Asymptotic Imaging of Perfectly Conducting Cracks

Abstract

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.

Structure Analysis of Single- And 2 Multi-Frequency Subspace Migrations in 3 Inverse Scattering Problems

“…Based on recent works in [4,6,19,20,24], it has been shown that subspace migration offers better results than the MUSIC and Kirchhoff migration. Specially, subspace migration can be applied to limited-view inverse scattering problems.…”

“…To find a good initial guess, alternative non-iterative reconstruction algorithms have been developed, such as the MUltiple SIgnal Classification (MUSIC)-type algorithm [5,23,25], linear sampling method [8,12], topological derivative strategy [3,6,18,21,22], linear-δ, vector and multipolarized approaches [29,30], and the multifrequency based algorithm such as Kirchhoff and subspace migrations [2,4,15,19,20,24]. Among them, although the multi-frequency based subspace migration has exhibited potential as a non-iterative imaging technique, a mathematical identification of its structure needs to be performed for its heuristical applications, which is the motivation behind.…”

A Topological Derivative Based Non-Iterative Electromagnetic Imaging of Perfectly Conducting Cracks

“…However, this method requires derivation of a complex Fréchet derivative, incurs large computational costs, and provides only a good initial guess regarding successful performance. Nevertheless, many practical experiments require starting with a good initial guess so alternative non-iterative imaging methods continue to be developed; e.g., the Multiple SIgnal Classification (MUSIC) algorithm [2], [3], linear sampling method [4], [5], and Kirchhoff migration [6], [7]. These appear to be fast and robust, and easily extend to the imaging of multiple cracks, but they still require a large number of incident fields with various directions and corresponding scattered field data.…”

Numerical Solution of the Scattering Problem for Acoustic Waves by a Two-Sided Crack in 2-Dimensional Space

“…But their approach seems not clear how to be applied for different impedances. On the other hand, we would like to point out the recent developments in connection with our studied problem and suggested techniques, specially for small cracks, see, e.g., [1,2].…”