0. 图像与卷积核¶
输入图像来自真实照片，Sobel 核会强调竖向边缘。卷积输出越大，表示该区域越像这个边缘模式。
In [2]:
# 使用真实花朵照片，配合 Sobel 核做边缘检测。
raw_conv_photo = load_sample_image("flower.jpg")
conv_photo = np.asarray(Image.fromarray(raw_conv_photo).resize((96, 96))) / 255.0
image = np.dot(conv_photo[..., :3], [0.299, 0.587, 0.114])
kernel = np.array([
[-1, 0, 1],
[-1, 0, 1],
[-1, 0, 1],
], dtype=float)
feature = correlate2d(image, kernel, mode="valid")
display(pd.DataFrame({
"图像": ["flower.jpg"],
"尺寸": [f"{image.shape[0]}x{image.shape[1]}"],
"灰度最小值": [float(image.min())],
"灰度最大值": [float(image.max())],
}).round(3))
display(pd.DataFrame(kernel.astype(int)))
| 图像 | 尺寸 | 灰度最小值 | 灰度最大值 | |
|---|---|---|---|---|
| 0 | flower.jpg | 96x96 | 0.024 | 0.851 |
| 0 | 1 | 2 | |
|---|---|---|---|
| 0 | -1 | 0 | 1 |
| 1 | -1 | 0 | 1 |
| 2 | -1 | 0 | 1 |
1. 卷积与池化¶
卷积逐窗口计算局部模式，池化保留局部最强响应。把三张图连起来看，能看到图像如何逐步变成特征。
In [3]:
# 展开中心区域的几个窗口，观察卷积值如何来自局部像素。
rows = []
for i in range(40, 46):
for j in range(40, 46):
window = image[i:i + 3, j:j + 3]
rows.append({
"位置": f"({i},{j})",
"窗口均值": window.mean(),
"左列均值": window[:, 0].mean(),
"右列均值": window[:, 2].mean(),
"卷积值": round(float((window * kernel).sum()), 3),
})
conv_df = pd.DataFrame(rows)
display(conv_df.head(10).round(3))
display(pd.DataFrame({
"特征图尺寸": [f"{feature.shape[0]}x{feature.shape[1]}"],
"最小响应": [float(feature.min())],
"最大响应": [float(feature.max())],
"平均绝对响应": [float(np.abs(feature).mean())],
}).round(3))
| 位置 | 窗口均值 | 左列均值 | 右列均值 | 卷积值 | |
|---|---|---|---|---|---|
| 0 | (40,40) | 0.404 | 0.441 | 0.356 | -0.255 |
| 1 | (40,41) | 0.353 | 0.415 | 0.289 | -0.379 |
| 2 | (40,42) | 0.369 | 0.356 | 0.462 | 0.318 |
| 3 | (40,43) | 0.361 | 0.289 | 0.332 | 0.132 |
| 4 | (40,44) | 0.373 | 0.462 | 0.325 | -0.410 |
| 5 | (40,45) | 0.320 | 0.332 | 0.302 | -0.092 |
| 6 | (41,40) | 0.380 | 0.415 | 0.335 | -0.239 |
| 7 | (41,41) | 0.324 | 0.391 | 0.247 | -0.432 |
| 8 | (41,42) | 0.332 | 0.335 | 0.413 | 0.236 |
| 9 | (41,43) | 0.295 | 0.247 | 0.223 | -0.073 |
| 特征图尺寸 | 最小响应 | 最大响应 | 平均绝对响应 | |
|---|---|---|---|---|
| 0 | 94x94 | -2.107 | 1.838 | 0.137 |
In [4]:
# 绘制输入、边缘响应和 2x2 max pooling。
positive_feature = np.maximum(feature, 0)
pool = np.array([
[positive_feature[i:i + 2, j:j + 2].max() for j in range(0, positive_feature.shape[1] - 1, 2)]
for i in range(0, positive_feature.shape[0] - 1, 2)
])
fig, axes = plt.subplots(2, 2, figsize=(8.4, 7.0))
for ax, data, title, cmap in zip(
axes.ravel(),
[conv_photo, image, feature, pool],
["真实照片", "灰度输入", "Sobel 边缘响应", "MaxPool 后特征"],
[None, "gray", "RdBu_r", "YlGnBu"],
):
im = ax.imshow(data, cmap=cmap)
ax.set_title(title, fontweight="bold")
ax.set_xticks([])
ax.set_yticks([])
fig.suptitle("真实照片 Sobel 卷积：局部边缘被提取为特征图", x=0.08, ha="left", fontsize=14, fontweight="bold", color="#0f172a")
plt.tight_layout()
plt.show()
