# Choice of Laplacian Filter for 2D Images

Context: I am solving a 2-dimensonal discrecte Poission equation ($$\mathbf{\nabla}^2x = f$$) by assembling a system of linear equations $$A x = b$$ and then solving it using numpy's integrated solver (np.linalg.solve). To assemble this system, I am currently using the well-known 5-point Laplace kernel:

I somehow felt that this is a very rough approximation for the divergence of $$x$$ and thought there should be better approximations. So I started researching and found the 9-point Laplace kernel:

My question: Besides the obvious increased computational requirements, is there any good reason as to why you should not use the 9-point over the 5-point Laplacian kernel? To me, it seems that it should be a better approximation in every way. Yet, most ressources you find online mention the 5-point kernel but very rarely talk about the 9-point kernel. Why is that?

• The 9-point kernel you hear most about has eight 1 values and a -8 in the middle. I don’t think I’ve seen this variation before. Commented Oct 14, 2022 at 2:09
• If rotation invariance is important, use Gaussian derivatives. Commented Oct 14, 2022 at 2:19
• Could you please review my answer?
– Royi
Commented Feb 25, 2023 at 8:38