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Comments



Next: Code II: Unknown Up: Image Restorations of High Previous: Construction of Algorithms

Comments

The use and performance of Code I will perhaps be clarified by the following remarks:

Initialization

The positions of the designated point sources () must be given as well as initial amplitudes for and . These are taken to be constant and such that = = .

Normalization

The integrated flux of the point sources () plus that of the distributed emission () is constrained to equal that of the observed image (). The allocation of flux between these two channels (initially 1:1, see above) is adjusted iteratively as is maximized.

Regularization

Applies only to the distributed emission. This implies that the converged point-source amplitudes are Maximum Likelihood estimates relative to the highly flexible model of background emission provided by . This latter aspect is to be compared with the mathematically simple background models used in standard photometric software packages.

Spatially-Variant PSFs

As implemented, Code I assumes a spatially-invariant PSF but only in order to use FFTs. As with the R-L algorithm, this assumption is not otherwise necessary.

Point Sources

These are modelled as single active pixels in a sea of zeroed pixels. If more accurate centering is necessary, sub-pixelation is readily introduced (Lucy and Baade 1989; White 1990). Such representations of delta functions facilitate the use of discrete FFTs and lead to simple code. Nevertheless, for highest accuracy, continuous positions should be used.

Floating Default

A ME restoration with uniform prior gives the most probable distribution of photons in the restored image for a specified goodness-of-fit to the observed image. Although intuitively appealing, this solution, as noted earlier, is photometrically biased as a result of redistributing photons from high to low intensity regions. Moreover, when the PSF is compact, this redistribution is physically inconsistent with the independence of different ``picture elements.'' However, for a floating default computed with a compact kernel, this bias is greatly reduced (e.g., Lucy 1994) as is this inconsistency.

Because the entropic form with floating default is probably not derivable by combinatorial arguments, it should be regarded as a useful mathematical device and not ascribed a more fundamental status than other regularization functions (Titterington 1985).

Non-Negativity

As implemented, Code I maximizes subject to the constraints for all and for all . There would be some merit in changing the second constraint to for all . This would then allow , which, though unphysical, avoids bias when estimating magnitudes near the noise limit, as happens, for example, in following the fading of a SN. An algorithm that achieves this has been constructed.

Non-Designated Sources

Given the resolution limit imposed on distributed emission, the question arises as to how well point sources are represented that are not designated as such. This is investigated in Fig. 1 where all emission derives from point sources but only one - the brightest - is designated as such. Despite the resolution limit, the restored background () shows sharp peaks at the positions of several stars. Accordingly, this model of the background could be used to identify and locate additional point sources and thus form a step in an iterative procedure that maximally decomposes the crowded field into individual stars, whose magnitudes are then derived from the final solution for .



Next: Code II: Unknown Up: Image Restorations of High Previous: Construction of Algorithms


rlw@sundog.stsci.edu
Mon Apr 18 15:23:11 EDT 1994