers = self.layers[::-1]
# for every layer, backwards
for i in range(len(reversed_layers) - 1):
layer = reversed_layers[i]
prev_layer = reversed_layers[i + 1]
# OutputLayer
if isinstance(layer, OutputLayer):
prev_a = prev_layer.get_prev_a()
weights_next, delta_next = layer._backpropagate(t_batch, prev_a, lam)
# FullyConnectedLayer
elif isinstance(layer, FullyConnectedLayer):
assert (
delta_next is not None
), "No OutputLayer to follow FullyConnectedLayer"
assert (
weights_next is not None
), "No OutputLayer to follow FullyConnectedLayer"
prev_a = prev_layer.get_prev_a()
weights_next, delta_next = layer._backpropagate(
weights_next, delta_next, prev_a, lam
)
# FlattenLayer
elif isinstance(layer, FlattenLayer):
assert (
delta_next is not None
), "No FullyConnectedLayer to follow FlattenLayer"
assert (
weights_next is not None
), "No FullyConnectedLayer to follow FlattenLayer"
delta_next = layer._backpropagate(weights_next, delta_next)
# Convolution2DLayer and Convolution2DLayerOPT
elif isinstance(layer, Convolution2DLayer):
assert (
delta_next is not None
), "No FlattenLayer to follow Convolution2DLayer"
delta_next = layer._backpropagate(delta_next)
# Pooling2DLayer
elif isinstance(layer, Pooling2DLayer):
assert delta_next is not None, "No Layer to follow Pooling2DLayer"
delta_next = layer._backpropagate(delta_next)
# Catch error
else:
raise NotImplementedError
def _compute_scores(
self,
scores: dict,
epoch: int,
X: np.ndarray,
t: np.ndarray,
X_val: np.ndarray,
t_val: np.ndarray,
) -> dict:
"""
Description:
------------
Computes scores such as training error, training accuracy, validation error
and validation accuracy for the CNN depending on if a validation set is used
and if the CNN performs classification or regression
Returns:
------------
scores (dict) a dictionary with "train_error", "train_acc", "val_error", val_acc" keys
that contain numpy arrays with float values of all accuracies/errors over all epochs.
Can be used to create plots. Also used to update the progress bar during training
"""
pred_train = self.predict(X)
cost_function_train = self.cost_func(t)
train_error = cost_function_train(pred_train)
scores["train_error"][epoch] = train_error
if X_val is not None and t_val is not None:
cost_function_val = self.cost_func(t_val)
pred_val = self.predict(X_val)
val_error = cost_function_val(pred_val)
scores["val_error"][epoch] = val_error
if self.pred_format != "Regression":
train_acc = self._accuracy(pred_train, t)
scores["train_acc"][epoch] = train_acc
if X_val is not None and t_val is not None:
val_acc = self._accuracy(pred_val, t_val)
scores["val_acc"][epoch] = val_acc
return scores
def _initialize_scores(self, epochs) -> dict:
"""
Description:
------------
Initializes scores such as training error, training accuracy, validation error
and validation accuracy for the CNN
Returns:
------------
A dictionary with "train_error", "train_acc", "val_error", val_acc" keys that
will contain numpy arrays with float values of all accuracies/errors over all epochs
when passed through the _compute_scores() function during fit()
"""
scores = dict()
train_errors = np.empty(epochs)
train_errors.fill(np.nan)
val_errors = np.empty(epochs)
val_errors.fill(np.nan)
train_accs = np.empty(epochs)
train_accs.fill(np.nan)
val_accs = np.empty(epochs)
val_accs.fill(np.nan)
scores["train_error"] = train_errors
scores["val_error"] = val_errors
scores["train_acc"] = train_accs
scores["val_acc"] = val_accs
return scores
def _initialize_weights(self, X: np.ndarray) -> None:
"""
Description:
------------
Initializes weights for all layers in CNN
Parameters:
------------
I X (np.ndarray) input of format [img, feature_maps, height, width]
"""
prev_nodes = X
for layer in self.layers:
prev_nodes = layer._reset_weights(prev_nodes)
def predict(self, X: np.ndarray, *, threshold=0.5) -> np.ndarray:
"""
Description:
------------
Predicts output of input X
Parameters:
------------
I X (np.ndarray) input [img, feature_maps, height, width]
"""
prediction = self._feedforward(X)
if self.pred_format == "Binary":
return np.where(prediction > threshold, 1, 0)
elif self.pred_format == "Multi-class":
class_prediction = np.zeros(prediction.shape)
for i in range(prediction.shape[0]):
class_prediction[i, np.argmax(prediction[i, :])] = 1
return class_prediction
else:
return prediction
def _accuracy(self, prediction: np.ndarray, target: np.ndarray) -> float:
"""
Description:
------------
Calculates accuracy of given prediction to target
Parameters:
------------
I prediction (np.ndarray): output of predict() fuction
(1s and 0s in case of classification, and real numbers in case of regression)
II target (np.ndarray): vector of true values (What the network should predict)
Returns:
------------
A floating point number representing the percentage of correctly classified instances.
"""
assert prediction.size == target.size
return np.average((target == prediction))
def _progress_bar(self, epoch: int, epochs: int, scores: dict) -> int:
"""
Description:
------------
Displays progress of training
"""
progression = epoch / epochs
epoch -= 1
print_length = 40
num_equals = int(progression * print_length)
num_not = print_length - num_equals
arrow = ">" if num_equals > 0 else ""
bar = "[" + "=" * (num_equals - 1) + arrow + "-" * num_not + "]"
perc_print = self._fmt(progression * 100, N=5)
line = f" {bar} {perc_print}% "
for key, score in scores.items():
if np.isnan(score[epoch]) == False:
value = self._fmt(score[epoch], N=4)
line += f"| {key}: {value} "
print(line, end="\r")
return len(line)
def _fmt(self, value: int, N=4) -> str:
"""
Description:
------------
Formats decimal numbers for progress bar
"""
if value > 0:
v = value
elif value < 0:
v = -10 * value
else:
v = 1
n = 1 + math.floor(math.log10(v))
if n >= N - 1:
return str(round(value))
# or overflow
return f"{value:.{N-n-1}f}"
Using the CNN codebase is very simple. We begin by initiating a CNN object, which takes a cost function, a scheduler and a seed as its arguments. If a scheduler is not provided, it will per default initiate an Adam scheduler with eta=1e-4, and if a seed is not provided, the CNN will not be seeded, meaning it will run with a different random seed every run. Below we demonstrate an initiation of our CNN.
adam_scheduler = Adam(eta=1e-3, rho=0.9, rho2=0.999)
cnn = CNN(cost_func=CostCrossEntropy, scheduler=adam_scheduler, seed=2023)
Now that we have our CNN object, we can begin to add layers to it! Many of the add_layer functions have default values, for example add_Convolution2DLayer() has a default v_stride and h_stride of 1. However, these can of course be set to any value you please. Note that the input channels of a subsequent convolutional layer must equal the previous convolutional layer's feature maps.
cnn.add_Convolution2DLayer(
input_channels=1,
feature_maps=1,
kernel_height=3,
kernel_width=3,
act_func=LRELU,
)
cnn.add_FlattenLayer()
cnn.add_FullyConnectedLayer(30, LRELU)
cnn.add_FullyConnectedLayer(20, LRELU)
cnn.add_OutputLayer(10, softmax)
Here we have created a CNN with the following architecture:
Now, before we can train the model, we need to load in our data. We will use the MNIST dataset and use 10000 \( 28 \times 28 images \).
from sklearn.datasets import fetch_openml
from sklearn.model_selection import train_test_split
def onehot(target: np.ndarray):
onehot = np.zeros((target.size, target.max() + 1))
onehot[np.arange(target.size), target] = 1
return onehot
# get dataset
dataset = fetch_openml("mnist_784", parser="auto")
mnist = dataset.data.to_numpy(dtype="float")[:10000, :]
# scale data
for i in range(mnist.shape[1]):
mnist[:, i] /= 255
# reshape to add single input channel to data shape [inputs, input_channels, height, width]
mnist = mnist.reshape(mnist.shape[0], 1, 28, 28)
# one hot encode target as we are doing multi-class classification
target = onehot(np.array([int(i) for i in dataset.target.to_numpy()[:10000]]))
# split into training and validation data
x_train, x_val, y_train, y_val = train_test_split(mnist, target)
Now we may train our model. Note that we can utilize regularization in the CNN by using the lam (lambda) parameter in fit(), and utilize different types of gradient descent by specifying the amount of batches via the batches parameter as shown below.
The functionfit() returns a score dictionary of the training error and accuracy (and validation error and accuracy if a validation set is provided) which can be used to plot the error and accuracy of the model over epochs.
scores = cnn.fit(
x_train,
y_train,
lam=1e-5,
batches=10,
epochs=100,
X_val=x_val,
t_val=y_val,
)
plt.plot(scores["train_acc"], label="Training")
plt.plot(scores["val_acc"], label="Validation")
plt.ylim([0.8,1])
plt.xlabel("Epochs")
plt.ylabel("Accuracy")
plt.legend()
plt.show()
Considering we only trained the model for 100 epochs without any tuning of the hyperparameters, this result is pretty good.
The codebase allows for great flexibility in CNN architectures. Pooling layers can be added before, inbetween or after convolutional layers, but due to the great optimizations made within Convolution2DLayerOPT, we recommend using the v_stride and h_stride parameters in add_Convolution2DLayer() to reduce the dimentionality of the problem as the pooling layer is slow in comparison. To use the unoptimized version of Convolution2DLayer, simply pass optimized=False as an argument in add_Convolution2DLayer().
If one wishes to perform binary classification using the CNN, simply use the cost function 'CostLogReg' when initializing the CNN and use 1 node at the OutputLayer.
Below we have created another, more untraditional architecture using our code to demonstrate its flexibility and different attributes such as asymmetric stride that might become useful when constructing your own CNN.
adam_scheduler = Adam(eta=1e-3, rho=0.9, rho2=0.999)
cnn = CNN(cost_func=CostCrossEntropy, scheduler=adam_scheduler, seed=2023)
cnn.add_Convolution2DLayer(
input_channels=1,
feature_maps=7,
kernel_height=7,
kernel_width=1,
act_func=LRELU,
)
cnn.add_PoolingLayer(
kernel_height=2,
kernel_width=2,
pooling="average",
)
cnn.add_PoolingLayer(
kernel_height=2,
kernel_width=2,
pooling="max",
)
cnn.add_Convolution2DLayer(
input_channels=7,
feature_maps=1,
kernel_height=4,
kernel_width=4,
v_stride=2,
h_stride=3,
act_func=LRELU,
optimized=False,
)
cnn.add_Convolution2DLayer(
input_channels=1,
feature_maps=1,
kernel_height=2,
kernel_width=2,
act_func=sigmoid,
optimized=True,
)
cnn.add_PoolingLayer(
kernel_height=2,
kernel_width=2,
pooling="max"
)
cnn.add_FlattenLayer()
cnn.add_FullyConnectedLayer(100, LRELU)
cnn.add_FullyConnectedLayer(10, sigmoid)
cnn.add_FullyConnectedLayer(101, identity)
cnn.add_OutputLayer(10, softmax)
Here we see the use of asymmetrical 1D kernels such as the \( 7 \times 1 \) kernel in the first convolutional layer, both max and average pooling, asymmetric stride in the unoptimized convolutional layer, more pooling, a flatten layer, a hidden layer with 100 nodes using LRELU, another hidden layer with 10 hidden nodes that uses the sigmoid activation function, and another hidden layer with 101 nodes which utilizes no activation function (identity). Finally, we arrive at the output layer with 10 nodes, which uses softmax as its activation function.
The stride parameter controls the distance between each convolution and the kernel/filter. If our image is padded, stride is the only parameter that determines the size of the output from a convolutional layer. However, if we decide not to perform any padding, the size of the output feature map depends on both the stride and kernel size. It is important to note that neither the stride nor the kernel has to be symmetrical. This means that we can use a rectangular filter if we choose, and the stride in the vertical direction (axis=0 in Python) does not need to be the same as the stride in the horizontal direction (axis=1 in Python). It may even be the case that asymmetric combinations of stride or kernel dimensions, or both, yield better results than symmetric values for these parameters.
def convolve(image, kernel, stride=1):
for i in range(2):
kernel = np.rot90(kernel)
k_half_height = kernel.shape[0] // 2
k_half_width = kernel.shape[0] // 2
conv_image = np.zeros(image.shape)
pad_image = padding(image, kernel)
for i in range(k_half_height, conv_image.shape[0] + k_half_height, stride):
for j in range(k_half_width, conv_image.shape[1] + k_half_width, stride):
conv_image[i - k_half_height, j - k_half_width] = np.sum(
pad_image[
i - k_half_height : i + k_half_height + 1, j - k_half_width : j + k_half_width + 1
]
* kernel
)
return conv_image
Despite the naive convolution algorithm shown above working finely, it is extremely slow, requiring approximately 20-30 seconds to process a single image. The time complexity of 2D convolution, which is O(NMnm), rapidly becomes a constraint and may, at worst, make computations infeasible. Consequently, optimizing the naive 2D convolution algorithm is a necessity, as the execution time of the algorithm significantly increases as the input data size expands. This can pose a bottleneck in applications that necessitate real-time processing of large data volumes, such as image and video processing, deep learning, and scientific simulations.
To address this issue, we shall present two widely used optimization techniques: the separable kernel approach and Fast Fourier Transform (FFT). Both of these methods can drastically reduce the computational complexity of convolution and enhance the overall efficiency of processing substantial data quantities. While we shall refrain from delving into the intricacies of these algorithms, we strongly encourage you to examine at least the application of FFT to optimize computations.
def conv2DSep(image, kernel, coef, stride=1, pad="zero"):
for i in range(2):
kernel = np.rot90(kernel)
# The kernel is quadratic, thus we only need one of its dimensions
half_dim = kernel.shape[0] // 2
ker1 = np.array(kernel[0, :])
ker2 = np.array(kernel[:, 0])
if pad == "zero":
conv_image = np.zeros(image.shape)
pad_image = padding(image, kernel)
else:
conv_image = np.zeros(
(image.shape[0] - kernel.shape[0], image.shape[1] - kernel.shape[1])
)
pad_image = image[:, :]
for i in range(half_dim, conv_image.shape[0] + half_dim, stride):
for j in range(half_dim, conv_image.shape[1] + half_dim, stride):
conv_image[i - half_dim, j - half_dim] = (
pad_image[
i - half_dim : i + half_dim + 1, j - half_dim : j + half_dim + 1
]
@ ker1
@ ker2.T
* coef
)
return conv_image
img_path = img_path = "data/IMG-2167.JPG"
image_of_cute_dog = imageio.imread(img_path, mode="L")
start_time = time.time()
filtered_image = conv2DSep(image_of_cute_dog, kernel=sobel_kernel, coef=1)
print(f'Time taken for convolution with seperated kernel on 128x128 image {time.time() - start_time}')
plt.imshow(filtered_image, cmap="gray", vmin=0, vmax=255, aspect="auto")
plt.show()
By taking advantage of the capabilities of separable kernels, we can effectively cut the computational expense of filtering an image in half. Yet, if we seek even more rapid processing, we can turn to the Fast Fourier Transform (FFT) algorithm provided by the numpy library. By utilizing FFT to transform the input image and filter into the frequency domain, we can perform convolution in this domain. This approach significantly reduces the number of operations needed and results in a marked speedup relative to other convolution techniques. In addition, it is worth noting that the FFT is widely regarded as one of the most critical algorithms developed to date, with applications ranging from digital signal processing to scientific computing.
start_time = time.time()
img_fft = np.fft.fft2(image_of_cute_dog)
kernel_fft = np.fft.fft2(sobel_kernel, s=image_of_cute_dog.shape)
conv_image = img_fft * kernel_fft
filtered_image = np.fft.ifft2(conv_image)
print(f'Time take for convolution in the fourier domain: {time.time() - start_time}')
plt.imshow(filtered_image.real, cmap="gray", vmin=0, vmax=255, aspect="auto")
plt.show()
It is evident that executing convolution in the Fourier domain yields the quickest computation time. Nonetheless, one should exercise caution, particularly when dealing with images of relatively small dimensions, as one of the other methods may prove to be more expeditious than FFT-enhanced convolution. The overhead involved in transferring both the image and filter into the Fourier domain, followed by their subsequent transformation back into the spatial domain, results in a minor inconvenience. Therefore, it is imperative to remain cognizant of this fact when utilizing FFT as the primary optimization technique.