Abstract. We consider the problem of locating perfectly conducting cracks and estimating their geometric features from multi-static response matrix measurements at a single or multiple frequencies. A main objective is to design specific crack detection rules and to analyze their receiver operating characteristics and the associated signal-to-noise ratios. In this paper we introduce an analytic framework that uses asymptotic expansions which are uniform with respect to the wavelength-to-crack size ratio in combination with a hypothesis test based formulation to construct specific procedures for detection of perfectly conducting cracks. A central ingredient in our approach is the use of random matrix theory to characterize the signal space associated with the multi-static response matrix measurements. We present numerical experiments to illustrate some of our main findings.

The imaging of a thin inclusion, with dielectric and/or magnetic contrasts with respect to the embedding homogeneous medium, is investigated. A MUSIC-type algorithm operating at a single time-harmonic frequency is developed in order to map the inclusion (that is, to retrieve its supporting curve) from scattered field data collected within the multi-static response matrix. Numerical experiments carried out for several types of inclusions (dielectric and/or magnetic ones, straight or curved ones), mostly single inclusions and also two of them close by as a straightforward extension, illustrate the pros and cons of the proposed imaging method.

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In this paper, we consider cracks with Dirichlet boundary conditions. We first derive an asymptotic expansion of the boundary perturbations that are due to the presence of a small crack. Based on this formula, we design a noniterative approach for locating a collection of small cracks. In order to do so, we construct a response matrix from the boundary measurements. The location and the length of the crack are estimated, respectively, from the projection onto the noise space and the first significant singular value of the response matrix. Indeed, the direction of the crack is estimated from the second singular vector. We then consider an extended crack with Dirichlet boundary conditions. We rigorously derive an asymptotic expansion for the boundary perturbations that are due to a shape deformation of the crack. To reconstruct an extended crack from many boundary measurements, we develop two methods for obtaining a good guess. Several numerical experiments show how the proposed techniques for imaging small cracks as well as those for obtaining good initial guesses toward reconstructing an extended crack behave.

Motivated from the application area of imaging of anti-personnel mines completely embedded in the homogeneous medium, the problem of noniterative imaging of thin dielectric inclusions buried within a dielectric halfspace is considered. For that purpose, an imaging algorithm operated at several frequencies is proposed. It is based on the asymptotic expansion formula of the scattering amplitude in the presence of the inclusions. Various numerical examples illustrate how the method behaves.

The topological derivative-based non-iterative imaging algorithm has demonstrated its applicability in limited-aperture inverse scattering problems. However, this has been confirmed through many experimental simulation results, and the reason behind this applicability has not been satisfactorily explained. In this paper, we identify the mathematical structure and certain properties of topological derivatives for the imaging of two-dimensional crack-like thin penetrable electromagnetic inhomogeneities that are completely embedded in a homogeneous material. To this end, we establish a relationship with an infinite series of Bessel functions of integer order of the first kind. Based on the derived structure, we discover a necessary condition for applying topological derivatives in limited-aperture inverse scattering problems, and thus confirm why topological derivatives can be applied. Furthermore, we analyze the structure of multi-frequency topological derivative, and identify why this improves the single-frequency topological derivative in limited-aperture inverse scattering problems. Various numerical simulations are conducted with noisy data, and the results support the derived structure and exhibit certain properties of single- and multi-frequency topological derivatives.

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