Perona-Malik Diffusion Equation

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Homework Statement

Implement the non-linear de-noising algorithm of Perona-Malik. Consider a noisy image, u, with pixel values referenced by u(i,j). Non-linear de-noising can be achieved by solving the following non-linear diffusion equation:

∇ · (g(∇u)∇u) = 0

with g(s) = ((K^2)v) / ((K^2) + |s|)

where v and K are parameters controlling the amount of diffusion.

Write down the linear system associated with this discretization.

My question is how do I do this? How can I take the gradient of a pixel? It isn't a function so how can I do partial derivative with respect to x and y to it?

 

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I have no idea what to do here. If I could have shown work I would have, but I have no work done. I just need a nudge in the right direction here.