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I'm trying to gain a better (mathematical) understanding of why inverse filtering is almost never the solution for correcting an image. From what I understand, we start with a discrete signal $s[n]$ that is the result of the true image $y[n]$ convolved by some kernel $A$ with some added noise $w[n]$. For discrete signals, this be converted to a matrix equation $s = Ay + w$, where $A$ is a circulant matrix representing the convolution. In Fourier Space, we have $s[f] = A[f]y[f] + w[f]$ which is much simpler.

Now, if we assume we know the kernel and its inverse exists, then we can approximately recover the true signal $y$ by inversion : $A^{-1} s = \hat s = y + A^{-1} w$. The problem becomes clear in the Fourier space, where inversion then becomes division, and if $A[f]$ is $0$ or very small at several frequencies this can lead to noise amplification. I am trying to get a better picture of this in the matrix representation.

I want to see if I can measure the noise amplification in this representation by trying to figure out an upper bound on the norm of $\hat s$ but I'm not sure if I'm going about it the right way. Calculating the norm of $\hat s$ I have $$|| \hat s|| = ||y + A^{-1} w||$$ $$||\hat s|| \leq ||y|| + ||A^{-1}||~||w||$$ Am I correct in assuming that the upper bound on the noise amplification is the norm of the inverse of the kernel?

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The noise amplification would be the ratio of the norms:

$\frac{\text{Norm of noise after inverse filter}}{\text{Norm of noise before inverse filter}} \leq \frac{||\mathbf{A}^{-1}||*||\mathbf{w}||}{||\mathbf{w}||}=||\mathbf{A}^{-1}||$

It seems your reasoning is correct that the norm of the inverse filter is an upper bound on the amplification.

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