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

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.