Inverse Filtering

If the degradation function is known or can be estimated, inverse filtering is one simple and direct approach to restore a degraded image:

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

The above equation is inverse filtering. However, even if we know the degradation function, we cannot recover the un-degraded image exactly. The quantity that decides whether the estimate is usable is the ratio $N/H$ as we can see below.

$$\hat{F}(u,v)=F(u,v)+\frac{N(u,v)}{H(u,v)}$$

• $N(u,v)$ is usually not known
• If $H(u,v)$ has zero or very small values at some frequencies, the ratio $N(u,v)/H(u,v)$ could dominate the estimate and make the result not reliable
• If $N/H$ is 0.01 the estimate is fine, but if $N/H$ is 100 the estimate is bad

Motion blur case

• For motion blur, $H(u,v)$ is a sinc function along some direction
• The inverse filter $H^{-}(u,v)$ becomes infinite at some frequencies
• Motion blur creates frequencies where $|H|$ is zero or very small, and dividing by those values turns weak noise into dominant artifacts
• An exact inverse can therefore be numerically and statistically unstable
• therefore we would want to target those zero points

Pseudo-inverse filter

In practice we tackle this by defining a generalized (or pseudo-) inverse filter:

$$H^{-}(u,v)=\{\begin{array}{ll}\frac{1}{H(u,v)},& |H(u,v)|>\epsilon \\ 0,& \text{otherwise}\end{array}$$

• The slide example uses $\epsilon =0.05$
• Restoring with different $\epsilon$ gives full inverse filter, cut off 40%, cut off 70%, cut off 85%

Reading the artifacts

• Ringing indicates sharp frequency truncation or boundary mismatch
• Grain amplification indicates division by small $|H|$
• A sharper image is not necessarily a more accurate restoration
• The noise amplification problem is better addressed by Wiener Filtering