Topological Imaging Methods for the Iterative Detection of Multiple Impedance Obstacles

In this paper, we investigate shape inversion algorithms based on the computation of iterated topological derivatives for the detection of multiple particles coated by a complex surface impedance in two- and three-dimensional acoustic media. New closed-form formulae for the topological derivative of the misfit functional are derived when an approximate set of unknown particles has already been recovered. Proofs rely on the computation of shape derivatives followed by the topological asymptotic analysis of a boundary integral equation formulation of the forward and adjoint problems. The relevance of the theoretical results is illustrated by various 2D and 3D experiments using monochromatic imaging algorithms either fully or partially based on topological derivatives.