Performance analysis of multi-frequency topological derivative for reconstructing perfectly conducting cracks

This paper concerns a fast, one-step iterative technique of imaging extended perfectly conducting cracks with Dirichlet boundary condition. In order to reconstruct the shape of cracks from scattered field data measured at the boundary, we introduce a topological derivative-based electromagnetic imaging functional operated at several nonzero frequencies. The structure of the imaging functionals is carefully analyzed by establishing relationships with infinite series of Bessel functions for the configurations of both symmetric and non-symmetric incident field directions. Identified structure explains why the application of incident fields with symmetric direction operated at multiple frequencies guarantees a successful reconstruction. Various numerical simulations with noise-corrupted data are conducted to assess the performance, effectiveness, robustness, and limitations of the proposed technique.