2D image convolution
但在CNN中，明明只是进行了correlation，但却称之为convolution，实在不解
下面， 将图像处理中的convolution重新整理记录
因为网络关于这部分的解释很多，这里直接借用其他 参考
“A convolution is done by multiplying a pixel's and its neighboring pixels color value by a matrix”, 这里的matrix就是convoluiton kernel （usually a small matrix of numbers）
这里假设图像是3*3，kernel也是3*3，实际计算中，有时为了使得卷积结果与原图像一致，会对原图像进行padding操作
原图像x：
| 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 2 | 3 | 0 |
| 0 | 4 | 5 | 6 | 0 |
| 0 | 7 | 8 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 |
| x(0,0) | x(0,1) | x(0,2) | x(0,3) | x(0,4) |
| x(1,0) | x(1,1) | x(1,2) | x(1,3) | x(1,4) |
| x(2,0) | x(2,1) | x(2,2) | x(2,3) | x(2,4) |
| x(3,0) | x(3,1) | x(3,2) | x(3,3) | x(3,4) |
| x(4,0) | x(4,1) | x(4,2) | x(4,3) | x(4,4) |
卷积核h：
| -1 | -2 | -1 |
| 0 | 0 | 0 |
| 1 | 2 | 1 |
| h(1,1) | h(1,2) | h(1,3) |
| h(2,1) | h(2,2) | h(2,3) |
| h(3,1) | h(3,2) | h(3,3) |
具体的过程为：
将h先上下翻转，再左右翻转，然后，与x进行correlation运算
| 1 | 2 | 1 |
| 0 | 0 | 0 |
| -1 | -2 | -1 |
| h(3,3) | h(3,2) | h(3,1) |
| h(2,3) | h(2,2) | h(2,1) |
| h(1,1) | h(1,2) | h(1,1) |
输出结果y:3*3
| x(0,0) | x(0,1) | x(0,2) | x(0,3) | x(0,4) |
| x(1,0) | x(1,1) | x(1,2) | x(1,3) | x(1,4) |
| x(2,0) | x(2,1) | x(2,2) | x(2,3) | x(2,4) |
| x(3,0) | x(3,1) | x(3,2) | x(3,3) | x(3,4) |
| x(4,0) | x(4,1) | x(4,2) | x(4,3) | x(4,4) |
依次覆盖，对应元素相乘
| h(3,3) | h(3,2) | h(3,1) |
| h(2,3) | h(2,2) | h(2,1) |
| h(1,1) | h(1,2) | h(1,1) |
y(1,1) = h(3,3) *x(0,0) + h(3,2) *x(0,1) + h(3,1) *x(0,2) +
h(2,3) *x(1,0) + h(2,2) *x(1,1) + h(2,1) *x(1,2) +
h(1,3) *x(2,0) + h(1,2) *x(2,1) + h(1,1) *x(2,2)
其他元素类似
注：In image processing, a kernel, convolution matrix, or mask is a small matrix useful for blurring, sharpening, embossing, edge-detection, and more. This is accomplished by means of convolution between a kernel and an image.