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

In practice, one useful solution is Wiener deconvolution http://en.wikipedia.org/wiki/Wiener_deconvolution which basically, in Fourier space, divides by the spectrum of the given filter. Zeros are handled by adding a fudge term: instead of 1/H(w), use H(w)/( abs(H(w))^2 + c) where H(w) is the discrete Fourier transform of the filter h(x) (add a 2nd dimension as you like) and "w" is supposed to be omega. The constant is chosen based on the noise level of the signal or image.