A discrepancy principle for Poisson data

In applications of imaging science, such as emissiontomography, fluorescence microscopy and optical/infrared astronomy,image intensity is measured via the counting of incident particles(photons, $\gamma$-rays, etc.). Fluctuations in the emission-countingprocess can be described by modeling the data as realizations of Poissonrandom variables (Poisson data). A maximum likelihood approach forimage reconstruction from Poisson data was proposed in the mideighties. Since the consequent maximization problem is, in general,ill-conditioned, various kinds of regularization were introduced in theframework of the so-called Bayesian paradigm. A modification ofthe well-known Tikhonov regularization strategy results, thedata-fidelity function being a generalized Kullback-Leiblerdivergence. Then a relevant issue is to find rules for selectinga proper value of the regularization parameter. In this paperwe propose a criterion, nicknamed discrepancy principle forPoisson data, that applies to both denoising and deblurring problemsand fits quite naturally the statistical properties of the data.The main purpose of the paper is to establish conditions, on thedata and the imaging matrix, ensuring that the proposed criteriondoes actually provide a unique value of the regularization parameterfor various classes of regularization functions. A few numericalexperiments are performed to demonstrate its effectiveness. Moreextensive numerical analysis and comparison with other proposedcriteria will be the object of future work.