Electromagnetic Inverse Scattering Research Articles

Far-field to Near-field Data Relations for the Inverse Electromagnetic Scattering Problem

This paper considers (in general form) the problem of recovering information (size and material parameters) about the scattering object from far-field measurements. The order of solution and functions of each equation for the fields inside and outside the scattering object are discussed. Using well-known mathematical theorems, a simple equation has been derived that connects the far-field data on one side to the near-field data on the other side. Consequently, this equation has been used in an optimization procedure to find the parameters of the dielectric cylinder.

Electromagnetic Inverse Scattering With Perceptual Generative Adversarial Networks

In this work, we introduce a learning-based method to achieve high-quality reconstructions for inverse scattering problems (ISPs). Particularly, the proposed method decouples the full-wave reconstruction model into two steps, including coarse imaging of dielectric profiles by the back-propagation scheme, and a resolution enhancement of coarse results as an image-to-image translation task solved by a novel perceptual generative adversarial network (PGAN). A perceptual adversarial (PA) loss, which is defined as a perceptual loss for the generator network using hidden layers from the discriminator network, is employed as a structural regularization in PGAN. The PA loss is further combined with the pixel-wise loss, and also possibly the adversarial loss, to enforce a multi-level match between the reconstructed image and its reference one. The adversarial training of the generator and discriminator networks ensures that the structural features of targets are dynamically learned by the generator. Numerical tests on both synthetic and experimental data verify that the proposed method is highly efficient and it achieves superior imaging results compared to other data-driven methods. The validation of the proposed PGAN on ISPs also provides a fast and high-precision way for solving other physics-related imaging problems.

Enhanced Deep Learning Approach Based on the Deep Convolutional Encoder–Decoder Architecture for Electromagnetic Inverse Scattering Problems

This letter proposes a novel deep learning (DL) approach to resolve the electromagnetic inverse scattering (EMIS) problems. The conventional approaches of resolving EMIS problems encounter assorted difficulties, such as high contrast, high computational cost, inevitable intrinsic nonlinearity, and strong ill-posedness. To surmount these difficulties, a novel DL approach is proposed based on a novel complex-valued deep fully convolutional neural network structure. The proposed complex-valued deep learning model for solving EMIS problems composes of an encoder network and its corresponding decoder network, followed by a final pixel-wise regression layer. The complex-valued encoder network extracts feature fragments from received scattered field data, while the role of the complex-valued decoder network is mapping the feature fragments to retrieve the final contrasts (permittivities) of dielectric scatterers. Hence, the proposed deep learning model functions as an “heterogeneous” transformation process, where measured scattering field data is converted into the corresponding contrasts of scatterers. As a consequence, the EMIS problem could be resolved accurately even for extremely high-contrast targets. Numerical benchmarks have illustrated the feasibility and accuracy of our proposed approach. The proposed approach opens a novel path for the deep learning-based real-time quantitative microwave imaging for high-contrast scatterers.

Two-Step Enhanced Deep Learning Approach for Electromagnetic Inverse Scattering Problems

In this letter, a new deep learning (DL) approach is proposed to solve the electromagnetic inverse scattering (EMIS) problems. The conventional methods for solving inverse problems face various challenges including strong ill-conditions, high contrast, expensive computation cost, and unavoidable intrinsic nonlinearity. To overcome these issues, we propose a new two-step machine learning based approach. In the first step, a complex-valued deep convolutional neural network is employed to retrieve initial contrasts (permittivities) of dielectric scatterers from measured scattering data. In the second step, the previously obtained contrasts are input into a complex-valued deep residual convolutional neural network to refine the reconstruction of images. Consequently, the EMIS problem can be solved with much higher accuracy even for high-contrast objects. Numerical examples have demonstrated the capability of the newly proposed method with the improved accuracy. The proposed DL approach for EMIS problem serves a new path for realizing real-time quantitative microwave imaging for high-contrast objects.

A Probabilistic Model for the Nonlinear Electromagnetic Inverse Scattering: TM Case

Electromagnetic inverse scattering (EMIS) is a noninvasive examination tool, which holds the promising potential in science, engineering, and military applications. In contrast to conventional tomography techniques, the inverse scattering is a quantitative superresolution imaging method since it is capable of accommodating more realistic interactions between the wavefield and the probed scene. In this paper, a full probabilistic formulation of the EMIS is presented for the first time, which is then solved by applying the well-known expectation maximization method. Afterward, the concept of the complex-valued alternating direction method of multipliers has been proposed as an alternative approach to solve the resulting nonlinear optimization problem. Finally, exemplary numerical and experimental results are provided to validate the proposed method.

Electromagnetic Inverse Scattering of Axially Moving Cylindrical Targets

Electromagnetic inverse scattering techniques are considered to reconstruct the permittivity and the velocity profiles of axially moving cylindrical targets. Two approaches are proposed. One of these is based on a two-step procedure. It is shown that it provides good approximations of the profiles to be reconstructed in a very efficient way, when the peak velocity is small with respect to the speed of light in vacuum. These features are obtained by neglecting the movement in the first step, which is devoted to the reconstruction of the geometric and dielectric properties of the cylindrical targets. The second approach is more classical and is defined as a global optimization problem. It is shown that the two procedures are in some way complementary in the sense that, when the two-step procedure can be applied, it is by far the best, with the general procedure giving large errors; vice versa, the two-step procedure is in trouble when the other one gives good results. In carrying out this analysis, some estimates on the effects of the movement on the scattered field components are deduced in the limit as the peak velocity goes to zero.

Electromagnetic Inverse Scattering Series Method for Positioning Three-Dimensional Targets in Near-Surface Two-Layer Medium With Unknown Dielectric Properties

Positioning 3-D targets buried in layered medium with electromagnetic waves has widespread applications, such as the detection of land mines. Most of the current electromagnetic inverse methods need to know the preknowledge of the dielectric properties for layered background medium, in order to accurately reconstruct concealed targets. However, this condition is hardly satisfied in real problems. In this letter, an electromagnetic inverse scattering series method (EISSM) is developed and derived for positioning 3-D targets buried in two-layered medium. The prominent advantage of the proposed method is that it does not need any prior information about the dielectric properties of the layered medium. The position error predicted by the EISSM is analyzed and discussed. Compared with the commonly used time-reversal-mirror technique, numerical simulations show that the EISSM is capable of positioning the target buried in two-layer medium with less error.

Properties of Broadband Non-Linear Lossy Materials Employed in the Electromagnetic Inverse Scattering during the Microchip Processing

A method has been designed and implemented to describe the optical properties of lossy materials as a continuous functions of a finite wave length spectrum, needed in analysis of the Maxwell-Material equations. Measurements of the index of refraction (N) and the absorption coefficient (K) over a limited spectral range are used as input data. The (complex) permittivity function is then represented as a sum of five types of terms: a plasma term, a conductivity term, several Debye poles, several symmetric Lorentz poles as well as several asymmetric extended Lorentz (“XLorentz”) poles. All these terms are particular solutions of the Lifshitz integral equation describing the dispersion relation of a mono-chromatic electromagnetic wave. This representation facilitates the numerical solution of broadband direct and the inverse scattering of electromagnetic waves for thin film stacks and composite physical structures, in particular those now employed by the microchip industry.

An FFT Twofold Subspace-Based Optimization Method for Solving Electromagnetic Inverse Scattering Problems

A fast Fourier transform (FFT) twofold subspace- based optimization method (TSOM) is proposed to solve electromagnetic inverse scattering problems. As mentioned in the original TSOM (Y. Zhong, et al, Inverse Probl., vol. 25, p. 085003, 2009), one is able to efficiently obtain a meaningful coarse result by constraining the induced current to a lower-dimensional subspace during the optimization, and use this result as the initial guess of the optimization with higher-dimensional current subspace. Instead of using the singular vectors to construct the current subspace as in the original TSOM, in this paper, we use discrete Fourier bases to construct a current subspace that is a good approximation to the original current subspace spanned by singular vectors. Such an approximation avoids the computationally burdensome singular value decomposition and uses the FFT to accomplish the construction of the induced current, which reduce the computational complexity and memory demand of the algorithm compared to the original TSOM. By using the new current subspace approximation, the proposed FFT-TSOM inherits the merits of the TSOM, better stability during the inversion and better robustness against noise compared to the SOM, and meanwhile has lower computational complexity than the TSOM. Numerical tests in the two-dimensional TM case and the three-dimensional one validate the algorithm.

A Hybrid Method of Solving the Electromagnetic Inverse Scattering Problem in Lossy Medium

A hybrid method of solving the electromagnetic inverse scattering problem in a lossy medium is proposed. The gradient search method is first used to determine the spatial distribution of the target, then the Monte Carlo method is selected to calculate the conductivity and permittivity of the scatterer. The results of numerical experiments verify that the hybrid method of combining the two ways is high-precision, efficient, and suited to global optimization.

A Sparsity Regularization Approach to the Electromagnetic Inverse Scattering Problem

We investigate solving the electromagnetic inverse scattering problem using the distorted Born iterative method (DBIM) in conjunction with a variable-selection approach known as the elastic net. The elastic net applies both ℓ1 and ℓ2 penalties to regularize the system of linear equations that result at each iteration of the DBIM. The elastic net thus incorporates both the stabilizing effect of the ℓ2 penalty with the sparsity encouraging effect of the ℓ1 penalty. The DBIM with the elastic net outperforms the commonly used ℓ2 regularizer when the unknown distribution of dielectric properties is sparse in a known set of basis functions. We consider two very different 3-D examples to demonstrate the efficacy and applicability of our approach. For both examples, we use a scalar approximation in the inverse solution. In the first example the actual distribution of dielectric properties is exactly sparse in a set of 3-D wavelets. The performances of the elastic net and ℓ2 approaches are compared to the ideal case where it is known a priori which wavelets are involved in the true solution. The second example comes from the area of microwave imaging for breast cancer detection. For a given set of 3-D Gaussian basis functions, we show that the elastic net approach can produce a more accurate estimate of the distribution of dielectric properties (in particular, the effective conductivity) within an anatomically realistic 3-D numerical breast phantom. In contrast, the DBIM with an ℓ2 penalty produces an estimate which suffers from multiple artifacts.

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.

Electromagnetic Transverse Electric Wave Inverse Scattering of a Partially Immersed Conductor by Steady-State Genetic Algorithm

In this article, we present a computational approach to the imaging of a partially immersed perfectly conducting cylinder by the steady-state genetic algorithm (GA). A conducting cylindrical section of unknown shape scatters the incident transverse electric wave in free space, while the scattered field is recorded outside. Based on the boundary condition and the measured scattered field, a set of nonlinear integral equations is derived, and the imaging problem is reformulated into an optimization problem. An improved steady-state GA is employed to search for the global extreme solution. Numerical results demonstrate that, even when the initial guess is far away from the exact one, good reconstruction can be obtained.

Fast Algorithm for Simulating 3‐D Electromagnetic Inverse Scattering in Horizontally Stratified Medium Via DTA

AbstractTwo kinds of approximation named DTA1 and DTA2 for integral equation are obtained according to different back substitution method on the basis of diagonal tensor approximation (DTA) in order to improve the efficiency for simulating 3‐D electromagnetic scattering and inverse scattering in horizontally stratified medium. The approximations can be used as the initial guess and pre‐conditioner in the stabilized biconjugate‐gradient fast Fourier transform (BCGS‐FFT) algorithm for computing integral equations, and hence a hybrid DTA‐BCGS algorithm with higher efficiency is formed. Numerical scattering examples have shown the high accuracy of DTA2 and the high efficiency of hybrid DTA‐BCGS algorithm, especially of hybrid DTA2‐BCGS. Since DTA2 has a much higher accuracy, it is combined with distorted Born iterative inversion method (DBIM) and a fast electromagnetic inverse scattering technique is formed to reconstruct 3‐D inhomogeneous objects. Numerical inverse scattering examples have demonstrated the efficiency of the inverse scattering technique in reconstructing arbitrary 3‐D inhomogeneous objects in horizontally stratified medium.

Solving Electromagnetic Inverse Scattering Problems by SVRMs: a Case of Study Towards Georadar Applications

In this paper, an heuristic approach based on Support Vector Regression Machines (SVRMs) is presented in order to solve a simple inverse scattering problem. Interesting results have been obtained, with a remarkable reduction of computational time. Future development of this works will interest the evaluation of the performances of SVRMs for detection of buried objects in stratifled media. This is the starting point to develop models for typical Georadar applications.

Computational Electromagnetism

The field of computational electromagnetism is dedicated to the design and analysis of numerical methods for the approximate solution of electromagnetic field problems. Since the exploitation of electromagnetic phenomena is one of the foundations of modern technology, computational electromagnetics is of tremendous industrial relevance: in a sense, it is peer to computational solid and fluid mechanics and huge research efforts are spent on developing and enhancing simulation methods and software for electromagnetic field computations. For a long time, computational electromagnetism remained a realm of engineering research with applied mathematics shunning the area. This was in stark contrast to elasticity and fluid mechanics, where mathematicians have been involved in the development of numerical methods from the very beginning. Maybe, the blame has to be laid on the incorrect belief of mathematicians who thought that the laws governing the behavior of electromagnetic fields basically boil down to well understood second-order elliptic problems. Fortunately, the past fifteen years have seen a real surge of mathematical research activities in the area of computational electromagnetism. This resulted in insights that have begun to have a big impact on the numerical methods used in engineering and industrial environments. A prominent example is the explanation of so-called spurious solutions that can arise when using continuous “nodal” finite elements for the discretization of certain electromagnetic boundary value or eigenvalue problems, respectively. Another example is the appreciation of so-called edge finite elements and the construction of multilevel iterative solvers for the low-frequency setting. Meanwhile, computational electromagnetism can claim to be a major area of numerical mathematics and scientific computing in its own right. This prompted us to ask the Mathematisches Forschungsinstitut Oberwolfach to host a one week workshop on computational electromagnetism, the first of its kind. Reflecting the growing importance of the subject, this workshop has been one of a series of events dedicated to mathematical issues in the computation of electromagnetic fields. We would like to mention, the NSF-CBMS Regional Conference in the Mathematical Sciences about “Numerical Methods in Forward and Inverse Electromagnetic Scattering”, held in Golden, CO, June 3–7, 2002 (from which the book [2] arose), and the “LMS Durham Symposium on Computational methods for Wave Propagation in Direct Scattering”, Durham, England, July 15–25, 2002 (see [1]). This Oberwolfach workshop brought together some 50 experts in computational electromagnetism. The majority of the participants were applied mathematicians, but a sizable number of people with a background in engineering also attended, as appropriate for a field with close ties to engineering and the applied sciences. Nevertheless, the workshop had a clear mathematical focus, emphasizing rigorous theory, principles and ideas. Throughout, the presentations matched these expectations. A total of 29 presentations were given, of which ten were survey lectures offering broader treatment of a particular subject. As is typical of an event that targets a specific area of application, it arose that a broad range of mathematical issues and techniques was addressed. Although it will certainly not do justice to many presentations, we will try categorize the talks as follows: We would like to add our personal impression that two families of methods have been received with particular interest during the workshop:

Electromagnetic Inverse Scattering of Multiple Perfectly Conducting Cylinders by Differential Evolution Strategy With Individuals in Groups (GDES)

The differential evolution strategy with individuals in groups (GDES) is proposed to solve the electromagnetic inverse scattering of multiple perfectly conducting cylinders with transverse magnetic wave incidence. The inverse problem is to locate the cylinders and reconstruct their contours, besides the determination on the number of cylinders. The governing electric field integral equations for the scattering problem are expressed as surface integrals over the cylinder contours. The scattering problem is solved using point-matching method with pulse basis and Dirac test functions. The inverse problem is cast into an optimization problem and solved using the GDES. Numerical reconstruction results show that both the quality of the reconstructed profile and the convergence performance are significantly improved, as compared to the original DES.

Image reconstruction from real scattering data using an iterative scheme with incorporated a priori information

The two-dimensional inverse electromagnetic-scattering problem of reconstructing the material properties of inhomogeneous lossy dielectric cylindrical objects is considered. The material properties are reconstructed from measured far-field scattering data, provided by the USAF Rome Laboratory Electromagnetic Measurement Facility in Ipswich. The targets in the Ipswich data set include perfectly electrically conducting (PEC) targets, penetrable (PEN) targets, and hybrid targets. A new method, which incorporates a priori information about the material properties, is proposed to solve the nonlinear inverse-scattering problem.

A New Algorithm in Electromagnetic Inverse Scattering Theory with an Application to Medical Imaging

We consider the electromagnetic inverse scattering problem of determining the index of refraction of an isotropic inhomogeneous medium from a knowledge of the total electric near field data. We consider both the case of scattering by an infinite inhomogeneous cylinder and a bounded inhomogeneous region in space. We conclude by giving an application of our method to the problem of imaging leukemia in the upper leg by means of microwaves. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.

On a Set of Continuous Wave Electromagnetic Inverse Scattering Boundary Conditions

The inverse problem of continuous wave electromagnetic scattering is considered. The shape and the material constituents of an unknown scatterer need to be determined from bistatic measurement data for a given transmitted field. Although methods have been outlined on how to determine the material constituents of a semi-transparent scatterer, no suitable method has been found which may be employed to simultaneously specify the shape and the material constituents of an unknown smooth, convex-shaped, close, and imperfectly conducting scatterer. Namely, such boundary conditions are sought which do not depend on either the shape or the material constituents of the scattering body and its enclosing surface, but allow one to specify the relevant characteristic parameters uniquely from the recovered near field.The main incentive of this study is to show that the unknown geometrical and material surface parameters can be determined from a set of equations which is derived from the Leontovich impedance boundary condition. The resulting inverse scattering boundary conditions are analyzed in detail and it is shown that in the perfectly conducting case these boundary conditions degenerate to those conditions investigated by Weston and Boerner.The theoretical analysis is verified by computation proving the importance of the result to the inverse theory of electromagnetic scattering.