I would like to spend the rest of my time showing an example of the kind of dilemma one can get into when doing restorations. In this particular case we were dealing with an FOC ultraviolet image of the center of M31, and a PC V-band image of the same region, scaled, rotated, and aligned. We wanted to take the quotient of the two images, to look for color effects. But first, of course, we had to get the light back where it belongs, by doing restorations.
Notice that for this problem one cannot use the sophisticated methods that have been shown to produce such handsome restorations. The degree to which they restore depends on , and the two images would get sharpened by different amounts, producing an erroneous quotient.
In doing the restorations we tried to treat the two images identically (except of course for using its own PSF for each image). What we did was to run ``wiener'' on the noisy FOC image, and extract the filter that was generated in the process. Then we did Fourier deconvolutions of each image by hand, using that same filter on each. (This left the PC image less than optimally filtered, but it should assure that each image has the identical amount of blurring.) The result showed a small, well-marked central peak that certainly represents a real ultraviolet excess at the center of M31. Surrounding this peak is a large, round, shallow depression. M31 is elongated, but this depressed ``bowl'' is round. The only component in the problem that is round is the PSF. Does the bowl represent a real UV feature of M31, or is it possible that the bowl is just an artifact that comes from the restoration process? We have tried using somewhat changed PSFs, but we haven't been able to make the bowl go away. We are going to keep trying such experiments, but I certainly hope that some one who is more expert than I am can make the suggestion that I have been lacking so far.
I have described a specific problem here, but instead I can pose an abstract problem that, as far as I know, has never been addressed. It goes as follows: ``Images A and B show the same extended object with identical placement, but were taken under different circumstances. Each is accompanied by its own PSF. Image A has low , while B has high . Wanted: a good estimator of the true arithmetic difference, pixel by pixel, between A and B, properly corrected for the different blurring in the two images.'' This is not at all a rare situation, and I think the problem is worthy of consideration.