Elastic energy regularization for inverse obstacle scattering problems
For inverse scattering problems, this method extends reconstruction to general shapes without star-shaped assumption, but the results are incremental with no quantitative SOTA comparison.
The paper introduces a shape manifold approach for inverse obstacle scattering that allows reconstruction of non-star-shaped curves, using bending energy for regularization and Möbius energy to avoid self-intersections. Numerical examples demonstrate feasibility.
By introducing a shape manifold as a solution set to solve inverse obstacle scattering problems we allow the reconstruction of general, not necessarily star-shaped curves. The bending energy is used as a stabilizing term in Tikhonov regularization to gain independence of the parametrization. Moreover, we discuss how self-intersections can be avoided by penalization with the Möbius energy and prove the regularizing property of our approach as well as convergence rates under variational source conditions. In the second part of the paper the discrete setting is introduced, and we describe a numerical method for finding the minimizer of the Tikhonov functional on a shape-manifold. Numerical examples demonstrate the feasibility of reconstructing non-star-shaped obstacles.