Q15Digital Image Processing

When an image is degraded by physical motion blur or a bad camera lens, the degradation is mathematically modeled as $G(u,v)=H(u,v)F(u,v)+N(u,v)$, where $H$ is the Degradation Function (the blur) and $N$ is random Additive Noise. The goal of restoration is to violently recover the original image $F(u,v)$.

The primitive mathematical approach is the Inverse Filter: $\hat{F}(u,v)=\frac{G(u,v)}{H(u,v)}=F(u,v)+\frac{N(u,v)}{H(u,v)}$.

This is a mathematical disaster. At high frequencies, the blur function $H(u,v)$ violently approaches exactly zero. When you divide the noise $N(u,v)$ by a number infinitesimally close to zero, the noise term violently explodes toward infinity, completely annihilating the image in a sea of chaotic white noise.

The Wiener Filter (Minimum Mean Square Error Filter) completely abandons direct division. It utilizes advanced probability mathematics to minimize the mean square error between the estimated image and the true image. The formula is:

$$\hat{F}(u,v)=\left[\frac{H^{\ast }(u,v)}{|H(u,v){|}^2+\frac{S_n(u,v)}{S_f(u,v)}}\right]G(u,v)$$

• Where $S_n$ is the Power Spectral Density of the Noise, and $S_f$ is the Power Spectral Density of the original image.
• The Genius: If the noise is extremely high, the fraction $\frac{S_n}{S_f}$ becomes massive, violently shrinking the entire multiplier toward zero, protecting the system from exploding.
• If the noise is zero ($S_n=0$), the formula mathematically simplifies perfectly back into the direct Inverse Filter.