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

One way is to write a matrix representation of convolution with your filter and then try to find some (regularized) inverse of this matrix.

For example the 1D [1,-1] filter is maybe the simplest discrete differential approximation that exists.

Its matrix will have two diagonals 1 on main diagonal and -1 on first off diagonal.

It's inverse will be integral filter. An IIR filter with infinite trail of 1s. Since our matrix has limited size this will in practice mean filling up the matrix with 1s everywhere on one side of the diagonal.