图像卷积

互相关运算

In [2]:
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

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

实现二维卷积层

In [4]:
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

卷积层的一个简单应用: 检测图像中不同颜色的边缘

In [5]:
X = torch.ones((6, 8))
X[:, 2:6] = 0
X
Out[5]:
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.]])
In [6]:
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

In [7]:
Y = corr2d(X, K)
Y
Out[7]:
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

In [8]:
corr2d(X.t(), K)
Out[8]:
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.]])

学习由X生成Y的卷积核

In [9]:
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}')
batch 2, loss 1.650
batch 4, loss 0.284
batch 6, loss 0.050
batch 8, loss 0.010
batch 10, loss 0.002

所学的卷积核的权重张量

In [10]:
conv2d.weight.data.reshape((1, 2))
Out[10]:
tensor([[ 0.9905, -0.9963]])