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The method adopted might be described as image restoration incorporating designated sources having specified shape and known positions. For astronomical images, the designated sources are naturally taken to be point sources (stars, QSOs, AGNs), so the specified shape is the delta function. Clearly, an observer can commonly, with high confidence, identify many objects in an image as point sources. Accordingly, it makes sense to develop codes that use this extra information and, at least for these designated point sources, thereby eliminate the ringing associated with failed attempts to recover delta functions.
In 1-D vector notation, the adopted model for , the
intensity distribution on the celestial sphere, is
Here represents distributed emission and
the
designated point sources. This latter representation is achieved by
defining
to be zero everywhere except at the locations
(
) of
point sources.
This model of the object can be mapped onto the image plane using the
equation of image formation, and the resulting prediction for
the intensity distribution in the image plane can then be compared to
the observed distribution
. The posed image
restoration problem is thus to determine
and
in
order to achieve an optimum fit of
to
.
An obvious first thought is to adopt the R-L procedure and
determine and
with an iterative scheme that
asymptotically yields Maximum Likelihood estimates subject to the
appropriate non-negativity and normalization constraints. But this
results in indeterminacy since an individual star can be represented
either in the
vector or by the corresponding single-pixel
peak added to the
vector, or indeed by any linear combination
of these extremes.
To eliminate this indeterminacy, we must impose a constraint on
that excludes single-pixel peaks. In other words, we must
impose a resolution limit on
that is somewhat larger than the
limit implied by the numerical discretization. This is achieved by
introducing regularization into the optimization problem for
determining
and
. Of the many possible
regularization terms (see, e.g., Titterington 1985), we choose the
entropic form
where and where the floating default
(Horne 1985, Lucy 1994)
Here the resolution kernel is a bell-shaped function
whose width is the required resolution limit. Structure in
on a smaller scale results in a decrease in the entropy
and so is
disfavored.