Parameter Selection by GCV and a $\chi^2$ test within Iterative Methods for $\ell_1$-regularized Inverse Problems

TL;DR

This paper develops methods to automatically select regularization parameters within iterative algorithms for sparse inverse problems, improving solution quality in image deblurring tasks by extending GCV and $\,\chi^2$ techniques.

Contribution

It extends GCV and $\,\chi^2$ methods for inner regularization problems in iterative schemes, enabling automatic parameter selection during the process.

Findings

Automatic parameter selection improves deblurring results.

Early iteration parameter estimation is effective for convergence.

Extensions of $\,\chi^2$ accommodate overdetermined regularization operators.

Abstract

$\ell_1$ regularization is used to preserve edges or enforce sparsity in a solution to an inverse problem. We investigate the Split Bregman and the Majorization-Minimization iterative methods that turn this non-smooth minimization problem into a sequence of steps that include solving an $\ell_2$-regularized minimization problem. We consider selecting the regularization parameter in the inner generalized Tikhonov regularization problems that occur at each iteration in these $\ell_1$ iterative methods. The generalized cross validation and $\chi^2$ degrees of freedom methods are extended to these inner problems. In particular, for the $\chi^2$ method this includes extending the $\chi^2$ result for problems in which the regularization operator has more rows than columns, and showing how to use the $A-$weighted generalized inverse to estimate prior information at each inner iteration. Numerical experiments for image deblurring problems demonstrate that it is more effective to select the regularization parameter automatically within the iterative schemes than to keep it fixed for all iterations. Moreover, an appropriate regularization parameter can be estimated in the early iterations and used fixed to convergence.