二维互相关运算

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import torch
from torch import nn
from d2l import torch as d2l

def corr2d(X, K):
  """计算二维互相关运算。"""
  h, w = K.shape
  Y = torch.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

验证上述二维互相关运算的输出：

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X = torch.tensor([[0.0, 1.0, 2.0], [3.0, 4.0, 5.0], [6.0, 7.0, 8.0]])
K = torch.tensor([[0.0, 1.0], [2.0, 3.0]])
corr2d(X, K)
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tensor([[19., 25.],
     [37., 43.]])

二维卷积层

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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

图像中目标的边缘检测

卷积层的一个简单应用：通过找到像素变化的位置，来检测图像中不同颜色的边缘。

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X = torch.ones((6, 8))
X[:, 2:6] = 0
X
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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.]])
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K = torch.tensor([[1.0, -1.0]])

输出Y中的1代表从白色到黑色的边缘，-1代表从黑色到白色的边缘

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Y = corr2d(X, K)
Y
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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.]])

卷积核K只可以检测垂直边缘

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corr2d(X.t(), K)
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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.]])

学习卷积核

学习由X生成Y的卷积核

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conv2d = nn.Conv2d(1, 1, kernel_size=(1, 2), bias=False)

X = X.reshape((1, 1, 6, 8))
Y = Y.reshape((1, 1, 6, 7))

for i in range(10):
    Y_hat = conv2d(X)
    l = (Y_hat - Y)2
    conv2d.zero_grad()
    l.sum().backward()
    conv2d.weight.data[:] -= 3e-2 * conv2d.weight.grad
    if (i + 1) % 2 == 0:
        print(f'batch {i+1}, loss {l.sum():.3f}')
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batch 2, loss 3.910
batch 4, loss 0.752
batch 6, loss 0.166
batch 8, loss 0.044
batch 10, loss 0.014

所学的卷积核的权重张量：

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conv2d.weight.data.reshape((1, 2))
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tensor([[ 1.0012, -0.9793]])