Minimization and parameter estimation for seminorm regularization models with I -divergence constraints

Abstract

In this papers we analyze the minimization of seminorms L · on R n under the constraint of a bounded I-divergence D(b, H·) for rather general linear operators H and L. The I-divergence is also known as Kullback-Leibler divergence and appears in many models in imaging science, in particular when dealing with Poisson data. Often H represents, e.g., a linear blur operator and L is some discrete derivative or frame analysis operator. We prove relations between the the parameters of I-divergence constrained and penalized problems without assuming the uniqueness of their minimizers. To solve the I-divergence constrained problem we apply first-order primal-dual algorithms which reduce the problem to the solution of certain proximal minimization problems in each iteration step. One of these proximation problems is an I-divergence constrained least squares problem which can be solved based on Morosov's discrepancy principle by a Newton method. Interestingly, the algorithm produces not only a sequence of vectors which converges to a minimizer of the constrained problem but also a sequence of parameters which convergences to a regularization parameter so that the corresponding penalized problem has the same solution as our constrained one. We demonstrate the performance of various algorithms for different image restoration tasks both for images corrupted by Poisson noise and multiplicative Gamma noise.

Epigraphical projection and proximal tools for solving constrained convex optimization problems

“…Various strategies were proposed in order to address the first question [26,27,28,29,30], but the computational cost of these methods is often high, especially when several regularization parameters have to be set. Alternatively, it has been recognized for a long time that incorporating constraints directly on the solutions [31,32,33,34,35] instead of considering regularized functions may often facilitate the choice of the involved parameters. Indeed, in a constrained formulation, the constraint bounds are usually related to some physical properties of the target solution or some knowledge of the degradation process, e.g.…”

Epigraphical Projection for Solving Least Squares Anscombe Transformed Constrained Optimization Problems

“…The classical Total Variation objective function [20] is obtained, as a special case, when Φ is an ℓ 2,1 norm and L corresponds to a discrete gradient operator. Constrained models based on the I-divergence have been considered in [5,22], where in the second paper special attention was paid to the relation between the parameters of the constrained and the penalized problem via discrepancy principles. Note that recently penalized versus constrained problems in a rather general form were handled in [2].…”

An epigraphical convex optimization approach for multicomponent image restoration using non-local structure tensor

“…Such a formulation may be preferred to a regularized one since it has been recognized for a long time that incorporating constraints directly on the solution often facilitates the choice of the involved parameters [1,2,3,4,5]. Indeed, the constraint bounds are usually related to some physical properties of the target solution or to some knowledge of the degradation process, e.g.…”