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