6.2  Convolutions for Images

Now that we understand how convolutional layers work in theory, we are ready to see how they work in practice. Building on our motivation of convolutional neural networks as efficient architectures for exploring structure in image data, we stick with images as our running example.

PyTorch

from d2l import torch as d2l
import torch
from torch import nn
from torch.nn import functional as F

TensorFlow

from d2l import tensorflow as d2l
import tensorflow as tf

JAX

from d2l import jax as d2l
from flax import nnx
import jax
from jax import numpy as jnp

MXNet

from d2l import mxnet as d2l
from mxnet import autograd, gluon, np, npx
from mxnet.gluon import nn
npx.set_np()

6.2.1 The Cross-Correlation Operation

Recall that strictly speaking, convolutional layers are a misnomer, since the operations they express are more accurately described as cross-correlations. Based on our descriptions of convolutional layers in Section 6.1, in such a layer, an input tensor and a kernel tensor are combined to produce an output tensor through a cross-correlation operation.

Let’s ignore channels for now and see how this works with two-dimensional data and hidden representations. In Figure 6.2.1, the input is a two-dimensional tensor with a height of 3 and width of 3. We mark the shape of the tensor as \(3 \times 3\) or (\(3\), \(3\)). The height and width of the kernel are both 2. The shape of the kernel window (or convolution window) is given by the height and width of the kernel (here it is \(2 \times 2\)).

Figure 6.2.1: Two-dimensional cross-correlation operation. The shaded portions are the first output element as well as the input and kernel tensor elements used for the output computation: \(0\times0+1\times1+3\times2+4\times3=19\).

In the two-dimensional cross-correlation operation, we begin with the convolution window positioned at the upper-left corner of the input tensor and slide it across the input tensor, both from left to right and top to bottom. When the convolution window slides to a certain position, the input subtensor contained in that window and the kernel tensor are multiplied elementwise and the resulting tensor is summed up yielding a single scalar value. This result gives the value of the output tensor at the corresponding location. Here, the output tensor has a height of 2 and width of 2 and the four elements are derived from the two-dimensional cross-correlation operation:

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

Note that along each axis, the output size is slightly smaller than the input size. Because the kernel has width and height greater than \(1\), we can only properly compute the cross-correlation for locations where the kernel fits wholly within the image, the output size is given by the input size \(n_\textrm{h} \times n_\textrm{w}\) minus the size of the convolution kernel \(k_\textrm{h} \times k_\textrm{w}\) via

\[(n_\textrm{h}-k_\textrm{h}+1) \times (n_\textrm{w}-k_\textrm{w}+1). \tag{6.2.1}\]

This is the case since we need enough space to “shift” the convolution kernel across the image. Later we will see how to keep the size unchanged by padding the image with zeros around its boundary so that there is enough space to shift the kernel. Next, we implement this process in the corr2d function, which accepts an input tensor X and a kernel tensor K and returns an output tensor Y.

PyTorch

def corr2d(X, K):
    """Compute 2D cross-correlation."""
    h, w = K.shape
    Y = d2l.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] = d2l.reduce_sum((X[i: i + h, j: j + w] * K))
    return Y

TensorFlow

def corr2d(X, K):
    """Compute 2D cross-correlation."""
    h, w = K.shape
    Y = tf.Variable(tf.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].assign(tf.reduce_sum(
                X[i: i + h, j: j + w] * K))
    return Y

JAX

def corr2d(X, K):
    """Compute 2D cross-correlation."""
    h, w = K.shape
    Y = jnp.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 = Y.at[i, j].set((X[i:i + h, j:j + w] * K).sum())
    return Y

MXNet

def corr2d(X, K):
    """Compute 2D cross-correlation."""
    h, w = K.shape
    Y = d2l.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] = d2l.reduce_sum((X[i: i + h, j: j + w] * K))
    return Y

We can construct the input tensor X and the kernel tensor K from Figure 6.2.1 to validate the output of the above implementation of the two-dimensional cross-correlation operation.

X = d2l.tensor([[0.0, 1.0, 2.0], [3.0, 4.0, 5.0], [6.0, 7.0, 8.0]])
K = d2l.tensor([[0.0, 1.0], [2.0, 3.0]])
corr2d(X, K)

PyTorch

tensor([[19., 25.],
        [37., 43.]])

TensorFlow

<tf.Variable 'Variable:0' shape=(2, 2) dtype=float32, numpy=
array([[19., 25.],
       [37., 43.]], dtype=float32)>

JAX

Array([[19., 25.],
       [37., 43.]], dtype=float32)

MXNet

array([[19., 25.],
       [37., 43.]])

6.2.2 Convolutional Layers

A convolutional layer cross-correlates the input and kernel and adds a scalar bias to produce an output. The two parameters of a convolutional layer are the kernel and the scalar bias. When training models based on convolutional layers, we typically initialize the kernels randomly, just as we would with a fully connected layer.

We are now ready to implement a two-dimensional convolutional layer based on the corr2d function defined above. In the __init__ constructor method, we declare weight and bias as the two model parameters. The forward propagation method calls the corr2d function and adds the bias.

PyTorch

class Conv2D(nn.Module):
    def __init__(self, kernel_size):
        super().__init__()
        self.weight = nn.Parameter(torch.rand(kernel_size))
        self.bias = nn.Parameter(torch.zeros(1))

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

TensorFlow

class Conv2D(tf.keras.layers.Layer):
    def __init__(self):
        super().__init__()

    def build(self, kernel_size):
        initializer = tf.random_normal_initializer()
        self.weight = self.add_weight(name='w', shape=kernel_size,
                                      initializer=initializer)
        self.bias = self.add_weight(name='b', shape=(1, ),
                                    initializer=initializer)

    def call(self, inputs):
        return corr2d(inputs, self.weight) + self.bias

JAX

class Conv2D(nnx.Module):
    def __init__(self, kernel_size, rngs=None):
        rngs = nnx.Rngs(d2l.get_key()) if rngs is None else rngs
        self.weight = nnx.Param(
            nnx.initializers.uniform()(rngs.params(), kernel_size))
        self.bias = nnx.Param(jnp.zeros(1))

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

MXNet

class Conv2D(nn.Block):
    def __init__(self, kernel_size):
        super().__init__()
        self.weight = gluon.Parameter('weight', shape=kernel_size)
        self.bias = gluon.Parameter('bias', shape=(1,))

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

In \(h \times w\) convolution or an \(h \times w\) convolution kernel, the height and width of the convolution kernel are \(h\) and \(w\), respectively. We also refer to a convolutional layer with an \(h \times w\) convolution kernel simply as an \(h \times w\) convolutional layer.

6.2.3 Object Edge Detection in Images

Let’s take a moment to parse 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 (\(1\)).

PyTorch

X = d2l.ones((6, 8))
X[:, 2:6] = 0
X

tensor([[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.]])

TensorFlow

X = tf.Variable(tf.ones((6, 8)))
X[:, 2:6].assign(tf.zeros(X[:, 2:6].shape))
X

<tf.Variable 'Variable:0' shape=(6, 8) dtype=float32, numpy=
array([[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.]], dtype=float32)>

JAX

X = jnp.ones((6, 8))
X = X.at[:, 2:6].set(0)
X

Array([[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.]], dtype=float32)

MXNet

X = d2l.ones((6, 8))
X[:, 2:6] = 0
X

array([[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.]])

Next, we construct a kernel K with a height of 1 and a width of 2. When we perform the cross-correlation operation with the input, if the horizontally adjacent elements are the same, the output is 0. Otherwise, the output is nonzero. Note that this kernel is a special case of a finite difference operator. At location \((i,j)\) it computes \(x_{i,j} - x_{i,j+1}\), i.e., it computes the difference between the values of horizontally adjacent pixels. This is a discrete approximation of the first derivative in the horizontal direction (up to a sign). After all, for a function \(f(i,j)\) its derivative is \(\partial_j f(i,j) = \lim_{\epsilon \to 0} \frac{f(i,j+\epsilon) - f(i,j)}{\epsilon}\), so the kernel output \(x_{i,j} - x_{i,j+1}\) approximates \(-\partial_j f(i,j)\). Let’s see how this works in practice.

K = d2l.tensor([[1.0, -1.0]])

We are ready to perform the cross-correlation operation with arguments X (our input) and K (our kernel). As you can see, we detect \(1\) for the edge from white to black and \(-1\) for the edge from black to white. All other outputs take value \(0\).

Y = corr2d(X, K)
Y

PyTorch

tensor([[ 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.]])

TensorFlow

<tf.Variable 'Variable:0' shape=(6, 7) dtype=float32, numpy=
array([[ 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.]], dtype=float32)>

JAX

Array([[ 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.]], dtype=float32)

MXNet

array([[ 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.]])

We can now apply the kernel to the transposed image. As expected, it vanishes. The kernel K only detects vertical edges.

corr2d(d2l.transpose(X), K)

PyTorch

tensor([[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.]])

TensorFlow

<tf.Variable 'Variable:0' shape=(8, 5) dtype=float32, numpy=
array([[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.]], dtype=float32)>

JAX

Array([[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.]], dtype=float32)

MXNet

array([[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.]])

6.2.4 Learning a Kernel

Designing an edge detector by finite differences [1, -1] is neat if we know this is precisely what we are looking for. However, as we look at larger kernels, and consider successive layers of convolutions, it might be impossible to specify precisely what each filter should be doing manually.

Now 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 as a random tensor. Next, in each iteration, we will use the squared error to compare Y with the output of the convolutional layer. We can then calculate the gradient to update the kernel. For the sake of simplicity, in the following we use the built-in class for two-dimensional convolutional layers and ignore the bias.

PyTorch

# Construct a two-dimensional convolutional layer with 1 output channel and a
# kernel of shape (1, 2). For the sake of simplicity, we ignore the bias here
conv2d = nn.LazyConv2d(1, kernel_size=(1, 2), bias=False)

# 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))
lr = 3e-2  # Learning rate

for i in range(10):
    Y_hat = conv2d(X)
    l = (Y_hat - Y) ** 2
    conv2d.zero_grad()
    l.sum().backward()
    # Update the kernel
    with torch.no_grad():
        conv2d.weight[:] -= lr * conv2d.weight.grad
    if (i + 1) % 2 == 0:
        print(f'epoch {i + 1}, loss {l.sum():.3f}')

epoch 2, loss 10.235
epoch 4, loss 3.103
epoch 6, loss 1.088
epoch 8, loss 0.415
epoch 10, loss 0.165

TensorFlow

# Construct a two-dimensional convolutional layer with 1 output channel and a
# kernel of shape (1, 2). For the sake of simplicity, we ignore the bias here
conv2d = tf.keras.layers.Conv2D(1, (1, 2), use_bias=False)

# The two-dimensional convolutional layer uses four-dimensional input and
# output in the format of (example, height, width, channel), where the batch
# size (number of examples in the batch) and the number of channels are both 1
X = tf.reshape(X, (1, 6, 8, 1))
Y = tf.reshape(Y, (1, 6, 7, 1))
lr = 3e-2  # Learning rate

for i in range(10):
    with tf.GradientTape() as g:
        Y_hat = conv2d(X)
        l = (Y_hat - Y) ** 2
    # Update the kernel
    update = tf.multiply(lr, g.gradient(l, conv2d.trainable_weights)[0])
    conv2d.kernel.assign(conv2d.kernel - update)
    if (i + 1) % 2 == 0:
        print(f'epoch {i + 1}, loss {tf.reduce_sum(l):.3f}')

epoch 2, loss 0.035
epoch 4, loss 0.010
epoch 6, loss 0.003
epoch 8, loss 0.001
epoch 10, loss 0.000

JAX

# Construct a two-dimensional convolutional layer with 1 output channel and a
# kernel of shape (1, 2). For the sake of simplicity, we ignore the bias here.
# Use a small-stddev normal init so the toy 10-step SGD has time to converge
# (Flax's default lecun_normal yields a much larger initial loss).
conv2d = nnx.Conv(1, 1, kernel_size=(1, 2), use_bias=False, padding='VALID',
                  kernel_init=nnx.initializers.normal(stddev=0.01),
                  rngs=nnx.Rngs(d2l.get_key()))

# The two-dimensional convolutional layer uses four-dimensional input and
# output in the format of (example, height, width, channel), where the batch
# size (number of examples in the batch) and the number of channels are both 1
X = X.reshape((1, 6, 8, 1))
Y = Y.reshape((1, 6, 7, 1))
lr = 3e-2  # Learning rate

def loss(model, X, Y):
    Y_hat = model(X)
    return ((Y_hat - Y) ** 2).sum()

for i in range(10):
    l, grads = nnx.value_and_grad(loss)(conv2d, X, Y)
    conv2d.kernel[...] -= lr * grads.kernel[...]
    if (i + 1) % 2 == 0:
        print(f'epoch {i + 1}, loss {l:.3f}')

epoch 2, loss 4.868
epoch 4, loss 0.817
epoch 6, loss 0.137
epoch 8, loss 0.023
epoch 10, loss 0.004

MXNet

# Construct a two-dimensional convolutional layer with 1 output channel and a
# kernel of shape (1, 2). For the sake of simplicity, we ignore the bias here
conv2d = nn.Conv2D(1, kernel_size=(1, 2), use_bias=False)
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)
lr = 3e-2  # Learning rate

for i in range(10):
    with autograd.record():
        Y_hat = conv2d(X)
        l = (Y_hat - Y) ** 2
    l.backward()
    # Update the kernel
    conv2d.weight.data()[:] -= lr * conv2d.weight.grad()
    if (i + 1) % 2 == 0:
        print(f'epoch {i + 1}, loss {float(l.sum()):.3f}')

epoch 2, loss 5.413
epoch 4, loss 0.909
epoch 6, loss 0.153
epoch 8, loss 0.026
epoch 10, loss 0.004

Note that the error has dropped to a small value after 10 iterations. Now we will take a look at the kernel tensor we learned.

PyTorch

d2l.reshape(conv2d.weight.data, (1, 2))

tensor([[ 0.9475, -1.0304]])

TensorFlow

d2l.reshape(conv2d.get_weights()[0], (1, 2))

<tf.Tensor: shape=(1, 2), dtype=float32, numpy=array([[ 1.0029018, -0.9985178]], dtype=float32)>

JAX

conv2d.kernel[...].reshape((1, 2))

Array([[ 0.98822   , -0.98883337]], dtype=float32)

MXNet

d2l.reshape(conv2d.weight.data(), (1, 2))

array([[ 0.98865455, -0.9871513 ]])

Indeed, the learned kernel tensor is close to the kernel tensor K we defined earlier.

6.2.5 Cross-Correlation and Convolution

Recall our observation from Section 6.1 of the correspondence between the cross-correlation and convolution operations. Here let’s continue to consider two-dimensional convolutional layers. What if such layers perform strict convolution operations as defined in Equation 6.1.6 instead of cross-correlations? In order to obtain the output of the strict convolution operation, we only need to flip the two-dimensional kernel tensor both horizontally and vertically, and then perform the cross-correlation operation with the input tensor.

Since kernels are learned from data in deep learning, the outputs of convolutional layers remain unaffected no matter such layers perform either the strict convolution operations or the cross-correlation operations.

To illustrate this, suppose that a convolutional layer performs cross-correlation and learns the kernel in Figure 6.2.1, which is here denoted as the matrix \(\mathbf{K}\). Assuming that other conditions remain unchanged, when this layer instead performs strict convolution, the learned kernel \(\mathbf{K}'\) will be the same as \(\mathbf{K}\) after \(\mathbf{K}'\) is flipped both horizontally and vertically. That is to say, when the convolutional layer performs strict convolution for the input in Figure 6.2.1 and \(\mathbf{K}'\), the same output in Figure 6.2.1 (cross-correlation of the input and \(\mathbf{K}\)) will be obtained.

In keeping with standard terminology in deep learning literature, we will continue to refer to the cross-correlation operation as a convolution even though, strictly-speaking, it is slightly different. Furthermore, we use the term element to refer to an entry (or component) of any tensor representing a layer representation or a convolution kernel.

6.2.6 Convolution as Matrix Multiplication

Every output element in Figure 6.2.1 is a dot product: the kernel, flattened into a vector of length \(k_\textrm{h} k_\textrm{w}\), multiplied with the input patch under the window, flattened the same way. If we extract each patch the sliding window visits, flatten it into a row, and stack the rows, we obtain a matrix with one row per output position. The entire cross-correlation then collapses into a single product of this patch matrix with the flattened kernel. The rearrangement is called im2col (it turns image patches into the columns, here rows, of a matrix).

This rewriting motivates one important family of convolution implementations. GPUs and other accelerators are built around fast dense matrix multiplication, and libraries often lower convolutions to explicit or implicit matrix products. Depending on the shapes and hardware, they may instead select direct, Winograd, FFT-based, or other specialized kernels. The explicit form costs memory: each input element is duplicated in up to \(k_\textrm{h} k_\textrm{w}\) rows. Let’s build the patch matrix for the input and kernel of Figure 6.2.1 and check that the matrix product reproduces corr2d.

X = d2l.tensor([[0.0, 1.0, 2.0], [3.0, 4.0, 5.0], [6.0, 7.0, 8.0]])
K = d2l.tensor([[0.0, 1.0], [2.0, 3.0]])
h, w = K.shape
p_h, p_w = X.shape[0] - h + 1, X.shape[1] - w + 1
patches = d2l.stack([d2l.reshape(X[i:i + h, j:j + w], (-1,))
                     for i in range(p_h) for j in range(p_w)])
Y_mat = d2l.reshape(d2l.matmul(patches, d2l.reshape(K, (-1, 1))), (p_h, p_w))
Y_mat, d2l.reduce_sum(d2l.abs(Y_mat - corr2d(X, K)))

PyTorch

(tensor([[19., 25.],
         [37., 43.]]),
 tensor(0.))

TensorFlow

(<tf.Tensor: shape=(2, 2), dtype=float32, numpy=
 array([[19., 25.],
        [37., 43.]], dtype=float32)>,
 <tf.Tensor: shape=(), dtype=float32, numpy=0.0>)

JAX

(Array([[19., 25.],
        [37., 43.]], dtype=float32),
 Array(0., dtype=float32))

MXNet

(array([[19., 25.],
        [37., 43.]]),
 array(0.))

The sum of absolute differences is zero: the one matrix product and the sliding-window loop compute the same output.

PyTorch

PyTorch exposes this rearrangement directly: F.unfold extracts the patch matrix (transposed, and with batch and channel dimensions), so any convolution can be written as unfold, matrix multiplication, and reshape.

PyTorch

X_batch = d2l.reshape(X, (1, 1, 3, 3))
patches_unfold = F.unfold(X_batch, kernel_size=(2, 2))[0].T
torch.allclose(patches_unfold, patches)

True

The same idea also works in the opposite direction: instead of unfolding the input, we can unroll the kernel into a sparse, banded matrix that multiplies the flattened image. The exercises explore this view.

6.2.7 Feature Map and Receptive Field

As described in Section 6.1.4, the convolutional layer output in Figure 6.2.1 is sometimes called a feature map, as it can be regarded as the learned representations (features) in the spatial dimensions (e.g., width and height) to the subsequent layer. In CNNs, for any element \(x\) of some layer, its receptive field refers to all the elements (from all the previous layers) that may affect the calculation of \(x\) during the forward propagation. Note that the receptive field may be larger than the actual size of the input.

Let’s continue to use Figure 6.2.1 to explain the receptive field. Given the \(2 \times 2\) convolution kernel, the receptive field of the shaded output element (of value \(19\)) is the four elements in the shaded portion of the input. Now let’s denote the \(2 \times 2\) output as \(\mathbf{Y}\) and consider a deeper CNN with an additional \(2 \times 2\) convolutional layer that takes \(\mathbf{Y}\) as its input, outputting a single element \(z\). In this case, the receptive field of \(z\) on \(\mathbf{Y}\) includes all the four elements of \(\mathbf{Y}\), while the receptive field on the input includes all the nine input elements. Thus, when any element in a feature map needs a larger receptive field to detect input features over a broader area, we can build a deeper network.

This layer-by-layer counting has a closed form. Consider a stack of \(L\) convolutional layers in which layer \(i\) has a \(k_i \times k_i\) kernel and stride \(s_i\) (padding affects only which outputs exist, not how far each one sees). One step at the input of layer \(i\) corresponds to \(\prod_{j=1}^{i-1} s_j\) steps at the original input, since every earlier layer with stride \(s_j\) multiplies the step size by \(s_j\). Layer \(i\)’s kernel spans \(k_i - 1\) steps of its own input, so it widens the receptive field by \((k_i - 1) \prod_{j=1}^{i-1} s_j\) input pixels. Starting from a single pixel and summing over layers, an element at the top of the stack has a receptive field of side length

\[ r = 1 + \sum_{i=1}^{L} \left( k_i - 1 \right) \prod_{j=1}^{i-1} s_j, \tag{6.2.2}\]

where the empty product for \(i = 1\) equals \(1\).

The most common case is \(L\) stacked \(3 \times 3\) layers with stride \(1\): each layer adds \(2\), so the stack sees \((2L + 1) \times (2L + 1)\) input pixels. Two such layers cover \(5 \times 5\), three cover \(7 \times 7\). This is why deep stacks of small kernels can replace single large ones: three \(3 \times 3\) layers match the receptive field of one \(7 \times 7\) layer with fewer parameters (\(27\) weights instead of \(49\) per input–output channel pair) and three nonlinearities instead of one. Strides enter through the product: after a stride-\(2\) layer (or a pooling step, Section 6.5), every later kernel counts double, so downsampling makes the receptive field grow geometrically with depth. We will use Equation 6.2.2 repeatedly when we analyze modern architectures in Chapter 7.

Equation Equation 6.2.2 gives the theoretical receptive field: the set of inputs that can affect an activation at all. Their influence is not uniform. Luo et al. (2016) measured the effective receptive field through gradients and found that it concentrates near the center of the theoretical region, with a roughly Gaussian profile in common randomly initialized and trained networks. Depth, dilation, and larger kernels enlarge the set of possible inputs; they do not guarantee that optimization will use all of it equally.

Receptive fields derive their name from neurophysiology. Experiments recording from the visual cortex of several animal species (Hubel and Wiesel 1959, 1962, 1968) found that its lower levels respond to edges and related shapes. Later, Field (1987) modeled these responses on natural images with what are, in effect, convolutional kernels. We reprint a key figure in Figure 6.2.2.

Figure 6.2.2: Figure and caption taken from Field (1987): An example of coding with six different channels. (Left) Examples of the six types of sensor associated with each channel. (Right) Convolution of the image in (Middle) with the six sensors shown in (Left). The response of the individual sensors is determined by sampling these filtered images at a distance proportional to the size of the sensor (shown with dots). This diagram shows the response of only the even symmetric sensors.

The correspondence extends to features computed by deeper layers of networks trained on image classification (Kuzovkin et al. 2018).

6.2.8 Summary

The core computation in a convolutional layer is cross-correlation. A nested loop states the operation directly; practical libraries select among implicit matrix multiplication, direct, Winograd, FFT-based, and specialized kernels. Locality creates extensive reuse of weights and overlapping input patches, but real performance depends on both arithmetic throughput and memory movement.

Convolutions can detect edges and lines, blur images, or sharpen them. In a neural network, the filters are learned from data rather than specified by hand. The language of receptive fields has a historical connection to visual neurophysiology, but that analogy is motivation rather than evidence that a particular network mirrors the brain.

6.2.9 Exercises

1. Construct an image X with diagonal edges.
   1. What happens if you apply the kernel K in this section to it?
   2. What happens if you transpose X?
   3. What happens if you transpose K?
2. Design some kernels manually.
   1. Given a directional vector \(\mathbf{v} = (v_1, v_2)\), derive an edge-detection kernel that detects edges orthogonal to \(\mathbf{v}\), i.e., edges in the direction \((v_2, -v_1)\).
   2. Derive a finite difference operator for the second derivative. What is the minimum size of the convolutional kernel associated with it? Which structures in images respond most strongly to it?
   3. How would you design a blur kernel? Why might you want to use such a kernel?
   4. What is the minimum size of a kernel to obtain a derivative of order \(d\)?
3. When you try to automatically find the gradient for the Conv2D class we created, what kind of error message do you see?
4. How do you represent a cross-correlation operation as a matrix multiplication by changing the input and kernel tensors?

PyTorch

Discussions

TensorFlow

Discussions

JAX

Discussions

MXNet

Discussions

Field, David J. 1987. “Relations Between the Statistics of Natural Images and the Response Properties of Cortical Cells.” JOSA A 4 (12): 2379–94. https://doi.org/10.1364/josaa.4.002379.

Hubel, David H, and Torsten N Wiesel. 1959. “Receptive Fields of Single Neurones in the Cat’s Striate Cortex.” Journal of Physiology 148 (3): 574–91. https://doi.org/10.1113/jphysiol.1959.sp006308.

Hubel, David H, and Torsten N Wiesel. 1962. “Receptive Fields, Binocular Interaction and Functional Architecture in the Cat’s Visual Cortex.” Journal of Physiology 160 (1): 106–54. https://doi.org/10.1113/jphysiol.1962.sp006837.

Hubel, David H, and Torsten N Wiesel. 1968. “Receptive Fields and Functional Architecture of Monkey Striate Cortex.” Journal of Physiology 195 (1): 215–43. https://doi.org/10.1113/jphysiol.1968.sp008455.

Kuzovkin, Ilya, Raul Vicente, Mathilde Petton, et al. 2018. “Activations of Deep Convolutional Neural Networks Are Aligned with Gamma Band Activity of Human Visual Cortex.” Communications Biology 1 (1): 1–12. https://doi.org/10.1038/s42003-018-0110-y.

Luo, Wenjie, Yujia Li, Raquel Urtasun, and Richard Zemel. 2016. “Understanding the Effective Receptive Field in Deep Convolutional Neural Networks.” Advances in Neural Information Processing Systems. https://arxiv.org/abs/1701.04128.