The other day, I asked this question 3x3x3 Laplace Kernel?, regarding what the 3x3x3 kernel was for applying a Laplacian convolution.

On that page, it mentions the kernels were "deduced by using discrete differential quotients." Does anyone know how this is done? For example, what if I need a 5x5x5 kernel? How do I "solve" for it? I tried looking in Matlab but didn't find anything.

The reason I want to do this is because if I have a 100x100x10 pixel image I want to apply a Laplacian convolution to, I can convolve with my 3x3x3 kernel, OR I can apply a Fourier transform to both, and then use point-wise multiplication to get the same solution, then use the inverse Fourier to get my result back, which should match the result I got using the direct convolution method. As I mentioned, to do the Fourier approach, I need two images - my original, and the "Laplacian" one. I figured this Laplacian one needs to be generated in the same way as the 3x3x3 one my previous question identified, but needs to also be the same dimensions as my real image (100x100x10 in this example) in order for the Fourier transform approach to work. So my question above is - how do I 'fill in' the 3-dimensional matrix that is my discrete Laplacian operator?

Sorry if this question doesn't make sense. In a very dumbed down sense, I just want to know how to solve for the 3x3x3 kernel myself, so that I can solve for 'larger kernels' like a 100x100x10 one.