Dominique Lesselier

Level set methods for inverse scattering

We give an overview of recent techniques which use a level set representation of shapes for solving inverse scattering problems. The main focus is on electromagnetic scattering using different popular models, such as for example Maxwells equations, TM-polarized and TE-polarized waves, impedance tomography, a transport equation or its diffusion approximation. These models are also representative of a broader class of inverse problems. Starting out from the original binary approach of Santosa for solving the corresponding shape reconstruction problem, we successively develop more recent generalizations, such as for example using colour or vector level sets. Shape sensitivity analysis and topological derivatives are discussed as well in this framework. Moreover, various techniques for incorporating regularization into the shape inverse problem using level sets are demonstrated, which also include the choice of subclasses of simple shapes, such as ellipsoids, for the inversion. Finally, we present various numerical examples in two dimensions and in three dimensions for demonstrating the performance of level set techniques in realistic applications.

Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set

We are concerned with the retrieval of the unknown cross section of a homogeneous cylindrical obstacle embedded in a homogeneous medium and illuminated by time-harmonic electromagnetic line sources. The dielectric parameters of the obstacle and embedding materials are known and piecewise constant. That is, the shape (here, the contour) of the obstacle is sufficient for its full characterization. The inverse scattering problem is then to determine the contour from the knowledge of the scattered field measured for several locations of the sources and/or frequencies. An iterative process is implemented: given an initial contour, this contour is progressively evolved such as to minimize the residual in the data fit. This algorithm presents two main important points. The first concerns the choice of the transformation enforced on the contour. We will show that this involves the design of a velocity field whose expression only requires the resolution of an adjoint problem at each step. The second concerns the use of a level-set function in order to represent the obstacle. This level-set function will be of great use to handle in a natural way splitting or merging of obstacles along the iterative process. The evolution of this level-set is controlled by a Hamilton-Jacobi-type equation which will be solved by using an appropriate finite-difference scheme. Numerical results of inversion obtained from both noiseless and noisy synthetic data illustrate the behaviour of the algorithm for a variety of obstacles.

MUSIC-Type Electromagnetic Imaging of a Collection of Small Three-Dimensional Inclusions

In this paper we consider the localization of a collection of small, three-dimensional bounded homogeneous inclusions via time-harmonic electromagnetic means, typically using arrays of electric or magnetic dipole transmitters and receivers with given polarization(s) at some distance from the collection, possibly also lying in the far field. The inclusions, somewhat apart or closely spaced, are buried within a homogeneous medium, and are of arbitrary contrast of permittivity, conductivity, and permeability vis-ag-vis this embedding medium. The problem is formulated as an inverse scattering problem for the full Maxwell equations and it involves a robust asymptotic modeling of the multistatic response matrix. No specific application is studied at this stage, but characterization of obstacles in subsoils, nondestructive evaluation of man-made structures, and medical imaging are primary fields of application envisaged. The proposed approach uses a MUSIC (multiple signal classification)-type algorithm, and it yields fast numbering, accurate localization, and estimates of the electromagnetic and geometric parameters (polarization tensors) of the inclusions. The mathematical machinery is detailed first, some specific attention being given to triaxial ellipsoidal inclusions and degenerate spherical shapes (for the latter known results are retrieved). Then, the viability of this algorithm—which would be easily extended to planarly layered environments by introduction of their Green’s functions—is documented by a variety of numerical results from synthetic, noiseless, and severely noisy field data.

A MUSIC Algorithm for Locating Small Inclusions Buried in a Half-Space from the Scattering Amplitude at a Fixed Frequency

In this paper a MUSIC (standing for multiple signal classification) algorithm for locating small inclusions buried in a half-space from the scattering amplitude at a fixed frequency is developed. The underlying application area which motivated this work is the imaging of antipersonnel mines from electromagnetic data, formulated as an inverse scattering problem for the Helmholtz equation. The algorithm makes use of an asymptotic expansion of the scattering amplitude. A derivation of the leading-order term in this asymptotic expansion and its application for designing a MUSIC type of algorithm are presented. The viability of this algorithm is documented by a variety of numerical results.

Diffraction tomography: contribution to the analysis of some applications in microwaves and ultrasonics

After a brief presentation of the principles of diffraction tomography, the authors focus on the applications they have investigated in the biomedical and non-destructive testing domains. Typical numerical and experimental results are presented and in their comments they state what they think are the current limitations of this approach and the possible opportunities for future work with this imaging technique.

Modified gradient approach to inverse scattering for binary objects in stratified media

We are concerned herein with inverse scattering problems in stratified media and aspect-limited data configurations. In such configurations, the sources and receivers of the probing waves are located in a medium different from the one which contains the object under test. This results in a lack of information which enhances the inherent ill-posedness of the inverse problem. To make the problem more tractable, we assume that the test object is homogeneous with known constitutive parameters so that the inverse problem consists of reconstructing its shape and location. This non-linear inverse problem is solved using the modified gradient method in which the a priori information is introduced as a binary constraint. A cooling parameter is introduced at the same time, which allows us to control the evolution of the iterative process. The effectiveness of this algorithm is studied for three different physical applications.

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.

Multistatic Response Matrix of a 3-D Inclusion in Half Space and MUSIC Imaging

Recent work on the retrieval of 3-D bounded dielectric and/or magnetic inclusions in free space is extended to burial in a half-space. Though emphasis is on the case of a single inclusion, enabling the derivation of closed-form mathematical results in illuminating fashion, the approach extends to the case of (an unknown number of) well-separated inclusions. Within an asymptotic field formulation derived from exact contrast-source vector integral formulations satisfied by the time-harmonic fields and using proper reciprocity relationships of the dyadic Greens functions, the multistatic response matrix of the inclusion is constructed from the leading-order term of the fields. Its singular value structure is analyzed in detail for a dielectric or a magnetic contrast, or both contrasts. This is performed in the case of a unique electric dipole array operated in the transmit/receive mode at a single frequency. A multiple signal classification-type algorithm follows from the decomposition, yielding a cost functional the magnitude of which peaks at the inclusion center. Numerical results illustrate the above as a function of the geometric and electromagnetic parameters of the configuration. Imaging of a spherical inclusion is in particular investigated from severely noisy synthetic data, as well as of two inclusions embedded within a non-conductive or conductive half space.

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.