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

¹ FaMAF-CIEM (CONICET), Universidad Nacional de Córdoba, Medina Allende s/n, X5000HUA, Córdoba, Argentina.
agnelli@famaf.unc.edu.ar
² Institute of Mathematics Statistics and Physics, Federal University of Rio Grande, Av. Italia km 8, 96201-900 Rio Grande, Brazil.
decezaromtm@gmail.com
³ Department of Mathematics, Federal University of St. Catarina, P.O. Box 476, 88040-900 Florianópolis, Brazil.
aleitao@mtm.ufsc.br; maicon.alves@ufsc.br

Received: 15 May 2015
Revised: 14 December 2015
Accepted: 14 January 2016

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 L¹ 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.