The Iteratively Regularized Gau{\ss}-Newton Method with Convex Constraints and Applications in 4Pi-Microscopy

TL;DR

This paper introduces an iterative regularization method based on a Newton-type approach with convex constraints, specifically designed for solving nonlinear ill-posed equations, motivated by applications in 4Pi microscopy.

Contribution

It develops a new Newton-type iterative regularization method incorporating convex constraints and analyzes its convergence under data errors, with applications to 4Pi microscopy.

Findings

Method converges with decreasing data errors

Effective in semi-blind deconvolution for microscopy

Validated on simulated and experimental 3D data

Abstract

This paper is concerned with the numerical solution of nonlinear ill-posed operator equations involving convex constraints. We study a Newton-type method which consists in applying linear Tikhonov regularization with convex constraints to the Newton equations in each iteration step. Convergence of this iterative regularization method is analyzed if both the operator and the right hand side are given with errors and all error levels tend to zero. Our study has been motivated by the joint estimation of object and phase in 4Pi microscopy, which leads to a semi-blind deconvolution problem with nonnegativity constraints. The performance of the proposed algorithm is illustrated both for simulated and for three-dimensional experimental data.