How to use 5x5, 7x7, ... matrix in Sobel filtering?

Question

infographics

Good afternoon!
I can't figure out where the coefficients in the matrices come from:
www.cim.mcgill.ca/~image529/TA529/Image529_99/assi...
please explain.
And how to expand the matrix to the size of 7x7 9x9 and so on?

Answer 1

In Wikipedia, it is beautifully painted: the Sobel operator is an ordinary gradient with weight coefficients. Obtained by convolution of two matrices: gradient in the desired direction (say, [-1/2 0 1/2] - Dx_{x,y} = (M_{x+1,y} - M_{x-1,y})/ 2) and weight coefficients (say, [1; 2; 1]/4 = [1/4;1/2;1/4]) → [-1/8 0 1/8; -1/4 0 1/4; -1/8 0 1/8]. Naturally, there is no strict gradient here - you are free to choose any weight coefficients you like.
You can also expand the gradient matrix. If we take a 5x1 kernel, we get: Dx_{x,y} = [(M_{x+1,y} - M_{x-1,y})/2) + (M_{x+2,y} - M_ {x-2,y})/4)]/2 → [-1/8 -1/4 0 1/4 1/8]. By weights we can take, for example, the matrix [1;2;4;2;1].
Similarly for matrices 7x7, 9x9 and more.

Answer 2

This is called the convolution kernel. https://ru.wikipedia.org/wiki/%D0%A1%D0%B2%D1%91%D...
Understand how the convolution of two functions works (or in the context of image processing - convolution of sequences) - no questions will.
In general, the Sobel operator, as far as I remember, uses only 3x3 kernels.