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Generalized cross‐validation as a stopping rule for the Richardson‐Lucy algorithm

The Richardson‐Lucy (R‐L) algorithm has been widely used to restore degraded astronomical images. This algorithm is nothing more than the expectation‐maximization (EM) algorithm applied to Poisson data. The R‐L method is iterative in nature and converges to a (possibly local) maximum of the likelihood function. Unfortunately, because of the ill‐conditioned nature of the problem, this maximum likelihood estimate may actually be a very poor restoration. One way to prevent degradation of the restoration is to stop the iteration before it reaches convergence. A number of methods have been proposed for determining the optimal stopping point‐the point that provides the best trade‐off between restoring the image and amplifying the noise. Cross‐validation (CV) has recently been proposed as an advantageous method for determining the optimal stopping point. We propose a different form of CV based on generalized cross‐validation (GCV) that overcomes some of the difficulties of CV. We derive a GCV‐based criterion for the R‐L algorithm that can be efficiently evaluated at each iteration. We present examples displaying the power of the stopping rule and discuss the abilities and shortcomings of the method.