Restoring a WF/PC Frame with a Space-Variant PSF

Let us now turn to the other main topic of this contribution, namely an algorithm for restoring a full WF/PC frame (or undersampled multi-frames) with its SV-PSF. The well-known ``brute-force'' SV-PSF restoration method is the ``sectioned'' algorithm (Trussel &Hunt 1978a, 1978b) which works as follows:

1.

Subdivide the observation into a regular set of patches (the ``sections'') with data .

2.

Restore each section under the hypothesis that there is a PSF which is space-invariant (SI) within patch .

3.

Sew the M restorations together.

The main problem with the simple sectioned approach is the danger of discontinuities occurring in the restoration at the patch boundaries, particularly when the PSF is asymmetric and changes its morphology, as in the case at least for WF/PC-1.

Here is an elegant way out: if one considers the patches as ``independent'' overlapping observations of the same field, each with its own space-invariant (SI) PSF, then one can simply use the Lucy-Hook ``co-addition'' algorithm to construct a single restored image from the multiple data patches . A spatially variable, ``fuzzy'' weighting function can be used in order to combine the correction factors belonging to the different patches

It is obvious that the formula above may be generalized to multiple input data frames.

Presently, it is an open question which weighting function to best use. A ``hard'' hat-shaped function may again lead to the problem of potential discontinuities at the section boundaries. Triangular or cosine bell-shaped weighting functions are better behaved in that respect. One could possibly ``derive'' a weighting function by cross-correlating a series of interpolating PSFs and use the correlation coefficient as a guide to the spatial variability of . Or one might apply a minimum mean-square-error criterion to optimize the choice of the functional form of the .

The proposed SV-PSF algorithm can be generalized in various ways. One could adapt the spatial density (and corresponding size) of the sections to the local signal-to-noise () level, since it does not make sense to restore low regions very accurately. Another way of saving CPU-time might be to use a denser patch grid where the PSF changes more rapidly, e.g., at the frame edges, or in areas of increased astronomical interest (White, pers. comm.).

Of course, the success of the proposed SV-PSF algorithm hinges upon the ability to produce suitable PSFs. Whether one has to rely on theoretical models or use one of the ``blind deconvolution'' PSF-estimation techniques (Lucy 1994) is currently an open question.

A final remark: The proposed SV-PSF algorithm is ideally suited to a coarse-grained vector-parallel computer architecture such as TMC's Connection Machine 5.