# Noise amplification with inverse filtering

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?

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