# Convolutions¶

Import packages for:

1. building a CNN from scratch;
2. using built-in architectures.
In [1]:
from mxnet import np, npx
from mxnet.gluon import nn
npx.set_np()


The 2D cross-correlation operator:

In [2]:
def corr2d(X, K):
h, w = K.shape
Y = np.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


For example, a two-dimensional cross-correlation operation. The shaded portions are the first output element and the input and kernel array elements used in its computation:

\begin{align*} 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{align*}

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

Out[3]:
array([[19., 25.],
[37., 43.]])

The convolutional layers

$\mathbf Y = \mathbf X \star \mathbf W + b$

In [4]:
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()


Check the output from the convolution layers.

In [5]:
def comp_conv2d(conv2d, X):
conv2d.initialize()
# Add batch and channel dimension.
X = X.reshape((1, 1) + X.shape)
Y = conv2d(X)
# Exclude the first two dimensions
return Y.reshape(Y.shape[2:])


In [6]:
X = np.random.uniform(size=(8, 8))
conv2d = nn.Conv2D(channels=1, kernel_size=3, padding=1, strides=2)
comp_conv2d(conv2d, X).shape

Out[6]:
(4, 4)
\begin{align} \text{ Output shape} & = \lfloor(n_h-k_h+p_h+s_h)/s_h\rfloor \times \lfloor(n_w-k_w+p_w+s_w)/s_w\rfloor \\ & = \lfloor(8 - 3 + 1 + 2) / 2\rfloor \times \lfloor(8 - 3 + 1 + 2) / 2\rfloor \\ & = (4, 4) \end{align}

A slightly more complicated example.

In [7]:
X = np.random.uniform(size=(8, 8))
conv2d = nn.Conv2D(1, kernel_size=(3, 5), padding=(0, 1), strides=(3, 4))
comp_conv2d(conv2d, X).shape

Out[7]:
(2, 2)
\begin{align} \text{ Output shape} & = \lfloor(n_h-k_h+p_h+s_h)/s_h\rfloor \times \lfloor(n_w-k_w+p_w+s_w)/s_w\rfloor \\ & = \lfloor(8 - 3 + 0 + 3)/3\rfloor \times \lfloor(8 - 5 + 1 + 4)/4\rfloor \\ & = (2, 2) \end{align}

# Pooling¶

A 2D pooling operator

In [8]:
def pool2d(X, pool_size, mode='max'):
p_h, p_w = pool_size
Y = np.zeros((X.shape[0] - p_h + 1, X.shape[1] - p_w + 1))
for i in range(Y.shape[0]):
for j in range(Y.shape[1]):
if mode == 'max':
Y[i, j] = np.max(X[i: i + p_h, j: j + p_w])
elif mode == 'avg':
Y[i, j] = X[i: i + p_h, j: j + p_w].mean()
return Y

X = np.array([[0, 1, 2], [3, 4, 5], [6, 7, 8]])
pool2d(X, (2, 2))

Out[8]:
array([[4., 5.],
[7., 8.]])

In [9]:
X = np.arange(16).reshape((1, 1, 4, 4))
print(X)
pool2d(X)

[[[[ 0.  1.  2.  3.]
[ 4.  5.  6.  7.]
[ 8.  9. 10. 11.]
[12. 13. 14. 15.]]]]

Out[9]:
array([[[[ 5.,  7.],
[13., 15.]]]])

Multiple channels pooling

In [10]:
X = np.concatenate((X, X + 1), axis=1)
print(X)
print("Input shape :", X.shape)

pool2d(X)

[[[[ 0.  1.  2.  3.]
[ 4.  5.  6.  7.]
[ 8.  9. 10. 11.]
[12. 13. 14. 15.]]

[[ 1.  2.  3.  4.]
[ 5.  6.  7.  8.]
[ 9. 10. 11. 12.]
[13. 14. 15. 16.]]]]
shape : (1, 2, 4, 4)

Out[10]:
array([[[[ 5.,  7.],
[13., 15.]],

[[ 6.,  8.],
[14., 16.]]]])