HyBR: A Method for Solving Ill-Posed Problems arising in Image Processing

Julianne Chung, Emory University

Problems arising in image processing applications are often ill-posed, meaning that noise in the data may give rise to significant errors in the computed solutions. Lanczos-hybrid regularization methods have been proposed as effective approaches for solving large-scale ill-posed inverse problems. Lanczos methods restrict the solution to lie in a Krylov subspace, but they are hindered by semi-convergence behavior, in that the quality of the solution first increases and then decreases. Hybrid methods apply a standard regularization technique, such as Tikhonov regularization, to the projected problem at each iteration. Thus, regularization in hybrid methods is achieved both by Krylov filtering and by appropriate choice of a regularization parameter at each iteration. We implement a weighted generalized cross validation (W-GCV) method for choosing the parameter. Using this method we demonstrate that the semi-convergence behavior of the Lanczos method can be overcome, making the solution less sensitive to the number of iterations.

Abstract Author(s): Julianne Chung and James Nagy