Samuel D. Stearns

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A treatment of adaptive signal processing featuring frequent use of examples.

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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.
It's easy to do using DFT.
But let's say I want to do it by convolution.
Now, that's the problem, Matlab symbolic isn't my specialty.
Assuming there's a 3X3 Inverse Filter it means convolution of the two will result in:

0 0 0

0 1 0

0 0 0

The problem is to create the right set of equations for that and solving it.
Doing it with symbols is easy to think yet I couldn't do it.

Any ideas? Thanks.

P.S. I'm not sure there an Inverse Filter for this one as it has zeros in its DTFT.

Moreover, someone should allow Latex in this forum the way MathOverflow has.

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.

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