Non-Linear Inverse Scattering via Sparsity Regularized Contrast Source Inversion

Two compressive sensing inspired approaches for the solution of non-linear inverse scattering problems are introduced and discussed. Differently from the sparsity promoting approaches proposed in most of the papers published in the literature, the two methods here tackle the problem in its full non-linearity, by adopting a contrast source inversion scheme. In the first approach, the <inline-formula><tex-math notation="LaTeX">${\ell _1}$</tex-math> </inline-formula>-norm of the unknown is added as a weighted penalty term to the contrast source cost functional. The second, and (to the best of our knowledge) completely original, approach enforces sparsity by constraining the solution of the non-linear problem into a convex set defined by the <inline-formula><tex-math notation="LaTeX">${\ell _1}$</tex-math></inline-formula>-norm of the unknown. A numerical assessment against a widely used benchmark example (the “Austria” profile) is given to assess the capabilities of the proposed approaches. Notably, the two approaches can be applied to any kind of basis functions and they can successfully tackle both reduced number of data (with respect to Nyquist sampling) and/or overcomplete dictionaries.