Won-Kwang Park

Imaging Schemes for Perfectly Conducting Cracks

We consider the problem of locating perfectly conducting cracks and estimating their geometric features from multistatic 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 multistatic response matrix measurements. We present numerical experiments to illustrate some of our main findings.

Asymptotic Imaging of Perfectly Conducting Cracks

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.

MUSIC-type imaging of a thin penetrable inclusion from its multi-static response matrix

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.

Electromagnetic MUSIC-type imaging of perfectly conducting, arc-like cracks at single frequency

We propose a non-iterative MUSIC (MUltiple SIgnal Classification)-type algorithm for the time-harmonic electromagnetic imaging of one or more perfectly conducting, arc-like cracks found within a homogeneous space R^2. The algorithm is based on a factorization of the Multi-Static Response (MSR) matrix collected in the far-field at a single, nonzero frequency in either Transverse Magnetic (TM) mode (Dirichlet boundary condition) or Transverse Electric (TE) mode (Neumann boundary condition), followed by the calculation of a MUSIC cost functional expected to exhibit peaks along the crack curves each half a wavelength. Numerical experimentation from exact, noiseless and noisy data shows that this is indeed the case and that the proposed algorithm behaves in robust manner, with better results in the TM mode than in the TE mode for which one would have to estimate the normal to the crack to get the most optimal results.

Reconstruction of thin electromagnetic inclusions by a level-set method

In this contribution, we consider a technique of electromagnetic imaging (at a single, non-zero frequency) which uses the level-set evolution method for reconstructing a thin inclusion (possibly made of disconnected parts) with either dielectric or magnetic contrast with respect to the embedding homogeneous medium. Emphasis is on the proof of the concept, the scattering problem at hand being so far based on a two-dimensional scalar model. To do so, two level-set functions are employed; the first one describes location and shape, and the other one describes connectivity and length. Speeds of evolution of the level-set functions are calculated via the introduction of Frechet derivatives of a least-square cost functional. Several numerical experiments on noiseless and noisy data as well illustrate how the proposed method behaves.

On the imaging of thin dielectric inclusions buried within a half-space

Motivated from the application area of imaging of anti-personnel mines completely embedded in the homogeneous medium, the problem of non-iterative imaging of thin dielectric inclusions buried within a dielectric half-space 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.

Topological derivative strategy for one-step iteration imaging of arbitrary shaped thin, curve-like electromagnetic inclusions

In this manuscript, a fast imaging of thin, curve-like electromagnetic inclusions completely hidden in the homogeneous domain with smooth boundaries is considered. By creating an electromagnetic inclusion of a small diameter and applying the asymptotic expansion formula in the existence of such an inclusion, the topological derivative is successfully derived. Based on this derivative, a one-step iteration imaging algorithm is designed by solving an adjoint problem. Various numerical experiments of single and multiple inclusions demonstrate the viability and limitation of the designed algorithm.

Multi-frequency subspace migration for imaging of perfectly conducting, arc-like cracks in full- and limited-view inverse scattering problems

Multi-frequency subspace migration imaging techniques are usually adopted for the non-iterative imaging of unknown electromagnetic targets, such as cracks in concrete walls or bridges and anti-personnel mines in the ground, in the inverse scattering problems. It is confirmed that this technique is very fast, effective, robust, and can not only be applied to full- but also to limited-view inverse problems if a suitable number of incidents and corresponding scattered fields are applied and collected. However, in many works, the application of such techniques is heuristic. With the motivation of such heuristic application, this study analyzes the structure of the imaging functional employed in the subspace migration imaging technique in two-dimensional full- and limited-view inverse scattering problems when the unknown targets are arbitrary-shaped, arc-like perfectly conducting cracks located in the two-dimensional homogeneous space. In contrast to the statistical approach based on statistical hypothesis testing, our approach is based on the fact that the subspace migration imaging functional can be expressed by a linear combination of the Bessel functions of integer order of the first kind. This is based on the structure of the Multi-Static Response (MSR) matrix collected in the far-field at nonzero frequency in either Transverse Magnetic (TM) mode (Dirichlet boundary condition) or Transverse Electric (TE) mode (Neumann boundary condition). The investigation of the expression of imaging functionals gives us certain properties of subspace migration and explains why multi-frequency enhances imaging resolution. In particular, we carefully analyze the subspace migration and confirm some properties of imaging when a small number of incident fields are applied. Consequently, we introduce a weighted multi-frequency imaging functional and confirm that it is an improved version of subspace migration in TM mode. Various results of numerical simulations performed on the far-field data affected by large amounts of random noise are similar to the analytical results derived in this study, and they provide a direction for future studies.

Multi-frequency topological derivative for approximate shape acquisition of curve-like thin electromagnetic inhomogeneities

Abstract In this paper, we investigate a non-iterative imaging algorithm based on the topological derivative in order to retrieve the shape of penetrable electromagnetic inclusions when their dielectric permittivity and/or magnetic permeability differ from those in the embedding (homogeneous) space. The main objective is the imaging of crack-like thin inclusions, but the algorithm can be applied to arbitrarily shaped inclusions. For this purpose, we apply multiple time-harmonic frequencies and normalize the topological derivative imaging function by its maximum value. In order to verify its validity, we apply it for the imaging of two-dimensional crack-like thin electromagnetic inhomogeneities completely hidden in a homogeneous material. Corresponding numerical simulations with noisy data are performed for showing the efficacy of the proposed algorithm.

Fast electromagnetic imaging of thin inclusions in half-space affected by random scatterers

We consider an inverse scattering problem wherein penetrable thin electromagnetic inclusions completely embedded in a half-space are surrounded by randomly distributed scatterers. A non-iterative algorithm for retrieving the shape of the inclusions is discussed. It is based on the fact that Multi-static Response (MSR) matrix data can be modeled via a rigorous asymptotic expansion formula of the scattering amplitude in the presence of the inclusions. Various numerical implementations show that the proposed algorithm performs satisfactorily for single and multiple thin inclusions, even with a fair number of random scatterers affecting the data.