On the identification of piecewise constant coefficients in optical diffusion tomography by level set

Abstract

In this paper, we propose a level set regularization approach combined with a split strategy for the simultaneous identification of piecewise constant diffusion and absorption coefficients from a finite set of optical tomography data (Neumann-to-Dirichlet data). This problem is a high nonlinear inverse problem combining together the exponential and mildly ill-posedness of diffusion and absorption coefficients, respectively. We prove that the parameter-to-measurement map satisfies sufficient conditions (continuity in the L 1 topology) to guarantee regularization properties of the proposed level set approach. On the other hand, numerical tests considering different configurations bring new ideas on how to propose a convergent split strategy for the simultaneous identification of the coefficients. The behavior and performance of the proposed numerical strategy is illustrated with some numerical examples.

A regularization method based on level sets and augmented Lagrangian for parameter identification problems with piecewise constant solutions

“…In [2] numerical experiments using SLS were shown for this parameter identification problem; we use here the same numerical setting as in [2] and compare there numerical results with the ones obtained in this article. It is worth mentioning that in [2] a splitting strategy was used in the reconstruction of the unknown pair of parameters (a similar splitting strategy is also used here). However, due to the augmented Lagrangian formulation used here, the number of iterations required for the convergence of our method is much smaller than the one in [2].…”

“…Here we propose a primal-dual iterative method for computing approximate solutions of (5); convergence of this algorithm is proven (see theorem 7). -We apply our numerical algorithm to a 2D diffuse optical tomography (DOT) benchmark problem [2,4,23]. In [2] numerical experiments using SLS were shown for this parameter identification problem; we use here the same numerical setting as in [2] and compare there numerical results with the ones obtained in this article.…”

“…-We apply our numerical algorithm to a 2D diffuse optical tomography (DOT) benchmark problem [2,4,23]. In [2] numerical experiments using SLS were shown for this parameter identification problem; we use here the same numerical setting as in [2] and compare there numerical results with the ones obtained in this article. It is worth mentioning that in [2] a splitting strategy was used in the reconstruction of the unknown pair of parameters (a similar splitting strategy is also used here).…”

“…in Ω (see lemma 2 (ii) below). Consequently, the inverse problem of solving (1), with data satisfying (2), is equivalent to the system of operator equations F(P pc (φ)) = y δ , K(φ) = 0.…”

“…In this manuscript, solutions φ δ α , α > 0 of (5) are computed using an augmented Lagrangian method. Furthermore, such approach defines a regularization method suitable for the inverse problem discussed in (1) and (2) with piecewise constant solutions (see section 3).…”