卷积的计算过程

卷积的计算过程

flyfish
包括手动计算，可视化使用torch.nn.Conv2d实现

示例

import torch
import torch.nn as nn

# 定义输入图像
input_image = torch.tensor([
    [1, 2, 3, 0, 1],
    [0, 1, 2, 3, 4],
    [2, 3, 0, 1, 2],
    [1, 2, 3, 4, 0],
    [0, 1, 2, 3, 4]
], dtype=torch.float32).unsqueeze(0).unsqueeze(0)  # 添加批次和通道维度
print(input_image.shape)

# 定义卷积核
conv_kernel = torch.tensor([
    [1, 0, -1],
    [1, 0, -1],
    [1, 0, -1]
], dtype=torch.float32).unsqueeze(0).unsqueeze(0)  # 添加输入和输出通道维度
print(conv_kernel.shape)
# 创建卷积层
conv_layer = nn.Conv2d(in_channels=1, out_channels=1, kernel_size=3, stride=1, padding=0, bias=False)

# 将卷积核的权重设置为自定义值
with torch.no_grad():
    conv_layer.weight = nn.Parameter(conv_kernel)

# 进行卷积操作
output_tensor = conv_layer(input_image)

# 打印输入图像
print("输入图像:")
print(input_image.squeeze().numpy())

# 打印卷积核
print("卷积核:")
print(conv_kernel.squeeze().numpy())

# 打印输出结果
print("输出结果:")
print(output_tensor.squeeze().detach().numpy())

torch.Size([1, 1, 5, 5])
torch.Size([1, 1, 3, 3])
# 输入图像:
[[1. 2. 3. 0. 1.]
 [0. 1. 2. 3. 4.]
 [2. 3. 0. 1. 2.]
 [1. 2. 3. 4. 0.]
 [0. 1. 2. 3. 4.]]
卷积核:
[[ 1.  0. -1.]
 [ 1.  0. -1.]
 [ 1.  0. -1.]]
输出结果:
[[-2.  2. -2.]
 [-2. -2. -1.]
 [-2. -2. -1.]]

输入图像和卷积核

输入图像 $I$:
[ 1 2 3 0 1 0 1 2 3 4 2 3 0 1 2 1 2 3 4 0 0 1 2 3 4 ]

​10210​21321​32032​03143​14204​ ​
卷积核 $K$:
[ 1 0 − 1 1 0 − 1 1 0 − 1 ]

[10−110−110−1]" role="presentation">⎡⎣⎢111000−1−1−1⎤⎦⎥$$\left[\begin{array}{ccc}1& 0& -1\\ 1& 0& -1\\ 1& 0& -1\end{array}\right]$$

​111​000​−1−1−1​ ​

手动计算卷积

我们将逐个计算每个位置的卷积结果：

1. 位置 (0, 0)： [ 1 2 3 0 1 2 2 3 0 ] ⊙ [ 1 0 − 1 1 0 − 1 1 0 − 1 ] = ( 1 ⋅ 1 + 2 ⋅ 0 + 3 ⋅ ( − 1 ) ) + ( 0 ⋅ 1 + 1 ⋅ 0 + 2 ⋅ ( − 1 ) ) + ( 2 ⋅ 1 + 3 ⋅ 0 + 0 ⋅ ( − 1 ) ) = ( 1 − 3 ) + ( − 2 ) + ( 2 ) = − 2
   [123012230]" role="presentation">⎡⎣⎢102213320⎤⎦⎥$$\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 2\\ 2& 3& 0\end{array}\right]$$
   \odot
   [10−110−110−1]" role="presentation">⎡⎣⎢111000−1−1−1⎤⎦⎥$$\left[\begin{array}{ccc}1& 0& -1\\ 1& 0& -1\\ 1& 0& -1\end{array}\right]$$
   = (1 \cdot 1 + 2 \cdot 0 + 3 \cdot (-1)) + (0 \cdot 1 + 1 \cdot 0 + 2 \cdot (-1)) + (2 \cdot 1 + 3 \cdot 0 + 0 \cdot (-1)) = (1 - 3) + (-2) + (2) \\= -2 ​102​213​320​ ​⊙ ​111​000​−1−1−1​ ​=(1⋅1+2⋅0+3⋅(−1))+(0⋅1+1⋅0+2⋅(−1))+(2⋅1+3⋅0+0⋅(−1))=(1−3)+(−2)+(2)=−2
2. 位置 (0, 1)： [ 2 3 0 1 2 3 3 0 1 ] ⊙ [ 1 0 − 1 1 0 − 1 1 0 − 1 ] = ( 2 ⋅ 1 + 3 ⋅ 0 + 0 ⋅ ( − 1 ) ) + ( 1 ⋅ 1 + 2 ⋅ 0 + 3 ⋅ ( − 1 ) ) + ( 3 ⋅ 1 + 0 ⋅ 0 + 1 ⋅ ( − 1 ) ) = 2 + ( 1 − 3 ) + ( 3 − 1 ) = 2
   [230123301]" role="presentation">⎡⎣⎢213320031⎤⎦⎥$$\left[\begin{array}{ccc}2& 3& 0\\ 1& 2& 3\\ 3& 0& 1\end{array}\right]$$
   \odot
   [10−110−110−1]" role="presentation">⎡⎣⎢111000−1−1−1⎤⎦⎥$$\left[\begin{array}{ccc}1& 0& -1\\ 1& 0& -1\\ 1& 0& -1\end{array}\right]$$
   = (2 \cdot 1 + 3 \cdot 0 + 0 \cdot (-1)) + (1 \cdot 1 + 2 \cdot 0 + 3 \cdot (-1)) + (3 \cdot 1 + 0 \cdot 0 + 1 \cdot (-1)) = 2 + (1 - 3) + (3 - 1) \\= 2 ​213​320​031​ ​⊙ ​111​000​−1−1−1​ ​=(2⋅1+3⋅0+0⋅(−1))+(1⋅1+2⋅0+3⋅(−1))+(3⋅1+0⋅0+1⋅(−1))=2+(1−3)+(3−1)=2
3. 位置 (0, 2)： [ 3 0 1 2 3 4 0 1 2 ] ⊙ [ 1 0 − 1 1 0 − 1 1 0 − 1 ] = ( 3 ⋅ 1 + 0 ⋅ 0 + 1 ⋅ ( − 1 ) ) + ( 2 ⋅ 1 + 3 ⋅ 0 + 4 ⋅ ( − 1 ) ) + ( 0 ⋅ 1 + 1 ⋅ 0 + 2 ⋅ ( − 1 ) ) = 3 − 1 + 2 − 4 − 2 = − 2
   [301234012]" role="presentation">⎡⎣⎢320031142⎤⎦⎥$$\left[\begin{array}{ccc}3& 0& 1\\ 2& 3& 4\\ 0& 1& 2\end{array}\right]$$
   \odot
   [10−110−110−1]" role="presentation">⎡⎣⎢111000−1−1−1⎤⎦⎥$$\left[\begin{array}{ccc}1& 0& -1\\ 1& 0& -1\\ 1& 0& -1\end{array}\right]$$
   = (3 \cdot 1 + 0 \cdot 0 + 1 \cdot (-1)) + (2 \cdot 1 + 3 \cdot 0 + 4 \cdot (-1)) + (0 \cdot 1 + 1 \cdot 0 + 2 \cdot (-1)) = 3 - 1 + 2 - 4 - 2 \\= -2 ​320​031​142​ ​⊙ ​111​000​−1−1−1​ ​=(3⋅1+0⋅0+1⋅(−1))+(2⋅1+3⋅0+4⋅(−1))+(0⋅1+1⋅0+2⋅(−1))=3−1+2−4−2=−2
4. 位置 (1, 0)： [ 0 1 2 2 3 0 1 2 3 ] ⊙ [ 1 0 − 1 1 0 − 1 1 0 − 1 ] = ( 0 ⋅ 1 + 1 ⋅ 0 + 2 ⋅ ( − 1 ) ) + ( 2 ⋅ 1 + 3 ⋅ 0 + 0 ⋅ ( − 1 ) ) + ( 1 ⋅ 1 + 2 ⋅ 0 + 3 ⋅ ( − 1 ) ) = − 2 + 2 + 1 − 3 = − 2
   [012230123]" role="presentation">⎡⎣⎢021132203⎤⎦⎥$$\left[\begin{array}{ccc}0& 1& 2\\ 2& 3& 0\\ 1& 2& 3\end{array}\right]$$
   \odot
   [10−110−110−1]" role="presentation">⎡⎣⎢111000−1−1−1⎤⎦⎥$$\left[\begin{array}{ccc}1& 0& -1\\ 1& 0& -1\\ 1& 0& -1\end{array}\right]$$
   = (0 \cdot 1 + 1 \cdot 0 + 2 \cdot (-1)) + (2 \cdot 1 + 3 \cdot 0 + 0 \cdot (-1)) + (1 \cdot 1 + 2 \cdot 0 + 3 \cdot (-1)) = -2 + 2 + 1 - 3 \\= -2 ​021​132​203​ ​⊙ ​111​000​−1−1−1​ ​=(0⋅1+1⋅0+2⋅(−1))+(2⋅1+3⋅0+0⋅(−1))+(1⋅1+2⋅0+3⋅(−1))=−2+2+1−3=−2
5. 位置 (1, 1)： [ 1 2 3 3 0 1 2 3 4 ] ⊙ [ 1 0 − 1 1 0 − 1 1 0 − 1 ] = ( 1 ⋅ 1 + 2 ⋅ 0 + 3 ⋅ ( − 1 ) ) + ( 3 ⋅ 1 + 0 ⋅ 0 + 1 ⋅ ( − 1 ) ) + ( 2 ⋅ 1 + 3 ⋅ 0 + 4 ⋅ ( − 1 ) ) = 1 − 3 + 3 − 1 + 2 − 4 = − 2
   [123301234]" role="presentation">⎡⎣⎢132203314⎤⎦⎥$$\left[\begin{array}{ccc}1& 2& 3\\ 3& 0& 1\\ 2& 3& 4\end{array}\right]$$
   \odot
   [10−110−110−1]" role="presentation">⎡⎣⎢111000−1−1−1⎤⎦⎥$$\left[\begin{array}{ccc}1& 0& -1\\ 1& 0& -1\\ 1& 0& -1\end{array}\right]$$
   =(1⋅1+2⋅0+3⋅(−1))+(3⋅1+0⋅0+1⋅(−1))+(2⋅1+3⋅0+4⋅(−1))=1−3+3−1+2−4=−2" role="presentation">=(1⋅1+2⋅0+3⋅(−1))+(3⋅1+0⋅0+1⋅(−1))+(2⋅1+3⋅0+4⋅(−1))=1−3+3−1+2−4=−2$$\begin{array}{r}\\ & =(1\cdot 1+2\cdot 0+3\cdot (-1))+(3\cdot 1+0\cdot 0+1\cdot (-1))+(2\cdot 1+3\cdot 0+4\cdot (-1))\\ & =1-3+3-1+2-4\\ & =-2\end{array}$$
   ​132​203​314​ ​⊙ ​111​000​−1−1−1​ ​​=(1⋅1+2⋅0+3⋅(−1))+(3⋅1+0⋅0+1⋅(−1))+(2⋅1+3⋅0+4⋅(−1))=1−3+3−1+2−4=−2​
6. 位置 (1, 2)： [ 2 3 4 0 1 2 3 4 0 ] ⊙ [ 1 0 − 1 1 0 − 1 1 0 − 1 ] = ( 2 ⋅ 1 + 3 ⋅ 0 + 4 ⋅ ( − 1 ) ) + ( 0 ⋅ 1 + 1 ⋅ 0 + 2 ⋅ ( − 1 ) ) + ( 3 ⋅ 1 + 4 ⋅ 0 + 0 ⋅ ( − 1 ) ) = − 2 − 2 + 3 = − 1
   [234012340]" role="presentation">⎡⎣⎢203314420⎤⎦⎥$$\left[\begin{array}{ccc}2& 3& 4\\ 0& 1& 2\\ 3& 4& 0\end{array}\right]$$
   \odot
   [10−110−110−1]" role="presentation">⎡⎣⎢111000−1−1−1⎤⎦⎥$$\left[\begin{array}{ccc}1& 0& -1\\ 1& 0& -1\\ 1& 0& -1\end{array}\right]$$
   \\ = (2 \cdot 1 + 3 \cdot 0 + 4 \cdot (-1)) + (0 \cdot 1 + 1 \cdot 0 + 2 \cdot (-1)) + (3 \cdot 1 + 4 \cdot 0 + 0 \cdot (-1)) \\ = -2 - 2 + 3 \\ = -1 ​203​314​420​ ​⊙ ​111​000​−1−1−1​ ​=(2⋅1+3⋅0+4⋅(−1))+(0⋅1+1⋅0+2⋅(−1))+(3⋅1+4⋅0+0⋅(−1))=−2−2+3=−1
7. 位置 (2, 0)： [ 2 3 0 1 2 3 0 1 2 ] ⊙ [ 1 0 − 1 1 0 − 1 1 0 − 1 ] = ( 2 ⋅ 1 + 3 ⋅ 0 + 0 ⋅ ( − 1 ) ) + ( 1 ⋅ 1 + 2 ⋅ 0 + 3 ⋅ ( − 1 ) ) + ( 0 ⋅ 1 + 1 ⋅ 0 + 2 ⋅ ( − 1 ) ) = 2 + ( 1 − 3 ) − 2 = − 2
   [230123012]" role="presentation">⎡⎣⎢210321032⎤⎦⎥$$\left[\begin{array}{ccc}2& 3& 0\\ 1& 2& 3\\ 0& 1& 2\end{array}\right]$$
   \odot
   [10−110−110−1]" role="presentation">⎡⎣⎢111000−1−1−1⎤⎦⎥$$\left[\begin{array}{ccc}1& 0& -1\\ 1& 0& -1\\ 1& 0& -1\end{array}\right]$$
   = (2 \cdot 1 + 3 \cdot 0 + 0 \cdot (-1)) + (1 \cdot 1 + 2 \cdot 0 + 3 \cdot (-1)) + (0 \cdot 1 + 1 \cdot 0 + 2 \cdot (-1)) \\= 2 + (1 - 3) - 2 \\= -2 ​210​321​032​ ​⊙ ​111​000​−1−1−1​ ​=(2⋅1+3⋅0+0⋅(−1))+(1⋅1+2⋅0+3⋅(−1))+(0⋅1+1⋅0+2⋅(−1))=2+(1−3)−2=−2
8. 位置 (2, 1)：$ [ 3 0 1 2 3 4 1 2 3 ] ⊙ [ 1 0 − 1 1 0 − 1 1 0 − 1 ] = ( 3 ⋅ 1 + 0 ⋅ 0 + 1 ⋅ ( − 1 ) ) + ( 2 ⋅ 1 + 3 ⋅ 0 + 4 ⋅ ( − 1 ) ) + ( 1 ⋅ 1 + 2 ⋅ 0 + 3 ⋅ ( − 1 ) ) = 3 − 1 + 2 − 4 + 1 − 3 = − 2
   [301234123]" role="presentation">⎡⎣⎢321032143⎤⎦⎥$$\left[\begin{array}{ccc}3& 0& 1\\ 2& 3& 4\\ 1& 2& 3\end{array}\right]$$
   \odot
   [10−110−110−1]" role="presentation">⎡⎣⎢111000−1−1−1⎤⎦⎥$$\left[\begin{array}{ccc}1& 0& -1\\ 1& 0& -1\\ 1& 0& -1\end{array}\right]$$
   \\= (3 \cdot 1 + 0 \cdot 0 + 1 \cdot (-1)) + (2 \cdot 1 + 3 \cdot 0 + 4 \cdot (-1)) + (1 \cdot 1 + 2 \cdot 0 + 3 \cdot (-1)) = 3 - 1 + 2 - 4 + 1 - 3 \\= -2 ​321​032​143​ ​⊙ ​111​000​−1−1−1​ ​=(3⋅1+0⋅0+1⋅(−1))+(2⋅1+3⋅0+4⋅(−1))+(1⋅1+2⋅0+3⋅(−1))=3−1+2−4+1−3=−2
9. 位置 (2, 2)： [ 0 1 2 3 4 0 2 3 4 ] ⊙ [ 1 0 − 1 1 0 − 1 1 0 − 1 ] = ( 0 ⋅ 1 + 1 ⋅ 0 + 2 ⋅ ( − 1 ) ) + ( 3 ⋅ 1 + 4 ⋅ 0 + 0 ⋅ ( − 1 ) ) + ( 2 ⋅ 1 + 3 ⋅ 0 + 4 ⋅ ( − 1 ) ) = − 2 + 3 + 2 − 4 = − 1
   [012340234]" role="presentation">⎡⎣⎢032143204⎤⎦⎥$$\left[\begin{array}{ccc}0& 1& 2\\ 3& 4& 0\\ 2& 3& 4\end{array}\right]$$
   \odot
   [10−110−110−1]" role="presentation">⎡⎣⎢111000−1−1−1⎤⎦⎥$$\left[\begin{array}{ccc}1& 0& -1\\ 1& 0& -1\\ 1& 0& -1\end{array}\right]$$
   \\= (0 \cdot 1 + 1 \cdot 0 + 2 \cdot (-1)) + (3 \cdot 1 + 4 \cdot 0 + 0 \cdot (-1)) + (2 \cdot 1 + 3 \cdot 0 + 4 \cdot (-1)) \\= -2 + 3 + 2 - 4 \\= -1 ​032​143​204​ ​⊙ ​111​000​−1−1−1​ ​=(0⋅1+1⋅0+2⋅(−1))+(3⋅1+4⋅0+0⋅(−1))+(2⋅1+3⋅0+4⋅(−1))=−2+3+2−4=−1

参数解释

conv_layer = nn.Conv2d(
    in_channels=3,        # 输入通道数
    out_channels=16,      # 输出通道数
    kernel_size=3,        # 卷积核大小
    stride=1,             # 步幅
    padding=1,            # 填充
    padding_mode='zeros', # 填充模式
    dilation=1,           # 空洞卷积
    groups=1,             # 组卷积
    bias=True             # 是否使用偏置
)

in_channels (int): 输入通道数。例如，对于RGB图像，in_channels 应为 3。
out_channels (int): 输出通道数，也就是卷积核的数量。
kernel_size (int or tuple): 卷积核的大小。如果是整数，表示卷积核的高度和宽度相等。如果是元组，表示 (高度, 宽度)。
stride (int or tuple, optional): 卷积操作中窗口滑动的步幅。如果是整数，表示高度和宽度的步幅相等。如果是元组，表示 (高度步幅, 宽度步幅)。默认值为 1。
padding (int or tuple, optional): 输入的每一边要填充的零的层数。如果是整数，表示高度和宽度的填充相等。如果是元组，表示 (高度填充, 宽度填充)。默认值为 0。
padding_mode (str, optional): 填充模式，可以是 'zeros', 'reflect', 'replicate' 或 'circular'。默认值为 'zeros'。
dilation (int or tuple, optional): 卷积核元素之间的间距。如果是整数，表示高度和宽度的间距相等。如果是元组，表示 (高度间距, 宽度间距)。默认值为 1。
groups (int, optional): 从输入通道到输出通道的阻塞连接数。默认值为 1。groups 可以用于实现深度可分离卷积。
bias (bool, optional): 如果设置为 True，则添加一个学习到的偏置。默认值为 True。

import numpy as np
import matplotlib.pyplot as plt
import matplotlib.patches as patches
from matplotlib.animation import FuncAnimation, PillowWriter

# 定义输入图像和卷积核
input_image = np.array([
    [1, 2, 3, 0, 1],
    [0, 1, 2, 3, 4],
    [2, 3, 0, 1, 2],
    [1, 2, 3, 4, 0],
    [0, 1, 2, 3, 4]
])

conv_kernel = np.array([
    [1, 0, -1],
    [1, 0, -1],
    [1, 0, -1]
])

# 输入图像和卷积核的尺寸
input_size = input_image.shape[0]
kernel_size = conv_kernel.shape[0]
output_size = input_size - kernel_size + 1

# 创建图形和轴
fig, ax = plt.subplots(figsize=(6, 6))

# 显示输入图像
im = ax.imshow(input_image, cmap='viridis')

# 初始化矩形框和文本
rect = patches.Rectangle((0, 0), kernel_size, kernel_size, linewidth=2, edgecolor='r', facecolor='none')
ax.add_patch(rect)
text = ax.text(0, 0, '', ha='center', va='center', color='white', fontsize=12)

# 动画更新函数
def update(frame):
    i, j = divmod(frame, output_size)
    sub_matrix = input_image[i:i+kernel_size, j:j+kernel_size]
    conv_result = np.sum(sub_matrix * conv_kernel)
    
    # 更新矩形框的位置
    rect.set_xy((j, i))
    
    # 更新文本的位置和内容
    text.set_position((j + kernel_size / 2, i + kernel_size / 2))
    text.set_text(f'{conv_result:.2f}')
    
    return im, rect, text

# 创建动画
ani = FuncAnimation(fig, update, frames=output_size * output_size, blit=True, repeat=False)

# 保存动画为 GIF 文件
ani.save('convolution_animation.gif', writer=PillowWriter(fps=1))

plt.show()

卷积的结果

[[-2.  2. -2.]
 [-2. -2. -1.]
 [-2. -2. -1.]]