A deconvolution problem …

An interesting paper from astro-ph today: namely, Konrad et al. presenting a method to “estimate the lensing potential from massive galaxy clusters”, at the heart of which lies an application of the “Richardson-Lucy algorithm”.  As a neophyte in the field of image deconvolution I might incautiously remark that the latter seems little more than a textbook example of the iterative method for approximation of bounded linear operators (a la Petryshyn 1963).  Nevertheless, I should be said that in the context of the three stage methodology proposed by the authors the power of this technique is most effectively demonstrated.

Two issues of terminology in their Section 2 bother me though.  The first is the attribution of the relation f(x)K(y|x) = g(y)K'(x|y) to “Bayes’ theorem” in the case that the x and y do not represent random variables under consideration in the problem; that is, the f(x) and K(y|x) just happen to be normalised and the g(y) and K'(x|y) are so assumed as well.  Somehow this doesn’t sit well with me, perhaps in part because when this manipulation of conditional densities is applied in the usual “Bayesian” context the assumption of well-behavedness (here boundedness) of the likelihood function, K'(x|y), feels more naturally left as implicit.  But, from a straw poll of my colleagues, it appears I’m alone on this one.

The other (more bothersome) issue of terminology relates to the description of their chi(r) term, used to control the convergence behaviour of the operator approximation, as a “prior”.  With the Richardson-Lucy algorithm giving a sort of maximum likelihood, non-parametric solution for f(x) the chi(x) used here seems more like what would be the prior predictive” if the model was in fact parametric … ?

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