The R-L algorithm (Richardson 1972; Lucy 1974) was initially derived from Bayes' theorem, but in fact its use is really justified by the subsequent discovery (Lucy 1974; Shepp and Vardi 1982) that every iteration increases the likelihood assigned to the observed image. Thus, in the usual case where no stars are designated point sources, the objective function that is asymptotically maximized by the R-L algorithm is
Knowing this, we can rewrite the R-L algorithm as an operation on .
The result for the increment
is
where the asterisk indicates that has been evaluated
by requiring that
Eq. (8) suggests an operational procedure for deriving potentially
effective algorithms: write down the function whose optimization
defines the restoration problem, then derive an iterative correction
scheme by applying Eq. (8).
With given by Eq. (4) and
by Eq. (2), the resulting algorithm
simplifies to
and
where is the R-L correction factor.