Penalized maximum likelihood image restoration with positivity constraints: multiplicative algorithms

Abstract

In this paper, we propose a general method to devise maximum likelihood penalized (regularized) algorithms with positivity constraints. Moreover, we explain how to obtain ‘product forms’ of these algorithms. The algorithmic method is based on Kuhn–Tucker first-order optimality conditions. Its application domain is not restricted to the cases considered in this paper, but it can be applied to any convex objective function with linear constraints. It is specially adapted to the case of objective functions with a bounded domain, which completely encloses the domain of the (linear) constraints. The Poisson noise case typical of this last situation and the Gaussian additive noise case are considered and they are associated with various forms of regularization functions, mainly quadratic and entropy terms. The algorithms are applied to the deconvolution of synthetic images blurred by a realistic point spread function similar to that of Hubble Space Telescope operating in the far-ultraviolet and corrupted by noise. The effect of the relaxation on the convergence speed of the algorithms is analysed. The particular behaviour of the algorithms corresponding to different forms of regularization functions is described. We show that the ‘prior’ image is a key point in the regularization and that the best results are obtained with Tikhonov regularization with a Laplacian operator. The analysis of the Poisson process and of a Gaussian additive noise leads to similar conclusions. We bring to the fore the close relationship between Tikhonov regularization using derivative operators, and regularization by a distance to a ‘default image’ introduced by Horne (Horne K 1985 Mon. Not. R. Astron. Soc. 213 129–41).

Forward-Backward Splitting with Bregman Distances

“…However, many problems in applications do not conform to these hypotheses, for example when and are Euclidean spaces and ψ is Boltzmann-Shannon entropy. This type of functions appears in many problems in image and signal processing, in statistics, and in machine learning [2,13,14,21,22,23]. Another difficulty in the implementation of (1.3) is that the operators (prox γ n ϕ ) n∈ are not always easy to evaluate.…”

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“…The aim of this iterative multiplicative algorithm is to minimize the negative log likelihood of data corrupted by a Gaussian noise process, that is the Euclidean distance between the noisy data and the convolutive model, subject to the non-negativity constraint of the successive estimates. The convergence of this algorithm was analyzed by De Pierro (1987) and in its relaxed form by Lantéri et al (2002).…”

“…The main drawback of the iterative algorithm, as well as all other non-regularized deconvolution algorithms, is that the deconvolution problem is an illposed problem (Bertero & Boccacci 1998), and as a consequence the reconstructed image becomes very noisy when the iteration number becomes too large. To avoid this problem, the classical ways of solving this problem are either to regularize explicitly (Lantéri et al 2002) the problem by introducing at some level a smoothness penalty, or to stop the iterative process before noise amplification that corresponds to a smoothing operation.…”

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“…5.1. About this second step, in a series of recent papers Lanteri et al [Lan01,Lan02] proposed a general approach, denoted as split gradient method (SGM), to the design of iterative methods for the minimization of a wide class of convex (and also non-convex) functionals of the following type:…”

“…Then, according to the split gradient method (SGM), proposed by Lanteri et al ([Lan01], [Lan02]), we write also the gradient of the regularization functional in a similar way:…”