Wiener Filtering

The noise amplification problem of the Inverse Filtering or pseudo-inverse filtering can be better addressed by Wiener filtering.

Wiener filtering incorporates both the degradation and statistical characteristics of noise to find an estimate of the un-degraded image such that the mean square error (MSE) between them is minimized:

$$e^2=E\left\{{\left(\hat{f}-f\right)}^2\right\}$$

Assumptions

Assume that the noise and the image are uncorrelated, one of them has zero mean, and that the gray levels in the estimate are a linear function of the gray levels in the degraded image. The MSE estimate of the un-degraded image is then:

$$\hat{F}(u,v)=H_W(u,v)G(u,v)$$

• $H_W(u,v)$ is known as the Wiener filter, also known as the minimum mean square error filter or least square error filter

The filter

$$H_W(u,v)=\frac{H^{\ast }(u,v)S_f(u,v)}{S_f(u,v)|H(u,v){|}^2+S_{\eta }(u,v)}=\frac{1}{H(u,v)}\left[\frac{|H(u,v){|}^2}{|H(u,v){|}^2+S_{\eta }(u,v)/S_f(u,v)}\right]$$

where:

$$\left[\frac{|H(u,v){|}^2}{|H(u,v){|}^2+S_{\eta }(u,v)/S_f(u,v)}\right]=W(u,v)$$

• $H(u,v)$ is the degradation function
• $H^{\ast }(u,v)$ is the complex conjugate of $H(u,v)$
• $|H(u,v){|}^2=H^{\ast }(u,v)H(u,v)$
• $S_f(u,v)=|F(u,v){|}^2$ is the power spectrum of the undegraded image
• $S_{\eta }(u,v)=|N(u,v){|}^2$ is the power spectrum of the noise
• $W(u,v)$ is the scale factor

Wiener is an Inverse filter with a scale factor

$$W(u,v)=\frac{|H(u,v){|}^2{S}_f(u,v)}{|H(u,v){|}^2{S}_f(u,v)+S_{\eta }(u,v)}$$

• The scale factor, $W(u,v)$ decreases with increasing noise power and increases with decreasing noise power
• The scale factor approaches 0 when the degradation function approaches 0
• When there is no noise, the scale factor becomes 1 and the Wiener filter reduces to the inverse filter
• If for some frequency there is no signal, the scale factor becomes 0, the Wiener filter reduces to 0 there, and the noise is suppressed
• If there is no degradation, $H(u,v)=1$, and the Wiener filter reduces to a smoothing filter

Approximation when the spectra are unknown

When both $S_f(u,v)$ and $S_{\eta }(u,v)$ are not known or cannot be well estimated, we can approximate the Wiener filter as:

$$H_W(u,v)=\frac{1}{H(u,v)}\left[\frac{|H(u,v){|}^2}{|H(u,v){|}^2+K}\right]$$

where $K$ is a specified constant.

Weiner Restoration

$$\boxed{\hat{F}(u,v)=H_w(u,v)G(u,v)}$$

This formula is the final restored image using the new weiner degradation function, $H_w(u,v)$.