Finite difference operators

1. Feb 6, 2014

pamparana

Hello all,

I hope this is the write sub-forum for this question. I have been looking at the Laplacian of a 2-D vector field. It is explained nicely by this Wikipedia article here. My question is more regarding how these operators work together.

So, in the case of the Laplacian, it tells me the divergence of the gradient field. One thing I have seen a lot of people do (in the field of image processing) is to use the following operator A'A in computing some sort of a smoothness criteria. Here 'A' is the discrete Laplacian operator in the matrix form and A' is its transpose (although the matrix should be symmetric). Can someone tell me what the operator A'A represents?

I would be very grateful for any help you can give me.

Thanks,
Luca

2. Feb 7, 2014

olivermsun

A'A is the discrete version of the biharmonic operator (Laplacian squared or Δ2). The Laplacian gives you the divergence of gradients, which is just a bunch of derivatives of derivatives (second derivatives, or curvature/acceleration of the function along each dimension).. The biharmonic tells you about fourth-order derivatives, so it's even more selective toward high spatial frequencies (which are one kind of "unsmoothness").