What does scipy.signal.convolve2d calculate? [duplicate]

Question

I am currently a bit confused by the output of

#!/usr/bin/env python

import scipy.signal

image = [[1, 2, 3, 4, 5, 6, 7],
         [8, 9, 10, 11, 12, 13, 14],
         [15, 16, 17, 18, 19, 20, 21],
         [22, 23, 24, 25, 26, 27, 28],
         [29, 30, 31, 32, 33, 34, 35],
         [36, 37, 38, 39, 40, 41, 42],
         [43, 44, 45, 46, 47, 48, 49]]

filter_kernel = [[-1, 1, -1],
                 [-2, 3, 1],
                 [2, -6, 0]]

res = scipy.signal.convolve2d(image, filter_kernel,
                              mode='same', boundary='fill', fillvalue=0)
print(res)

It was

[[  -2   -8   -7   -6   -5   -4   28]
 [   3   -7  -10  -13  -16  -19   14]
 [ -18  -28  -31  -34  -37  -40    0]
 [ -39  -49  -52  -55  -58  -61  -14]
 [ -60  -70  -73  -76  -79  -82  -28]
 [ -81  -91  -94  -97 -100 -103  -42]
 [-101  -61  -63  -65  -67  -69  -57]]

I expected the top left element to be 3*1 + 1*2 + (-6) *8 + 0*9 = -43 (ommitting the padded zeros).

I thought this would expand the matrix image \in R^{7x7} to R^{9x9} by adding one 0 to the left / right and top / bottom. Then I thought the filter_kernel would be calculated by "sliding" it over the image. At each position, the numbers from the image are point-wise multiplied with the numbers from the kernel. The nine products are the summed up and written into res.

However, it is -2. Obviously, something different happens.

Answer 1

Convolution reverses the direction of one of the functions it works on. Check The definition on Wikipedia: one function is parameterized with τ and the other with -τ. The same applies to 2D convolution.

You need to mirror the kernel to get the expected resut:

filter_kernel = [[0, -6, 2],
                 [1, 3, -2],
                 [-1, 1, -1]]

res = scipy.signal.convolve2d(image, filter_kernel,
                              mode='same', boundary='fill', fillvalue=0)
print(res[0, 0])
# -43