Amelie Litman

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.

Theoretical and computational aspects of 2-D inverse profiling

The authors discuss two techniques for solving two-dimensional (2D) inverse scattering problems by parameterizing the scattering configuration, and determining the optimum value of the parameters by minimizing a cost function involving the known scattered-field data. The computation of the fields in each estimated configuration is considered as an auxiliary problem. To improve the efficiency of these computations, the CGFFT iterative scheme is combined with a special extrapolation procedure that is valid for problems with a varying physical parameter such as frequency, angle of incidence, or contrast. Further, they analyze the dynamic range and the resolution of linearized schemes. To obtain an acceptable resolution for an object with a large contrast with respect to the surrounding medium, multiple-frequency information is used. Finally, the availability of a fast-forward solver was an incentive to consider nonlinear optimization. In particular, the authors use a quasi-Newton algorithm at only twice the computational cost of the distorted-wave Born iterative scheme.

Drift correction for scattering measurements

The authors propose a method to correct for drift errors which occur when performing three-dimensional scattering field measurements. This method has the advantages of being fast, without loss of information and with no need of a priori information on the scatterer. It is based on the properties of limited spatial bandwidth of the scattered field.

A Two-Step Procedure for Characterizing Obstacles Under a Rough Surface From Bistatic Measurements

A two-step electromagnetic detection procedure is proposed to characterize a dielectric obstacle buried at low depth under a rough surface from single-frequency and multistatic data. First, we have developed, in the framework of the small perturbation theory, a correlation procedure of the scattered field, which enables us to recover an estimation of the roughness profile. This method is tested for various cases with synthetic data provided by a rigorous boundary integral solver. Second, the obtained surface profile is introduced into the numerical simulation due to a finite-element code. An iterative process is then used, based on a level-set formulation, to obtain the shape of the buried target. The influence of the prior step on the accuracy of the reconstruction of the target is studied via various criteria and for different configurations.

Testing inversion algorithms against experimental data: 3D targets

One of the strengths of Institut Fresnel in Marseille, France, is the tight coupling which exists between experiments and theory. This is how the idea came about, inspired by the Ipswich database [1-4], of designing a new database containing results of controlled scattering experiments and making it available to the inverse problems community, thus giving researchers a further opportunity to test and validate their inversion algorithms against reliable experimental data. The experiments were carried out in the anechoic chamber of the Centre Commun de Ressources Micro-Ondes (CCRM), managed for this topic by the researchers of Institut Fresnel. This anechoic chamber is one of the microwave measurement setups that the institute is developing. A large effort has been made to be able to measure the scattered fields of 3D targets, in a meaningful yet computationally affordable way. This has led to the consideration of objects that, especially at the lower frequencies, were small as compared to the wavelength, thus posing the challenge of performing measurements characterized by a very low signal-to-noise ratio. Hence, several improvements have been obtained on the measurement system itself, as well as on the data processing part [8-10], making it now possible to extend the database to 3D targets, which is herein presented.

Multiple-frequency distorted-wave Born approach to 2D inverse profiling

The present paper deals with the inversion from experimental data provided by Institut Fresnel, France. The distorted-wave Born iterative approach is applied to the reconstruction of two lossless configurations involving dielectric circular cylinders. The dynamic range and the resolution of this scheme are governed by the operating frequency. For a low frequency, the dynamic range is large and the resolution is limited; raising the frequency improves the resolution at the cost of dynamic range. To obtain a high resolution for a large contrast, scattered-field information at multiple frequencies can be used. This is demonstrated for two cases where a direct inversion does not lead to convergence.

Phaseless imaging with experimental data: facts and challenges

Two-dimensional target characterization using inverse profiling approaches with total-field phaseless data is discussed. Two different inversion schemes are compared. In the first one, the intensity-only data are exploited in a minimization scheme, thanks to a proper definition of the cost functional. Specific normalization and starting guess are introduced to avoid the need for global optimization methods. In the second scheme [J. Opt. Soc. Am. A21, 622 (2004)], one exploits the field properties and the theoretical results on the inversion of quadratic operators to derive a two-step solution strategy, wherein the (complex) scattered fields embedded in the available data are retrieved first and then a traditional inverse scattering problem is solved. In both cases, the analytical properties of the fields allow one to properly fix the measurement setup and identify the more convenient strategy to adopt. Also, indications on the number and types of sources and receivers to be used are given. Results from experimental data show the efficiency of these approaches and the tools introduced.

Small Dielectric Spheres with High Refractive Index as New Multifunctional Elements for Optical Devices

The future of ultra-fast optical communication systems is inevitably connected with progress in optical circuits and nanoantennas. One of the key points of this progress is the creation of elementary components of optical devices with scattering diagrams tailored for redirecting the incident light in a desired manner. Here we demonstrate theoretically and experimentally that a small, simple, spatially homogeneous dielectric subwavelength sphere with a high refractive index and low losses (as some semiconductors in the visible or near infrared region) exhibits properties allowing to utilize it as a new multifunctional element for the mentioned devices. This can be achieved by taking advantage of the coherent effects between dipolar and multipolar modes, which produce anomalous scattering effects. The effects open a new way to control the directionality of the scattered light. The directional tuning can be obtained in a practical way just by a change in the frequency of the incident wave, and/or by a well-chosen diameter of the sphere. Dielectric nanoparticles with the required optical properties in the VIS-NIR may be now readily fabricated. These particles could be an efficient alternative to the widely discussed scattering units with a more complicated design.

Two-dimensional inverse profiling problem using phaseless data

We discuss the characterization of two-dimensional targets based on their diffracted intensity. The target characterization is performed by minimizing an adequate cost functional, combined with a level-set representation if the target is homogeneous. One key issue in this minimization is the choice of an updating direction, which involves the gradient of the cost functional. This gradient can be evaluated using a fictitious field, the solution of an adjoint problem in which receivers act as sources with a specific amplitude. We explore the Born approximation for the adjoint field and compare various approaches for a wide variety of objects.

On the Calibration of a Multistatic Scattering Matrix Measured by a Fixed Circular Array of Antennas

The calibration of the multistatic scattering matrix plays an important part in the construction of a quantitative microwave imaging system. For scattering measurement applications, the calibration must be performed on the amplitude and on the phase of the flelds of interest. When the antennas are not completely identical, as for example with a multiplexed antennas array, a speciflc calibration procedure must be constructed. In the present work, we explain how a complex calibration matrix can be deflned which takes advantage of the geometrical organization of the antennas. Indeed, for arrays of antennas positioned on a circle, the inherent symmetries of the conflguration can be fully exploited by means of an adequate reorganization of the multistatic scattering matrix. In addition, the reorganization permits to detect antenna pairs which are not properly functioning and to estimate the signal-to-noise ratio. Experimental results obtained within a cylindrical cavity enclosed by a metallic casing are provided to assess the performance of the proposed calibration procedure.This calibration protocol, which is described here in detail, has already been applied to provide quantitative images of dielectric targets (1,2).