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Nested iterative algorithms for convex constrained image recovery problems. (English) Zbl 1186.68520

Summary: The objective of this paper is to develop methods for solving image recovery problems subject to constraints on the solution. More precisely, we will be interested in problems which can be formulated as the minimization over a closed convex constraint set of the sum of two convex functions \(f\) and \(g\), where \(f\) may be nonsmooth and \(g\) is differentiable with a Lipschitz-continuous gradient. To reach this goal, we derive two types of algorithms that combine forward-backward and Douglas-Rachford iterations. The weak convergence of the proposed algorithms is proved. In the case when the Lipschitz-continuity property of the gradient of \(g\) is not satisfied, we also show that, under some assumptions, it remains possible to apply these methods to the considered optimization problem by making use of a quadratic extension technique. The effectiveness of the algorithms is demonstrated for two wavelet-based image restoration problems involving a signal-dependent Gaussian noise and a Poisson noise, respectively.

MSC:

68U10 Computing methodologies for image processing
65K10 Numerical optimization and variational techniques
65N21 Numerical methods for inverse problems for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65F22 Ill-posedness and regularization problems in numerical linear algebra
90C25 Convex programming
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