Deriving the Inverse Filter of Image Convolution Kernel

Question

Does anyone has an idea how to calculate the Inverse of a 2-D filter?

Let\'s say I have a 3x3 filter:
0 1 0
1 1 1
0 1 0
I want to find it\'s inverse.

Answer 1

For a separable filter (i.e., one in which you can pull out a horizontal and vertical filter which can be applied in any order and give the same result as the composite 2-D filter), you could attempt to calculate the inverse of each on individually.

The inverse of a filter H(w) is G(w)=1/H(w), so one way to do it is to take the impulse response (the h[n] time-domain coeffs) and inverse-DFT them. It's not always easy to get an analytic expression for such a filter, so you could either compute it numerically (approximating to the desired precision) or do adaptive inverse filtering, as Steve suggested. See Widrow and Stearn's Adaptive Signal Processing for more info on this latter method.