Section: New Results

Sampling methods for inverse scattering problems

Factorization Method for Periodic Penetrable Media

Participant : Armin Lechleiter.

Imaging periodic penetrable scattering objects is of interest for non-destructive testing of photonic devices. The problem is motivated from the decreasing size of periodic structures in photonic devices, together with an increasing demand in fast non-destructive testing. In this project, we consider the problem of imaging a periodic penetrable structure from measurements of scattered electromagnetic waves. Qualitative inverse scattering techniques are particularly attractive here since they do not use time consuming optimization techniques for reconstruction but rather directly transform measured data into a picture of the scattering object. We show that the Factorization method can be used as an algorithm for imaging of a special class of periodic dielectric structures known as diffraction gratings. Our sampling method computes a picture of the shape of the periodic structure from measured near-field data in a rapid way [38] .

Noise Subspace Methods for Inverse Scattering Problems

Participant : Armin Lechleiter.

The MUSIC algorithm is a well-known imaging technique in signal processing for determining the location of emitters from sensors with arbitrary locations and directional characteristics in a noisy environment. Recent research in the inverse scattering literature has sought to apply this technique to determine not only the location, but the shape of extended scatterers. In a joint work with Tilo Arens and D. Russell Luke we relate the MUSIC algorithm to the Factorization method and show that MUSIC is actually applicable to inverse scattering problems without constraint on the object size. These results are also extended to scattering from cracks. With explicit constructions in hand, we are also able to provide error and stability estimates for practical implementations in noisy environments with limited data. In particular, we address the relation of the spectral properties of the continuous far field operator to those of the discrete version used implicitly in numerical examples appearing in the literature [8] .

The RG-LSM method applied to urban infrastructure imaging

Participant : Houssem Haddar.

The RG-LSM algorithm has been introduced by Colton-Haddar in 2005 as a reformulation of the linear sampling method in the cases where measurements consist of Cauchy data at a given surface, by using the concept of reciprocity gap. The main advantage of this algorithm is to avoid the need of computing the background Green tensor (as required by classical sampling methods) as well as the Dirichlet-to-Neumann map for the probed medium (as required by sampling methods for impedance tomography problems). This method is for instance well suited for medical imaging techniques using microwaves (to detect tumors and malignancies characterized by strong variation in dielectric properties). However, in many other practical applications, like imaging of embedded facilities in the soil or mine detection, the required data at the interface cannot be easily obtained and one has only access to measurements of the scattered wave in the air. In order to overcome this limitation we proposed to couple the RG-LSM algorithm with a continuation method that would provide the Cauchy data from the scattered field. We showed that the obtained scheme has the same convergence properties as RG-LSM with exact data and remains competitive with respect to classical approaches. Preliminary numerical results in a 2-D configuration confirmed these conclusions and also gave further insight on the sampling resolution: Due to the ill-posedness of the first step, only the propagative part of the wave is well reconstructed, which may results in poor approximations of the field. However, the second step (RG-LSM) seems not being affected by this error and therefore is the reconstruction of the target. In a joint work with O. Ozdemir we first extended this approach to the case of rough interfaces [37] . Motivated by microwave imaging experiments, we are currently investigating the cases where the inclusions are buried under thin rough layers for which the use of generalized interface conditions would be appropriate. A long time prospective of this work is to tackle the 3-D electromagnetic case.

Inverse scattering from screens with impedance boundary conditions

Participants : Houssem Haddar, Yosra Boukari.

We are interested in solving the inverse problem of determining a screen (or a crack) from multi-static measurements of electromagnetic (or acoustic) scattered field at a given frequency. An impedance boundary condition is assumed to be verified at both faces of the screen. We extended in a first step the use of the linear sampling method and the reciprocity-gap sampling method to retrieve the shape of the screen and we are currently analyzing the accuracy of these methods with respect to the impedances values as well as using this analysis to derive a priori estimates on the impedances values. This work is pursued in collaboration with F. Ben Hassen.

Sampling methods with time dependent data

Participants : Houssem Haddar, Armin Lechleiter.

In collaboration with P. Monk and Q. Chen from the University of Delaware, we extended the use of sampling methods to inverse scattering problems with time dependent data. We considered in this first investigation the scalar problem and obstacles with Dirichlet boundary conditions. Motivated by ground penetrating radar experiments, we treated the case of near field scattered data generated by incident point sources with causal pulses. We first formulate the sampling algorithm using appropriate convolution in time. The causality assumption introduces additional difficulty is carrying out usual theoretical analysis of the method since one cannot rely on the use of Fourier transform. We provide a factorization of the sampling operator using retarded potentials which are then analyzed with the help of a Fourier-Laplace analysis. We also performed preliminary numerical simulations where the sampling equation is solved using truncated singular value decomposition. The obtained numerical results show good reconstructions and provide a satisfactory validation of our approach.

Transmission Eigenvalues and their application to the identification problem

Participants : Anne Cossonnière, Houssem Haddar.

The so-called interior transmission problem plays an important role in the study of inverse scattering problems from (anisotropic) inhomogeneities. Solutions to this problem associated with singular sources can be used for instance to establish uniqueness for the imaging of anisotropic inclusions from muti-static data at a fixed frequency. It is also well known that the injectivity of the far field operator used in sampling methods is equivalent to the uniqueness of solutions to this problem. The frequencies for which this uniqueness fails are called transmission eigenvalues. We are currently developing approaches where these frequencies can be used in identifying (qualitative informations on) the medium properties. Our research on this topic is mainly done in the framework of the associate team ISIP http://www-direction.inria.fr/international/PHP/Networks/LiEA.php with the University of Delaware. Three contributions have been accomplished:

• On the Determination of Dirichlet or Transmission Eigenvalues from Far Field Data . In this joint work with F. Cakoni and D. Colton we show that the Herglotz wave function with kernel the Tikhonov regularized solution of the far field equation becomes unbounded as the regularization parameter tends to zero iff the wavenumber k belongs to a discrete set of values. When the scatterer is such that the total field vanishes on the boundary, these values correspond to the square root of Dirichlet eigenvalues for - . When the scatterer is a non absorbing inhomogeneous medium these values correspond to so-called transmission eigenvalues. This work provides for instance a theoretical justification of the algorithm that localises the transmission eigenvalues based on the behavior of the solution to the far field equation with respect to the frequency [33] .

• The Interior Transmission Problem For regions with Cavities. In this joint work with F. Cakoni and D. Colton we considered the interior transmission problem in the case when the inhomogeneous medium has cavities, i.e. regions in which the index of refraction is the same as the host medium. In this case we establish the Fredholm property for this problem and show that transmission eigenvalues exist and form a discrete set. We also derive Faber-Krahn type inequalities for the transmission eigenvalues [12] .

• The existence of an infinite discrete set of transmission eigenvalues . This problem was open for a long time. Jointly with F. Cakoni, D. Gintides we prove the existence of an infinite discrete set of transmission eigenvalues corresponding to the scattering problem for isotropic as well as anisotropic inhomogeneous media for the Helmholtz and Maxwell's equations. Our discussion also includes the case of the interior transmission problem for an inhomogeneous medium with cavities, i.e. subregions with contrast zero [13] .

The main topic of the PhD thesis of A. Cossonnière is to extend some of the results obtained above (for the scalar problem) to the Maxwell's problem. In this perspective, theoretcial results related to solutions of the interior transmission problem for medium with cavities and existence of transmission eigenvalues have been obtained. During September-December 2009, A. Cossonnière visited the UDEL and studied the continuity of transmission eigenvalues with respect to the medium properties. Parallel to this work, G. Giorgi, who started in 2009 a PhD thesis co-directed by H. Haddar and M. Piana begun investigating (during a three months training at the DEFI team) a new procedure to improve the lower bound on medium index from observed transmission eigenvalue based on ideas inspired by the work of Cakoni-Gintides-Haddar mentioned above.