A sparsity regularization and total variation based computational framework for the inverse medium problem in scattering

We present a computational framework for the inverse medium problem in scattering, i. e. we look at discretization, reconstruction and numerical performance. The Helmholtz equation in two and three dimensions is used as a physical model of scattering. Point sources as well as plane waves are taken into account as incident fields. Further, near and far field measurements are considered. For the reconstruction of the medium, we set up a variational regularization scheme. The underlying paradigm is, roughly speaking, to minimize the discrepancy between the reconstruction and measured data while, at the same time, taking into account various structural a-priori information via suitable penalty terms. In particular, the involved penalty terms are designed to promote information expected in real-world environments. To this end, a combination of sparsity promoting terms, total variation, and physical bounds of the inhomogeneous medium, e. g. positivity constraints, is employed in the regularization penalty. A primal-dual algorithm is used to solve the minimization problem related to the variational regularization. The computational feasibility, performance and efficiency of the proposed approach is demonstrated for synthetic as well as experimentally measured data (from Institute Fresnel) in two and three dimensions.