Regularization of the factorization method applied to 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 is 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.

Regularization of the Factorization Method with Applications to Inverse Scattering

“…In this section, we will discuss the theoretical frame work that was developed in [14] for the regularized factorization method. The analysis here generalizes the main result in [16].…”

“…a Gelfand triple of Hilbert spaces. Now in [14] it is proven that the operator A has a spectral decomposition provided that either X is a complex Hilbert space or the bilinear form (x, y) −→ y , Ax X×X * for any x, y ∈ X is symmetric. Under these assumptions we have that…”

“…Here we will assume that φ(t; α) : 0, λ 1 → R ≥0 satisfies that for all 0 < t ≤ λ 1 lim α→0 φ(t; α) = 1 and φ(t; α) ≤ C reg for all α > 0. Now, do to the fact that A is positive and compact we have that there is a bounded linear 'square root' operator denoted [14] for details. We also obtain that [14] which is proven in a similar was as Picard's Criteria (see for e.g.…”

“…Now, do to the fact that A is positive and compact we have that there is a bounded linear 'square root' operator denoted [14] for details. We also obtain that [14] which is proven in a similar was as Picard's Criteria (see for e.g. Theorem 1.28 of [5]).…”

“…In this paper, we will discuss a regularized version of the factorization method as well as it's applications scattering theory. We will briefly review the theoretical framework that was developed in [14]. The factorization method (see for e.g.…”