image-processing - Convolution theory vs implementation

Question

I study convolution in image processing as it is a part of the curriculum, I understand the theory and the formula but I am confused about its implementation.

The formula is:

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What I understand

The convolution kernel is flipped both horizontally and vertically then the values in the kernel are multiplied by the corresponding pixel values, the results are summed, divided by "row x column" to get the average, and then finally this result is the value of the pixel at the center of the kernel location.

Confusion in implementation

When I run the example convolution program from my course material and insert as input a 3x3 convolution kernel where:

1st row: (0, 1, 0)

2nd row: (0, 0, 0)

3rd row: (0, 0, 0)

The processed image is shifted down by one pixel, where I expected it to shift upwards by one pixel. This result indicates that no horizontal or vertical flipping is done before calculating (as if it is doing correlation).

I thought there might be a fault in the program so I looked around and found that Adobe Flex 3 and Gimp are doing this as well.

I don't understand, is there something that I missed to notice?

Appreciate any help or feedback.

Answer 1

我猜您尝试的程序实现了相关性而不是卷积。

我已经使用该ImageFilter函数在 Mathematica 中尝试了您的过滤器，结果按预期向上移动：

结果：

我也在 Octave（一个开源的 Matlab 克隆）中尝试过它：

imfilter([1,1,1,1,1;
          2,2,2,2,2;
          3,3,3,3,3;
          4,4,4,4,4;
          5,5,5,5,5],
         [0,1,0;
          0,0,0;
          0,0,0],"conv")

（“conv”表示卷积 -imfilter的默认值是相关性）。结果：

   2   2   2   2   2
   3   3   3   3   3
   4   4   4   4   4
   5   5   5   5   5
   0   0   0   0   0

请注意，最后一行是不同的。这是因为不同的实现使用不同的填充（默认情况下）。Mathematica 对 使用常量填充ImageConvolve，对 没有填充ListConvolve。Octaveimfilter使用零填充。

另请注意（如 belisarius 所述）卷积的结果可以比源图像更小、相同或更大。（我在 Matlab 和 IPPI 文档中阅读了术语“有效”、“相同大小”和“完全”卷积，但我不确定这是否是标准术语）。这个想法是可以执行求和

• 仅在内核完全位于图像内部的源图像像素上。在这种情况下，结果小于源图像。
• 在每个源像素上。在这种情况下，结果与源图像具有相同的大小。这需要在边界处填充
• 在内核的任何部分位于源图像内的每个像素上。在这种情况下，结果图像大于源图像。这也需要在边界处填充。

Answer 2

请注意：

结果是：

因此，“移动”不是真实的，因为尺寸会受到影响。