A general framework for nonlinear regularized Krylov-based image restoration

Abstract. This paper introduces a new approach to computing an approximate solution of Tikhonov-regularized large-scale ill-posed problems with a general nonlinear regularization operator. The iterative method applies a sequence of projections onto generalized Krylov subspaces using a semi-implicit approach to deal with the nonlinearity in the regularization term. A suitable value of the regularization parameter is determined by the discrepancy principle. Computed examples illustrate the performance of the method applied to the restoration of blurred and noisy images

On Krylov projection methods and Tikhonov regularization

In the framework of large-scale linear discrete ill-posed problems, Krylov projection methods represent an essential tool since their development, which dates back to the early 1950\u2019s. In recent years, the use of these methods in a hybrid fashion or to solve Tikhonov regularized problems has received great attention especially for problems involving the restoration of digital images. In this paper we review the fundamental Krylov-Tikhonov techniques based on Lanczos bidiagonalization and the Arnoldi algorithms. Moreover, we study the use of the unsymmetric Lanczos process that, to the best of our knowledge, has just marginally been considered in this setting. Many numerical experiments and comparisons of different methods are presented

A new steplength selection for scaled gradient methods with application to image deblurring

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 well-known 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 most recent m back gradients 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 non-negatively 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

Optimization Methods for Inverse Problems

Optimization plays an important role in solving many inverse problems. Indeed, the task of inversion often either involves or is fully cast as a solution of an optimization problem. In this light, the mere non-linear, non-convex, and large-scale nature of many of these inversions gives rise to some very challenging optimization problems. The inverse problem community has long been developing various techniques for solving such optimization tasks. However, other, seemingly disjoint communities, such as that of machine learning, have developed, almost in parallel, interesting alternative methods which might have stayed under the radar of the inverse problem community. In this survey, we aim to change that. In doing so, we first discuss current state-of-the-art optimization methods widely used in inverse problems. We then survey recent related advances in addressing similar challenges in problems faced by the machine learning community, and discuss their potential advantages for solving inverse problems. By highlighting the similarities among the optimization challenges faced by the inverse problem and the machine learning communities, we hope that this survey can serve as a bridge in bringing together these two communities and encourage cross fertilization of ideas.Comment: 13 page