A class of regularizations based on nonlinear isotropic diffusion for inverse problems

TL;DR

This paper introduces an adaptive nonlinear isotropic diffusion regularization method for inverse problems, improving reconstruction quality for piecewise smooth solutions by incorporating prior edge information and addressing non-convexity issues.

Contribution

It develops an adaptive NID regularization technique that varies during iterations, enhancing inverse problem solutions with prior edge information and non-convex functional management.

Findings

Convergence and well-posedness are theoretically established.

Validated through CT simulation experiments.

Improves reconstruction of piecewise smooth images.

Abstract

Building on the well-known total-variation (TV), this paper develops a general regularization technique based on nonlinear isotropic diffusion (NID) for inverse problems with piecewise smooth solutions. The novelty of our approach is to be adaptive (we speak of A-NID) i.e. the regularization varies during the iterates in order to incorporate prior information on the edges, deal with the evolution of the reconstruction and circumvent the limitations due to the non-convexity of the proposed functionals. After a detailed analysis of the convergence and well-posedness of the method, this latter is validated by simulations perfomed on computerized tomography (CT).