Dive into Deep Learning
Table Of Contents
Dive into Deep Learning
Table Of Contents

Convolutions for Images

Now that we understand how to design convolutional networks in theory, let’s see how this works in practice. For the next chapters we will stick to dealing with images, since they constitute one of the most common use cases for convolutional networks. That is, we will discuss the two-dimensional case (image height and width). We begin with the cross-correlation operator that we introduced in the previous section. Strictly speaking, convolutional networks are a slight misnomer (but for notation only), since the operations are typically expressed as cross correlations.

The Cross-Correlation Operator

In a convolutional layer, an input array and a correlation kernel array output an array through a cross-correlation operation. Let’s see how this works for two dimensions. As shown below, the input is a two-dimensional array with a height of 3 and width of 3. We mark the shape of the array as \(3 \times 3\) or (3, 3). The height and width of the kernel array are both 2. This array is also called a kernel or filter in convolution computations. The shape of the kernel window (also known as the convolution window) depends on the height and width of the kernel, which is \(2 \times 2\).

Two-dimensional.cross-correlation.operation..The.shaded.portions.are.the.first.output.element.and.the.input.and.kernel.array.elements.used.in.its.computation:.\ :math:`0\times0+1\times1+3\times2+4\times3=19`..

Two-dimensional.cross-correlation.operation..The.shaded.portions.are.the.first.output.element.and.the.input.and.kernel.array.elements.used.in.its.computation:.\(0\times0+1\times1+3\times2+4\times3=19\)..

In the two-dimensional cross-correlation operation, the convolution window starts from the top-left of the input array, and slides in the input array from left to right and top to bottom. When the convolution window slides to a certain position, the input subarray in the window and kernel array are multiplied and summed by element to get the element at the corresponding location in the output array. The output array has a height of 2 and width of 2, and the four elements are derived from a two-dimensional cross-correlation operation:

\[\begin{split}0\times0+1\times1+3\times2+4\times3=19,\\ 1\times0+2\times1+4\times2+5\times3=25,\\ 3\times0+4\times1+6\times2+7\times3=37,\\ 4\times0+5\times1+7\times2+8\times3=43.\end{split}\]

Note that the output size is smaller than the input. In particular, the output size is given by the input size \(H \times W\) minus the size of the convolutional kernel \(h \times w\) via \((H-h+1) \times (W-w+1)\). This is the case since we need enough space to ‘shift’ the convolutional kernel across the image (later we will see how to keep the size unchanged by padding the image with zeros around its boundary such that there’s enough space to shift the kernel). Next, we implement the above process in the corr2d function. It accepts the input array X with the kernel array K and outputs the array Y.

In [1]:
from mxnet import autograd, nd
from mxnet.gluon import nn

def corr2d(X, K):  # This function has been saved in the gluonbook package for future use.
    h, w = K.shape
    Y = nd.zeros((X.shape[0] - h + 1, X.shape[1] - w + 1))
    for i in range(Y.shape[0]):
        for j in range(Y.shape[1]):
            Y[i, j] = (X[i: i + h, j: j + w] * K).sum()
    return Y

We can construct the input array X and the kernel array K of the figure above to validate the output of the two-dimensional cross-correlation operation.

In [2]:
X = nd.array([[0, 1, 2], [3, 4, 5], [6, 7, 8]])
K = nd.array([[0, 1], [2, 3]])
corr2d(X, K)
Out[2]:

[[19. 25.]
 [37. 43.]]
<NDArray 2x2 @cpu(0)>

Convolutional Layers

The convolutional layer cross-correlates the input and kernels and adds a scalar bias to get the output. The model parameters of the convolutional layer include the kernel and scalar bias. When training the model, we usually randomly initialize the kernel and then continuously iterate the kernel and bias.

Next, we implement a custom two-dimensional convolutional layer based on the corr2d function. In the __init__ constructor function, we declare weight and bias as the two model parameters. The forward computation function forward directly calls the corr2d function and adds the bias. Just like the \(h \times w\) cross-correlation we also refer to convolutional layers as \(h \times w\) dimensional convolutions.

In [3]:
class Conv2D(nn.Block):
    def __init__(self, kernel_size, **kwargs):
        super(Conv2D, self).__init__(**kwargs)
        self.weight = self.params.get('weight', shape=kernel_size)
        self.bias = self.params.get('bias', shape=(1,))

    def forward(self, x):
        return corr2d(x, self.weight.data()) + self.bias.data()

Object Edge Detection in Images

Let’s look at a simple application of a convolutional layer: detecting the edge of an object in an image by finding the location of the pixel change. First, we construct an ‘image’ of \(6\times 8\) pixels. The middle four columns are black (0), and the rest are white (1).

In [4]:
X = nd.ones((6, 8))
X[:, 2:6] = 0
X
Out[4]:

[[1. 1. 0. 0. 0. 0. 1. 1.]
 [1. 1. 0. 0. 0. 0. 1. 1.]
 [1. 1. 0. 0. 0. 0. 1. 1.]
 [1. 1. 0. 0. 0. 0. 1. 1.]
 [1. 1. 0. 0. 0. 0. 1. 1.]
 [1. 1. 0. 0. 0. 0. 1. 1.]]
<NDArray 6x8 @cpu(0)>

Next we construct a kernel K with a height and width of 1 and 2. When this performs the cross-correlation operation with the input, if the horizontally adjacent elements are the same, the output is 0. Otherwise, the output is non-zero.

In [5]:
K = nd.array([[1, -1]])

Enter X and our designed kernel K to perform the cross-correlation operations. As you can see, we will detect 1 for the edge from white to black and -1 for the edge from black to white. The rest of the outputs are 0.

In [6]:
Y = corr2d(X, K)
Y
Out[6]:

[[ 0.  1.  0.  0.  0. -1.  0.]
 [ 0.  1.  0.  0.  0. -1.  0.]
 [ 0.  1.  0.  0.  0. -1.  0.]
 [ 0.  1.  0.  0.  0. -1.  0.]
 [ 0.  1.  0.  0.  0. -1.  0.]
 [ 0.  1.  0.  0.  0. -1.  0.]]
<NDArray 6x7 @cpu(0)>

Let’s apply the kernel to the transposed ‘image’. As expected, it vanishes. The kernel K only detects vertical edges.

In [7]:
corr2d(X.T, K)
Out[7]:

[[0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0.]]
<NDArray 8x5 @cpu(0)>

Learning a Kernel

Designing an edge detector by finite differences [1, -1] is neat if we know what we are looking for. However, as we look at larger kernels, or possibly multiple layers, it is next to impossible to specify such filters manually. Let’s see whether we can learn the kernel that generated Y from X by looking at the (input, output) pairs only. We first construct a convolutional layer and initialize its kernel into a random array. Next, in each iteration, we use the squared error to compare Y and the output of the convolutional layer, then calculate the gradient to update the weight. For the sake of simplicity, the convolutional layer here ignores the bias.

We previously constructed the Conv2D class. However, since we used single-element assignments, Gluon has some trouble finding the gradient. Instead, we use the built-in Conv2D class provided by Gluon below.

In [8]:
# Construct a convolutional layer with 1 output channel (channels will be introduced in the following section) and a kernel array shape of (1, 2).
conv2d = nn.Conv2D(1, kernel_size=(1, 2))
conv2d.initialize()

# The two-dimensional convolutional layer uses four-dimensional input and output in the format of (example channel, height, width), where the batch size (number of examples in the batch) and
# the number of channels are both 1.
X = X.reshape((1, 1, 6, 8))
Y = Y.reshape((1, 1, 6, 7))

for i in range(10):
    with autograd.record():
        Y_hat = conv2d(X)
        l = (Y_hat - Y) ** 2
    l.backward()
    # For the sake of simplicity, we ignore the bias here.
    conv2d.weight.data()[:] -= 3e-2 * conv2d.weight.grad()
    if (i + 1) % 2 == 0:
        print('batch %d, loss %.3f' % (i + 1, l.sum().asscalar()))
batch 2, loss 4.949
batch 4, loss 0.831
batch 6, loss 0.140
batch 8, loss 0.024
batch 10, loss 0.004

As you can see, the error has dropped to a relatively small value after 10 iterations. Now we will take a look at the kernel array we learned.

In [9]:
conv2d.weight.data().reshape((1, 2))
Out[9]:

[[ 0.9895    -0.9873705]]
<NDArray 1x2 @cpu(0)>

We find that the kernel array we learned is very close to the kernel array K we defined earlier.

Cross-correlation and Convolution

Recall the observation from the previous section that cross-correlation and convolution are equivalent. In the figure above it is easy to see this correspondence. Simply flip the kernel from the bottom left to the top right. In this case the indexing in the sum is reverted, yet the same result can be obtained. In keeping with standard terminology with deep learning literature we will continue to refer to the cross-correlation operation as a convolution even though it is stricly speaking something slightly different.

Summary

  • The core computation of a two-dimensional convolutional layer is a two-dimensional cross-correlation operation. In its simplest form, this performs a cross-correlation operation on the two-dimensional input data and the kernel, and then adds a bias.
  • We can design a kernel to detect edges in images.
  • We can learn the kernel through data.

Problems

  1. Construct an image X with diagonal edges.
    • What happens if you apply the kernel K to it?
    • What happens if you transpose X?
    • What happens if you transpose K?
  2. When you try to automatically find the gradient for the Conv2D class we created, what kind of error message do you see?
  3. How do you represent a cross-correlation operation as a matrix multiplication by changing the input and kernel arrays?
  4. Design some kernels manually.
    • What is the form of a kernel for the second derivative?
    • What is the kernel for the Laplace operator?
    • What is the kernel for an integral?
    • What is the minimum size of a kernel to obtain a derivative of degree \(d\)?

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