Q15Digital Image Processing

Image restoration is treated as an inverse problem: given a degraded observation, recover the best possible estimate of the original scene using a mathematical model of how the degradation occurred.

In the spatial domain, the degradation process is modeled as $g(x,y)=h(x,y)\ast f(x,y)+n(x,y)$, where $f(x,y)$ is the original (unknown) image, $h(x,y)$ is the degradation function (e.g., blur caused by motion or defocus, represented as a convolution kernel), $n(x,y)$ is additive noise, and $g(x,y)$ is the observed degraded image. $\ast$ denotes spatial convolution.

Using the convolution theorem, this model is more conveniently expressed in the frequency domain as $G(u,v)=H(u,v)F(u,v)+N(u,v)$, where $H(u,v)$ is the Fourier transform of the degradation function (the optical transfer function).

The goal of image restoration is to obtain an estimate $\hat{f}(x,y)$ of the original image, given only $g(x,y)$ and some knowledge of $H$ and the statistical properties of $n(x,y)$. When $H$ is known or can be estimated (e.g., from the physics of the imaging system), restoration reduces to designing a filter to invert its effect while controlling noise amplification, as in inverse filtering or the Wiener filter. When noise is dominant and $H$ is identity, the problem reduces to pure spatial denoising using filters such as the mean, median, or adaptive filters.

A naive approach would simply divide in the frequency domain, $\hat{F}(u,v)=G(u,v)/H(u,v)$, known as inverse filtering. This works perfectly in the absence of noise, but fails catastrophically in practice because at frequencies where $H(u,v)$ is small (which is common at high frequencies for typical blur functions), dividing by it massively amplifies whatever noise $N(u,v)$ is present at that frequency, since the noise term becomes $N(u,v)/H(u,v)$. This is precisely the motivation for the Wiener filter, which incorporates the noise-to-signal power ratio to suppress this amplification at frequencies where $H$ is small, providing a statistically optimal balance between deblurring and noise control.