Numerical treatment of the Mumford-Shah model for the inversion and segmentation of X-ray tomography data

The goal of this work is to identify a density function of a physical body from a given X-ray data. The mathematical relation between parameter and data is described by the Radon transform. We propose a piecewise smooth Mumford-Shah model for the simultaneous inversion and segmentation of the tomography data. In our approach the functional variable is eliminated by solving a classical variational problem for each fixed geometry. The solution is then inserted in the Mumford-Shah cost functional leading to a geometrical optimization problem for the singularity set. The resulting shape optimization problem is solved using shape sensitivity calculus and propagation of shape variables in the level-set form. The optimality system for the fixed geometry has the form of a coupled system of integro-differential equations on variable and irregular domains. A new finite difference method-based approach for the solution of the optimality system is presented. Here a standard five-point stencil is used on regular points of an underlying uniform grid and modifications of the standard stencil are made at points close to the boundary. The optimality system is solved iteratively. Numerical experiments are presented.