Ann Franchois

An efficient hybrid MLFMA-FFT solver for the volume integral equation in case of sparse 3D inhomogeneous dielectric scatterers

Electromagnetic scattering problems involving inhomogeneous objects can be numerically solved by applying a Method of Moments discretization to the volume integral equation. For electrically large problems, the iterative solution of the resulting linear system is expensive, both computationally and in memory use. In this paper, a hybrid MLFMA-FFT method is presented, which combines the fast Fourier transform (FFT) method and the High Frequency Multilevel Fast Multipole Algorithm (MLFMA) in order to reduce the cost of the matrix-vector multiplications needed in the iterative solver. The method represents the scatterers within a set of possibly disjoint identical cubic subdomains, which are meshed using a uniform cubic grid. This specific mesh allows for the application of FFTs to calculate the near interactions in the MLFMA and reduces the memory cost considerably, since the aggregation and disaggregation matrices of the MLFMA can be reused. Additional improvements to the general MLFMA framework, such as an extention of the FFT interpolation scheme of Sarvas et al. from the scalar to the vectorial case in combination with a more economical representation of the radiation patterns on the lowest level in vector spherical harmonics, are proposed and the choice of the subdomain size is discussed. The hybrid method performs better in terms of speed and memory use on large sparse configurations than both the FFT method and the HF MLFMA separately and it has lower memory requirements on general large problems. This is illustrated on a number of representative numerical test cases.

Three-dimensional quantitative microwave imaging from measured data with multiplicative smoothing and value picking regularization

This paper presents reconstructions of four targets from the 3D Fresnel database. The electromagnetic inverse scattering problem is treated as a nonlinear optimization problem for the complex permittivity in an investigation domain. The goal of this paper is to explore the achievable reconstruction quality when such a quantitative inverse scattering approach is employed on real world measurements, using only single-frequency data. Two regularization techniques to reduce the ill-possedness of the inverse scattering problem are compared. The first one is a multiplicative smoothing regularization, applied directly to the cost function, which yields smoothed reconstructions of the homogeneous Fresnel targets without much experimentation to determine the regularization parameter. The second technique is the recently proposed value picking (VP) regularization which is particularly suited for the class of piecewise (quasi-)homogeneous targets, such as those of the Fresnel database. In contrast to edge-preserving regularization methods, VP regularization does not operate on the spatial distribution of permittivity values, but it clusters them around some reference values, the VP values, in the complex plane. These VP values are included in the cost function as auxiliary optimization variables and their number can be gradually increased using a stepwise relaxed VP regularization scheme. Both regularization strategies are incorporated in a Gauss–Newton minimization framework with line search. It is shown that the reconstruction quality using single-frequency Fresnel data is good when using multiplicative smoothing and even better when using the VP regularization. In particular, the completely blind reconstruction of the mystery target in the database provides us with a detailed quantitative image of a plausible object.

A New Value Picking Regularization Strategy—Application to the 3-D Electromagnetic Inverse Scattering Problem

The nonlinear electromagnetic inverse scattering problem of reconstructing a possibly quasi-piecewise constant inhomogeneous complex permittivity profile is solved by iterative minimization of a pixel-based data fit cost function. Because of the ill-posedness it is necessary to introduce some form of regularization. Many authors apply a smoothing constraint on the reconstructed permittivity profile, but such regularization smooths away sharp edges. In this paper, a simple yet effective regularization strategy, the value picking (VP) regularization, is proposed. This new technique is capable of reconstructing piecewise constant permittivity profiles without degrading the edges. It is based on the knowledge that only a few different permittivity values occur in such profiles, the values of which need not be known in advance. VP regularization does not impose this a priori information in a strict sense, such that it can be applied also to profiles that are only approximately piecewise constant. The VP regularization is introduced in the solution of the inverse problem by adding a choice function to the data fit cost function for every permittivity unknown in the discretized problem. When minimized, the choice function forces the corresponding permittivity unknown to be close to one member of a set of auxiliary variables, the VP values, which are continuously updated throughout the iterations. To minimize the regularized cost function, a half quadratic Gauss-Newton optimization technique is presented. Finally, a stepwise relaxed VP regularization scheme is proposed, in which the number of VP values is gradually increased. This scheme is tested with synthetic and measured scattering data, obtained from inhomogeneous 3D targets, and is shown to achieve high reconstruction quality.

A Full-Wave 2.5D Volume Integral Equation Solver for 3D Millimeter-Wave Scattering by Large Inhomogeneous 2D Objects

A two-and-a-half dimensional full-wave forward solver to compute the three-dimensional (3D) electromagnetic field scattered by an infinitely long inhomogeneous (lossy) dielectric cylinder with arbitrary cross-sectional shape under a given 3D time-harmonic illumination is presented. The relevant set of linear equations, obtained after performing a spatial Fourier transform of the fields along the axial direction and applying a method of moments discretization to the two-dimensional contrast source volume integral equation, is solved iteratively with a stabilized biconjugate gradient fast Fourier transform method. In this way, objects with cross-sectional dimensions of several to many wavelengths can be handled in a very fast way. Furthermore, a vectorial 3D Gaussian beam illumination, usually employed in active millimeter-wave (mm-wave) imaging systems, is implemented using a complex source Gaussian beam formulation. The validity of the method is proved by comparison with analytic results and with the results from a full-wave 3D solver. Finally, the method is applied to a millimeter-wave imaging example, showing the scattering from an object hidden under clothing on a human body.

Quantitative microwave tomography from sparse measurements using a robust huber regularizer

In statistical theory, the Huber function yields robust estimations reducing the effect of outliers. In this paper, we employ the Huber function as regularization in a challenging inverse problem: quantitative microwave imaging. Quantitative microwave tomography aims at estimating the permittivity profile of a scattering object based on measured scattered fields, which is a nonlinear, ill-posed inverse problem. The results on 3D data sets are encouraging: the reconstruction error is reduced and the permittivity profile can be estimated from fewer measurements compared to state-of-the art inversion procedures.

Piecewise Smoothed Value Picking Regularization Applied to 2-D TM and TE Inverse Scattering

The Stepwise Relaxed Value Picking (SRVP) regularization technique, proposed earlier for the iterative reconstruction of piecewise (quasi-)homogeneous objects, is a non-spatial technique, whereby the reconstruction unknowns are clustered around a limited number of-a-priori unknown-reference values. Artifacts have been observed in some 2-D and 3D complex permittivity reconstructions. This paper therefore combines the non-spatial SRVP technique with a spatial smoothing technique, whereby the reference values provided by the former-in each iteration-are employed by the latter to define separate smoothing regions. This way edges are preserved, since the spatial smoothing constraints in the cost function are active within but not across the region boundaries. This combined technique, denoted as Stepwise Relaxed Piecewise Smoothed Value Picking (SRPSVP) regularization, is formulated for the 2.5D microwave inverse scattering problem and is illustrated with reconstructions from the Institut Fresnel 2-D scattering database.

Weakly Convex Discontinuity Adaptive Regularization for Microwave Imaging

Reconstruction of inhomogeneous dielectric objects from microwave scattering is a nonlinear and ill-posed inverse problem. In this communication, we develop a new class of weakly convex discontinuity adaptive (WCDA) models as a regularization for quantitative microwave tomography. We show that this class includes the Huber regularizer and we show how to combine these methods with electromagnetic solvers operating on the complex permittivity profile. 2D reconstructions of objects from the Institute Fresnel database and experimental data at a single frequency demonstrate the effectiveness of the proposed regularization even when employing far less transmitters and receivers than available in the database.

3D Microwave Tomography with Huber Regularization applied to Realistic Numerical Breast Phantoms

Quantitative active microwave imaging for breast cancer screening and therapy monitoring applications requires adequate reconstruction algorithms, in particular with regard to the nonlinearity and ill-posedness of the inverse problem. We employ a fully vectorial three-dimensional nonlinear inversion algorithm for reconstructing complex permittivity profiles from multi-view single-frequency scattered field data, which is based on a Gauss-Newton optimization of a regularized cost function. We tested it before with various types of regularizing functions for piecewise-constant objects from Institut Fresnel and with a quadratic smoothing function for a realistic numerical breast phantom. In the present paper we adopt a cost function that includes a Huber function in its regularization term, relying on a Markov Random Field approach. The Huber function favors spatial smoothing within homogeneous regions while preserving discontinuities between contrasted tissues. We illustrate the technique with 3D reconstructions from synthetic data at 2GHz for realistic numerical breast phantoms from the University of Wisconsin-Madison UWCEM online repository: we compare Huber regularization with a multiplicative smoothing regularization and show reconstructions for various positions of a tumor, for multiple tumors and for different tumor sizes, from a sparse and from a denser data configuration.

Weakly convex discontinuity adaptive regularization for 3D quantitative microwave tomography

We present an analysis of weakly convex discontinuity adaptive (WCDA) models for regularizing three-dimensional (3D) quantitative microwave imaging. In particular, we are concerned with complex permittivity reconstructions from sparse measurements such that the reconstruction process is significantly accelerated. When dealing with such a highly underdetermined problem, it is crucial to employ regularization, relying in this case on prior knowledge about the structural properties of the underlying permittivity profile: we consider piecewise homogeneous objects. We present a numerical study on the choice of the potential function parameter for the Huber function and for two selected WCDA functions, one of which (the Leclerc‐CauchyLorentzian function) is designed to be more edge-preserving than the other (the Leclerc‐Huber function). We evaluate the effect of reducing the number of (simulated) scatteredfield data on the reconstruction quality. Furthermore, reconstructions from subsampled single-frequency experimental data from the 3D Fresnel database illustrate the effectiveness of WCDA regularization.

Three-dimensional quantitative microwave imaging of realistic numerical breast phantoms using Huber regularization

Breast tumor detection with microwaves is based on the difference in dielectric properties between normal and malignant tissues. The complex permittivity reconstruction of inhomogeneous dielectric biological tissues from microwave scattering is a nonlinear, ill-posed inverse problem. We proposed to use the Huber regularization in our previous work where some preliminary results for piecewise constant objects were shown. In this paper, we employ the Huber function as regularization in the even more challenging 3D piecewise continuous case of a realistic numerical breast phantom. The resulting reconstructions of complex permittivity profiles indicate potential for biomedical imaging.