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Performance analysis of maximum likelihood methods for regularization problems with nonnegativity constraints

In many numerical applications, for instance in image deconvolution, the nonnegativity of the computed solution is required. When a problem of deconvolution is formulated in a statistical frame, the recorded image is seen as the realization of a random process, where the nature of the noise is taken into account. This formulation leads to the maximization of a likelihood function which depends on the statistical property assumed for the noise. In this paper we revisit, under this unifying statistical approach, some iterative methods coupled with suitable strategies for enforcing nonnegativity and other ones which instead naturally embed nonnegativity. For all these methods we carry out a comparative study taking into account several performance indicators. The reconstruction accuracy, the computational cost, the consistency with the discrepancy principle (a common technique for guessing the best regularization parameter) and the sensitivity to this choice are compared in a simulation context, by means of an extensive experimentation on both 1D and 2D problems.