557 lines
20 KiB
Plaintext
557 lines
20 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "7d2e90ba",
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"metadata": {
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"origin_pos": 0
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},
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"source": [
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"# 图像卷积\n",
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":label:`sec_conv_layer`\n",
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"\n",
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"上节我们解析了卷积层的原理，现在我们看看它的实际应用。由于卷积神经网络的设计是用于探索图像数据，本节我们将以图像为例。\n",
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"\n",
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"## 互相关运算\n",
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"\n",
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"严格来说，卷积层是个错误的叫法，因为它所表达的运算其实是*互相关运算*（cross-correlation），而不是卷积运算。\n",
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"根据 :numref:`sec_why-conv`中的描述，在卷积层中，输入张量和核张量通过(**互相关运算**)产生输出张量。\n",
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"\n",
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"首先，我们暂时忽略通道（第三维）这一情况，看看如何处理二维图像数据和隐藏表示。在 :numref:`fig_correlation`中，输入是高度为$3$、宽度为$3$的二维张量（即形状为$3 \\times 3$）。卷积核的高度和宽度都是$2$，而卷积核窗口（或卷积窗口）的形状由内核的高度和宽度决定（即$2 \\times 2$）。\n",
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"\n",
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"\n",
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":label:`fig_correlation`\n",
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"\n",
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"在二维互相关运算中，卷积窗口从输入张量的左上角开始，从左到右、从上到下滑动。\n",
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"当卷积窗口滑动到新一个位置时，包含在该窗口中的部分张量与卷积核张量进行按元素相乘，得到的张量再求和得到一个单一的标量值，由此我们得出了这一位置的输出张量值。\n",
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"在如上例子中，输出张量的四个元素由二维互相关运算得到，这个输出高度为$2$、宽度为$2$，如下所示：\n",
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"\n",
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"$$\n",
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"0\\times0+1\\times1+3\\times2+4\\times3=19,\\\\\n",
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"1\\times0+2\\times1+4\\times2+5\\times3=25,\\\\\n",
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"3\\times0+4\\times1+6\\times2+7\\times3=37,\\\\\n",
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"4\\times0+5\\times1+7\\times2+8\\times3=43.\n",
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"$$\n",
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"\n",
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"注意，输出大小略小于输入大小。这是因为卷积核的宽度和高度大于1，\n",
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"而卷积核只与图像中每个大小完全适合的位置进行互相关运算。\n",
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"所以，输出大小等于输入大小$n_h \\times n_w$减去卷积核大小$k_h \\times k_w$，即：\n",
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"\n",
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"$$(n_h-k_h+1) \\times (n_w-k_w+1).$$\n",
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"\n",
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"这是因为我们需要足够的空间在图像上“移动”卷积核。稍后，我们将看到如何通过在图像边界周围填充零来保证有足够的空间移动卷积核，从而保持输出大小不变。\n",
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"接下来，我们在`corr2d`函数中实现如上过程，该函数接受输入张量`X`和卷积核张量`K`，并返回输出张量`Y`。\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "1bd2b0f5",
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"metadata": {
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"execution": {
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"iopub.execute_input": "2023-08-18T07:07:26.587988Z",
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"iopub.status.busy": "2023-08-18T07:07:26.587419Z",
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"iopub.status.idle": "2023-08-18T07:07:28.559553Z",
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"shell.execute_reply": "2023-08-18T07:07:28.558681Z"
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},
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"origin_pos": 2,
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"tab": [
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"pytorch"
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]
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},
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"outputs": [],
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"source": [
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"import torch\n",
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"from torch import nn\n",
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"from d2l import torch as d2l"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "16abe7ca",
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"metadata": {
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"execution": {
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"iopub.execute_input": "2023-08-18T07:07:28.563668Z",
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"iopub.status.busy": "2023-08-18T07:07:28.562986Z",
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"iopub.status.idle": "2023-08-18T07:07:28.569424Z",
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"shell.execute_reply": "2023-08-18T07:07:28.568319Z"
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},
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"origin_pos": 4,
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"tab": [
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"pytorch"
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]
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},
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"outputs": [],
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"source": [
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"def corr2d(X, K): #@save\n",
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" \"\"\"计算二维互相关运算\"\"\"\n",
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" h, w = K.shape\n",
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" Y = torch.zeros((X.shape[0] - h + 1, X.shape[1] - w + 1))\n",
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" for i in range(Y.shape[0]):\n",
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" for j in range(Y.shape[1]):\n",
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" Y[i, j] = (X[i:i + h, j:j + w] * K).sum()\n",
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" return Y"
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]
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},
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{
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"cell_type": "markdown",
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"id": "e2adaedd",
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"metadata": {
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"origin_pos": 6
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},
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"source": [
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"通过 :numref:`fig_correlation`的输入张量`X`和卷积核张量`K`，我们来[**验证上述二维互相关运算的输出**]。\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"id": "6f84e512",
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"metadata": {
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"execution": {
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"iopub.execute_input": "2023-08-18T07:07:28.572958Z",
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"iopub.status.busy": "2023-08-18T07:07:28.572449Z",
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"iopub.status.idle": "2023-08-18T07:07:28.604854Z",
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"shell.execute_reply": "2023-08-18T07:07:28.603813Z"
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},
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"origin_pos": 7,
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"tab": [
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"pytorch"
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]
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"tensor([[19., 25.],\n",
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" [37., 43.]])"
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]
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},
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"execution_count": 3,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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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]])\n",
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"K = torch.tensor([[0.0, 1.0], [2.0, 3.0]])\n",
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"corr2d(X, K)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "e93ccf40",
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"metadata": {
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"origin_pos": 8
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},
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"source": [
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"## 卷积层\n",
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"\n",
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"卷积层对输入和卷积核权重进行互相关运算，并在添加标量偏置之后产生输出。\n",
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"所以，卷积层中的两个被训练的参数是卷积核权重和标量偏置。\n",
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"就像我们之前随机初始化全连接层一样，在训练基于卷积层的模型时，我们也随机初始化卷积核权重。\n",
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"\n",
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"基于上面定义的`corr2d`函数[**实现二维卷积层**]。在`__init__`构造函数中，将`weight`和`bias`声明为两个模型参数。前向传播函数调用`corr2d`函数并添加偏置。\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"id": "450def67",
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"metadata": {
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"execution": {
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"iopub.execute_input": "2023-08-18T07:07:28.610672Z",
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"iopub.status.busy": "2023-08-18T07:07:28.609819Z",
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"iopub.status.idle": "2023-08-18T07:07:28.615602Z",
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"shell.execute_reply": "2023-08-18T07:07:28.614632Z"
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},
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"origin_pos": 10,
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"tab": [
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"pytorch"
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]
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},
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"outputs": [],
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"source": [
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"class Conv2D(nn.Module):\n",
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" def __init__(self, kernel_size):\n",
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" super().__init__()\n",
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" self.weight = nn.Parameter(torch.rand(kernel_size))\n",
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" self.bias = nn.Parameter(torch.zeros(1))\n",
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"\n",
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" def forward(self, x):\n",
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" return corr2d(x, self.weight) + self.bias"
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]
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},
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{
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"cell_type": "markdown",
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"id": "d361e4c7",
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"metadata": {
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"origin_pos": 13
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},
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"source": [
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"高度和宽度分别为$h$和$w$的卷积核可以被称为$h \\times w$卷积或$h \\times w$卷积核。\n",
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"我们也将带有$h \\times w$卷积核的卷积层称为$h \\times w$卷积层。\n",
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"\n",
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"## 图像中目标的边缘检测\n",
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"\n",
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"如下是[**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**)。\n",
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"首先，我们构造一个$6\\times 8$像素的黑白图像。中间四列为黑色（$0$），其余像素为白色（$1$）。\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"id": "dee1bc79",
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"metadata": {
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"execution": {
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"iopub.execute_input": "2023-08-18T07:07:28.620077Z",
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"iopub.status.busy": "2023-08-18T07:07:28.619277Z",
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"iopub.status.idle": "2023-08-18T07:07:28.626719Z",
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"shell.execute_reply": "2023-08-18T07:07:28.625746Z"
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},
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"origin_pos": 14,
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"tab": [
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"pytorch"
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]
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"tensor([[1., 1., 0., 0., 0., 0., 1., 1.],\n",
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" [1., 1., 0., 0., 0., 0., 1., 1.],\n",
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" [1., 1., 0., 0., 0., 0., 1., 1.],\n",
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" [1., 1., 0., 0., 0., 0., 1., 1.],\n",
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" [1., 1., 0., 0., 0., 0., 1., 1.],\n",
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" [1., 1., 0., 0., 0., 0., 1., 1.]])"
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]
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},
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"execution_count": 5,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"X = torch.ones((6, 8))\n",
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"X[:, 2:6] = 0\n",
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"X"
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]
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},
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{
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"cell_type": "markdown",
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"id": "ea455932",
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"metadata": {
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"origin_pos": 16
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},
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"source": [
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"接下来，我们构造一个高度为$1$、宽度为$2$的卷积核`K`。当进行互相关运算时，如果水平相邻的两元素相同，则输出为零，否则输出为非零。\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"id": "d042bda0",
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"metadata": {
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"execution": {
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"iopub.execute_input": "2023-08-18T07:07:28.630101Z",
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"iopub.status.busy": "2023-08-18T07:07:28.629606Z",
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"iopub.status.idle": "2023-08-18T07:07:28.634133Z",
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"shell.execute_reply": "2023-08-18T07:07:28.633165Z"
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},
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"origin_pos": 17,
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"tab": [
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"pytorch"
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]
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},
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"outputs": [],
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"source": [
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"K = torch.tensor([[1.0, -1.0]])"
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]
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},
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{
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"cell_type": "markdown",
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"id": "19635ba4",
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"metadata": {
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"origin_pos": 18
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},
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"source": [
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"现在，我们对参数`X`（输入）和`K`（卷积核）执行互相关运算。\n",
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"如下所示，[**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**]，其他情况的输出为$0$。\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"id": "36de9e2a",
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"metadata": {
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"execution": {
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"iopub.execute_input": "2023-08-18T07:07:28.639056Z",
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"iopub.status.busy": "2023-08-18T07:07:28.638505Z",
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"iopub.status.idle": "2023-08-18T07:07:28.646532Z",
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"shell.execute_reply": "2023-08-18T07:07:28.645509Z"
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},
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"origin_pos": 19,
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"tab": [
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"pytorch"
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]
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"tensor([[ 0., 1., 0., 0., 0., -1., 0.],\n",
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" [ 0., 1., 0., 0., 0., -1., 0.],\n",
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" [ 0., 1., 0., 0., 0., -1., 0.],\n",
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" [ 0., 1., 0., 0., 0., -1., 0.],\n",
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" [ 0., 1., 0., 0., 0., -1., 0.],\n",
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" [ 0., 1., 0., 0., 0., -1., 0.]])"
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]
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},
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"execution_count": 7,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"Y = corr2d(X, K)\n",
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"Y"
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]
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},
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{
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"cell_type": "markdown",
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"id": "9f3991ae",
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"metadata": {
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"origin_pos": 20
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},
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"source": [
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"现在我们将输入的二维图像转置，再进行如上的互相关运算。\n",
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"其输出如下，之前检测到的垂直边缘消失了。\n",
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"不出所料，这个[**卷积核`K`只可以检测垂直边缘**]，无法检测水平边缘。\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"id": "0a754b2d",
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"metadata": {
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"execution": {
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"iopub.execute_input": "2023-08-18T07:07:28.651371Z",
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"iopub.status.busy": "2023-08-18T07:07:28.650819Z",
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"iopub.status.idle": "2023-08-18T07:07:28.658419Z",
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"shell.execute_reply": "2023-08-18T07:07:28.657436Z"
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},
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"origin_pos": 21,
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"tab": [
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"pytorch"
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]
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"tensor([[0., 0., 0., 0., 0.],\n",
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" [0., 0., 0., 0., 0.],\n",
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" [0., 0., 0., 0., 0.],\n",
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" [0., 0., 0., 0., 0.],\n",
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" [0., 0., 0., 0., 0.],\n",
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" [0., 0., 0., 0., 0.],\n",
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" [0., 0., 0., 0., 0.],\n",
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" [0., 0., 0., 0., 0.]])"
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]
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},
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"execution_count": 8,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"corr2d(X.t(), K)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "18ceafe9",
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"metadata": {
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"origin_pos": 22
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},
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"source": [
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"## 学习卷积核\n",
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"\n",
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"如果我们只需寻找黑白边缘，那么以上`[1, -1]`的边缘检测器足以。然而，当有了更复杂数值的卷积核，或者连续的卷积层时，我们不可能手动设计滤波器。那么我们是否可以[**学习由`X`生成`Y`的卷积核**]呢？\n",
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"\n",
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"现在让我们看看是否可以通过仅查看“输入-输出”对来学习由`X`生成`Y`的卷积核。\n",
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"我们先构造一个卷积层，并将其卷积核初始化为随机张量。接下来，在每次迭代中，我们比较`Y`与卷积层输出的平方误差，然后计算梯度来更新卷积核。为了简单起见，我们在此使用内置的二维卷积层，并忽略偏置。\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"id": "2b423578",
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"metadata": {
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"execution": {
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"iopub.execute_input": "2023-08-18T07:07:28.662260Z",
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"iopub.status.busy": "2023-08-18T07:07:28.661527Z",
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"iopub.status.idle": "2023-08-18T07:07:28.681412Z",
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"shell.execute_reply": "2023-08-18T07:07:28.680192Z"
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},
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"origin_pos": 24,
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"tab": [
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"pytorch"
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]
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},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"epoch 2, loss 6.422\n",
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"epoch 4, loss 1.225\n",
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"epoch 6, loss 0.266\n",
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"epoch 8, loss 0.070\n",
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"epoch 10, loss 0.022\n"
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]
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}
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],
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"source": [
|
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"# 构造一个二维卷积层，它具有1个输出通道和形状为（1，2）的卷积核\n",
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"conv2d = nn.Conv2d(1,1, kernel_size=(1, 2), bias=False)\n",
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"\n",
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"# 这个二维卷积层使用四维输入和输出格式（批量大小、通道、高度、宽度），\n",
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"# 其中批量大小和通道数都为1\n",
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"X = X.reshape((1, 1, 6, 8))\n",
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"Y = Y.reshape((1, 1, 6, 7))\n",
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"lr = 3e-2 # 学习率\n",
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"\n",
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"for i in range(10):\n",
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" Y_hat = conv2d(X)\n",
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" l = (Y_hat - Y) ** 2\n",
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" conv2d.zero_grad()\n",
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" l.sum().backward()\n",
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" # 迭代卷积核\n",
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" conv2d.weight.data[:] -= lr * conv2d.weight.grad\n",
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" if (i + 1) % 2 == 0:\n",
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" print(f'epoch {i+1}, loss {l.sum():.3f}')"
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]
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},
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{
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"cell_type": "markdown",
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"id": "37744bcf",
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"metadata": {
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"origin_pos": 27
|
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},
|
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"source": [
|
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"在$10$次迭代之后，误差已经降到足够低。现在我们来看看我们[**所学的卷积核的权重张量**]。\n"
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]
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},
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{
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"cell_type": "code",
|
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"execution_count": 10,
|
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"id": "b40515e8",
|
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"metadata": {
|
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"execution": {
|
|
"iopub.execute_input": "2023-08-18T07:07:28.684721Z",
|
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"iopub.status.busy": "2023-08-18T07:07:28.684428Z",
|
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"iopub.status.idle": "2023-08-18T07:07:28.691507Z",
|
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"shell.execute_reply": "2023-08-18T07:07:28.690512Z"
|
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},
|
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"origin_pos": 29,
|
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"tab": [
|
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"pytorch"
|
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]
|
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},
|
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"outputs": [
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{
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"data": {
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"text/plain": [
|
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"tensor([[ 1.0010, -0.9739]])"
|
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]
|
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},
|
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"execution_count": 10,
|
|
"metadata": {},
|
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"output_type": "execute_result"
|
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}
|
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],
|
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"source": [
|
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"conv2d.weight.data.reshape((1, 2))"
|
|
]
|
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},
|
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{
|
|
"cell_type": "markdown",
|
|
"id": "366d2c4f",
|
|
"metadata": {
|
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"origin_pos": 32
|
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},
|
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"source": [
|
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"细心的读者一定会发现，我们学习到的卷积核权重非常接近我们之前定义的卷积核`K`。\n",
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"\n",
|
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"## 互相关和卷积\n",
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"\n",
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"回想一下我们在 :numref:`sec_why-conv`中观察到的互相关和卷积运算之间的对应关系。\n",
|
|
"为了得到正式的*卷积*运算输出，我们需要执行 :eqref:`eq_2d-conv-discrete`中定义的严格卷积运算，而不是互相关运算。\n",
|
|
"幸运的是，它们差别不大，我们只需水平和垂直翻转二维卷积核张量，然后对输入张量执行*互相关*运算。\n",
|
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"\n",
|
|
"值得注意的是，由于卷积核是从数据中学习到的，因此无论这些层执行严格的卷积运算还是互相关运算，卷积层的输出都不会受到影响。\n",
|
|
"为了说明这一点，假设卷积层执行*互相关*运算并学习 :numref:`fig_correlation`中的卷积核，该卷积核在这里由矩阵$\\mathbf{K}$表示。\n",
|
|
"假设其他条件不变，当这个层执行严格的*卷积*时，学习的卷积核$\\mathbf{K}'$在水平和垂直翻转之后将与$\\mathbf{K}$相同。\n",
|
|
"也就是说，当卷积层对 :numref:`fig_correlation`中的输入和$\\mathbf{K}'$执行严格*卷积*运算时，将得到与互相关运算 :numref:`fig_correlation`中相同的输出。\n",
|
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"\n",
|
|
"为了与深度学习文献中的标准术语保持一致，我们将继续把“互相关运算”称为卷积运算，尽管严格地说，它们略有不同。\n",
|
|
"此外，对于卷积核张量上的权重，我们称其为*元素*。\n",
|
|
"\n",
|
|
"## 特征映射和感受野\n",
|
|
"\n",
|
|
"如在 :numref:`subsec_why-conv-channels`中所述， :numref:`fig_correlation`中输出的卷积层有时被称为*特征映射*（feature map），因为它可以被视为一个输入映射到下一层的空间维度的转换器。\n",
|
|
"在卷积神经网络中，对于某一层的任意元素$x$，其*感受野*（receptive field）是指在前向传播期间可能影响$x$计算的所有元素（来自所有先前层）。\n",
|
|
"\n",
|
|
"请注意，感受野可能大于输入的实际大小。让我们用 :numref:`fig_correlation`为例来解释感受野：\n",
|
|
"给定$2 \\times 2$卷积核，阴影输出元素值$19$的感受野是输入阴影部分的四个元素。\n",
|
|
"假设之前输出为$\\mathbf{Y}$，其大小为$2 \\times 2$，现在我们在其后附加一个卷积层，该卷积层以$\\mathbf{Y}$为输入，输出单个元素$z$。\n",
|
|
"在这种情况下，$\\mathbf{Y}$上的$z$的感受野包括$\\mathbf{Y}$的所有四个元素，而输入的感受野包括最初所有九个输入元素。\n",
|
|
"因此，当一个特征图中的任意元素需要检测更广区域的输入特征时，我们可以构建一个更深的网络。\n",
|
|
"\n",
|
|
"## 小结\n",
|
|
"\n",
|
|
"* 二维卷积层的核心计算是二维互相关运算。最简单的形式是，对二维输入数据和卷积核执行互相关操作，然后添加一个偏置。\n",
|
|
"* 我们可以设计一个卷积核来检测图像的边缘。\n",
|
|
"* 我们可以从数据中学习卷积核的参数。\n",
|
|
"* 学习卷积核时，无论用严格卷积运算或互相关运算，卷积层的输出不会受太大影响。\n",
|
|
"* 当需要检测输入特征中更广区域时，我们可以构建一个更深的卷积网络。\n",
|
|
"\n",
|
|
"## 练习\n",
|
|
"\n",
|
|
"1. 构建一个具有对角线边缘的图像`X`。\n",
|
|
" 1. 如果将本节中举例的卷积核`K`应用于`X`，会发生什么情况？\n",
|
|
" 1. 如果转置`X`会发生什么？\n",
|
|
" 1. 如果转置`K`会发生什么？\n",
|
|
"1. 在我们创建的`Conv2D`自动求导时，有什么错误消息？\n",
|
|
"1. 如何通过改变输入张量和卷积核张量，将互相关运算表示为矩阵乘法？\n",
|
|
"1. 手工设计一些卷积核。\n",
|
|
" 1. 二阶导数的核的形式是什么？\n",
|
|
" 1. 积分的核的形式是什么？\n",
|
|
" 1. 得到$d$次导数的最小核的大小是多少？\n"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "c9adecf6",
|
|
"metadata": {
|
|
"origin_pos": 34,
|
|
"tab": [
|
|
"pytorch"
|
|
]
|
|
},
|
|
"source": [
|
|
"[Discussions](https://discuss.d2l.ai/t/1848)\n"
|
|
]
|
|
}
|
|
],
|
|
"metadata": {
|
|
"language_info": {
|
|
"name": "python"
|
|
},
|
|
"required_libs": []
|
|
},
|
|
"nbformat": 4,
|
|
"nbformat_minor": 5
|
|
} |