Regularization of the Factorization Method applied to diffuse optical tomography

TL;DR

This paper introduces a regularized Factorization Method to improve stability and accuracy in imaging unknown regions within diffuse optical tomography, especially when dealing with compact data operators.

Contribution

It develops a new regularization technique for the Factorization Method, enhancing its applicability to diffuse optical tomography with minimal prior information.

Findings

The regularized method effectively images internal structures.

Numerical examples demonstrate improved stability.

Applicable to two-dimensional diffuse optical tomography.

Abstract

In this paper, we develop a new regularized version of the Factorization Method for positive operators mapping a complex Hilbert Space into it's dual space. The Factorization Method uses Picard's Criteria to define an indicator function to image an unknown region. In most applications, the data operator is compact which gives that the singular values can tend to zero rapidly which can cause numerical instabilities. The regularization of the Factorization Method presented here seeks to avoid the numerical instabilities in applying Picard's Criteria. This method allows one to image the interior structure of an object with little a priori information in a computationally simple and analytically rigorous way. Here we will focus on an application of this method to diffuse optical tomography where will prove that this method can be used to recover an unknown subregion from the Dirichlet-to-Neumann mapping. Numerical examples will be presented in two dimensions.