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Reversible jump Markov chain Monte Carlo for Bayesian deconvolution of point sources

In this article, we address the problem of Bayesian deconvolution of point sources in nuclear imaging under the assumption of Poissonian statistics. The observed image is the result of the convolution by a known point spread function of an unknown number of point sources with unknown parameters. To detect the number of sources and estimate their parameters we follow a Bayesian approach. However, instead of using a classical low level prior model based on Markov random fields, we prose a high-level model which describes the picture as a list of its constituent objects, rather than as a list of pixels on which the data are recorded. More precisely, each source is assumed to have a circular Gaussian shape and we set a prior distribution on the number of sources, on their locations and on the amplitude and width deviation of the Gaussian shape. This high-level model has far less parameters than a Markov random field model as only s small number of sources are usually present. The Bayesian model being defined, all inference is based on the resulting posterior distribution. This distribution does not admit any closed-form analytical expression. We present here a Reversible Jump MCMC algorithm for its estimation. This algorithm is tested on both synthetic and real data.