Nonconvex and nonsmooth total variation regularization method for diffuse optical tomography based on RTE *

Abstract

Abstract Diffuse optical tomography with near-infrared light is a promising technique for noninvasive study of the functional characters of human tissues. Mathematically, it is a seriously ill-posed parameter identification problem. For the purpose of better providing both segmentation and piecewise constant approximation of the underlying solution, nonconvex nonsmooth total variation based regularization functional is considered in this paper. We first give a theoretical study on the well-posedness of solutions corresponding to this minimization problem in the Banach space of piecewise constant functions. Moreover, our theoretical results show that the minimizers corresponding to a sequence nonconvex nonsmooth potential functions which converge to the 0–1 functions, can be used to approximate the solution to the weak Mumford–Shah regularization. Then from the numerical side, we propose a double graduated nonconvex Gauss–Newton algorithm to solve this nonconvex nonsmooth regularization. All illustrations and numerical experiments give a flavor of the possibilities offered by the minimizers of the proposed algorithm.