Thanks Jack - very interesting and a bit scary!

Robert,

The approach that Jack takes (in the frequency domain) is pretty close to what I do (in the spatial domain). We both assume (based on both theory and empirical evidence) that the cascade of blur sources will usually result in a Gaussian type of PSF.

Jack takes a medium response (MTF50) as pivot point on the actual MTF curve, and calculates the corresponding MTF (at that point) of a pure Gaussian blur function, he calculates the required sigma. In principle that's fine, although one might also try to find a sigma that minimizes the absolute difference between the actual MTF and that of the pure Gaussian over a wider range. Although it's a reasonable single point optimization, maybe MTF50 is not the best pivot point, maybe e.g. MTF55 or MTF45 would give an overall better match, who knows.

My approach is also trying to fit a Gaussian (Edge-Spread function) to the actual data, but does so on two points (10% and 90% rise) on the edge profile in the spatial domain. That may result in a slightly different optimization, e.g. in case of veiling glare which raises the dark tones more than the light tones, also on the slanted edge transition profile. My webtool attempts to minimize the absolute difference between

*the entire* edge response and the Gaussian model. It therefore attempts to make a better overall edge profile fit, which usually is most difficult for the dark edge, due to veiling glare which distorts the Gaussian blur profile. That also gives an indication of how much of a role the veiling glare plays in the total image quality, and how it complicates a successful resolution restoration because it reduces the lower frequencies of the MTF response. BTW, Topaz Detail can be used to adjust some of that with the large detail control.

I thought I would check out what happens using Bart's deconvolution, based on the correct radius and then increasing it progressively, and this is what happens:

The left-hand image has the correct radius of 1.06, the one at the right has a radius of 4. As you can see, all that happens is that there is a significant overshoot on the MTF at 4 (this overshoot increases progressively from a radius of about 1.4).

The MTF remains roughly Gaussian unlike the one in your article … and there is no sudden transition around the Nyquist frequency or shoot off to infinity as the radius increases. Are these effects due to division by zero(ish) in the frequency domain … or to something else?

Jack's model is purely mathematical, and as such allows to predict the effects of full restoration in the frequency domain. However, anything that happens above the Nyquist frequency (0.5 cycles/pixel) folds back (mirrors) to the below Nyquist range and manifests itself as aliasing in the spatial domain (so you won't see it as an amplification above Nyquist in the actually sharpened version, but as a boost below Nyquist).

Also, since the actual signal's MTF near the Nyquist frequency is very low, there is little detail (with low S/N ratio) left to reconstruct, so there will be issues with noise amplification. MTF curves need to be interpreted, because actual images are not the same as simplified mathematical models (simple numbers do not tell the whole story, they just show a certain aspect of it, like spatial frequency response of a system in an MTF, and an separation of aliasing due to sub-pixel phase effects of fine detail).

Cheers,

Bart