The most difficult practical problem to be solved when restoring HST images is usually not the choice of a restoration algorithm, but rather finding a good point-spread function (PSF). Even the most sophisticated algorithm cannot produce a decent restored image using a bad PSF, and even the least-sophisticated algorithm will do a pretty good job given a good PSF.
The problem is that the PSF changes as a function of wavelength, position in the camera field of view (for the WF/PC, WFPC2, and FOC with COSTAR), and time. There are enough variables that it is not practical to maintain a library of high-quality, observed PSFs covering all possibilities. Computed PSFs using programs such as Tiny Tim (Krist 1993) present a more attractive alternative because they can be computed at any wavelength and camera position, they are noise-free, and they can be computed on sub-sampled pixel grids. Unfortunately the PSFs computed with existing optical modeling programs are usually in relatively poor agreement with the observations, so that they are useful only for deconvolving observations with fairly low signal-to-noise ratios. We are pursuing a number of possible improvements for calculated PSFs, including more sophisticated optical models (Redding et al. 1994) and restoration methods that can adjust the PSF to get better restored images (see the discussion of blind deconvolution in § 6.)
Another promising approach is to use phase retrieval to develop better optical models for HST. In this paper I briefly introduce the topic of phase retrieval and show some results that have been obtained for HST images.