A New Steplength Selection for Scaled Gradient Methods with Application to Image Deblurring

Abstract

Gradient methods are frequently used in large scale image deblurring problems since they avoid the onerous computation of the Hessian matrix of the objective function. Second order information is typically sought by a clever choice of the steplength parameter defining the descent direction, as in the case of the wellknown Barzilai and Borwein rules. In a recent paper, a strategy for the steplength selection approximating the inverse of some eigenvalues of the Hessian matrix has been proposed for gradient methods applied to unconstrained minimization problems. In the quadratic case, this approach is based on a Lanczos process applied every m iterations to the matrix of the gradients computed in the previous m iterations, but the idea can be extended to a general objective function. In this paper we extend this rule to the case of scaled gradient projection methods applied to constrained minimization problems, and we test the effectiveness of the proposed strategy in image deblurring problems in both the presence and the absence of an explicit edge-preserving regularization term.

A Variable Metric Forward-Backward Method with Extrapolation

“…In this section, borrowing the ideas in [10], we prove that the iterates {x (k) } converge to a minimizer of F , when the parameters sequences are chosen as in (27) with a > 2 and the scaling matrices sequence satisfies some additional assumption. Before giving the main result, we need to prove some technical lemmas.…”

“…Finally, the sequence {β k } employed in the definition of the extrapolation step for both FISTA and SFBEM has been chosen as in (27) with a = 2.1 in order to ensure the convergence of the sequence of the iterates.…”

“…F (x (k) ) − F * = O 1 k (see [3,8]). However, in spite of the theoretical convergence rate, a suitable combination of the stepsize parameter α k and the scaling matrix D k can allow method (4) to reach practical performances which are comparable with those of (2)-(3) [8,27].…”

On the convergence of a linesearch based proximal-gradient method for nonconvex optimization

“…The parameter µ bounding the diagonal entries of D k is set to 10 10 . Once computed the matrix D k , the stepsize parameter α k is chosen using a recent strategy proposed in [52] and based on the approximation of the eigenvalues of the Hessian matrix of the objective function by means of a Lanczos-like process (see also [43] for more details in the unconstrained case). In our problem, for a fixed positive integer m (in our experiments we consider m = 3), one has to: a) Define the matrices…”

A comparison of edge-preserving approaches for differential interference contrast microscopy

“…Future work will concern the application of the proposed strategies to real acquisitions and the reformulation of the minimization problem in terms of the specimen's transmission function e −iφ . This would lead to a standard regularized least-squares problem restricted to a nonconvex feasible set, which would require a generalization of the (VM)ILA approach able to account for nonconvex projections and to exploit the steplength selection rule proposed by Fletcher [25] in the presence of constraints [45,46].…”