Nonlinear Inverse Scattering Problems Research Articles

A New Scheme for Solving Highly Nonlinear Inverse Scattering Problems With Noise Disturbance

A New Scheme for Solving Highly Nonlinear Inverse Scattering Problems With Noise Disturbance

On Phase Information for Deep Neural Networks to Solve Full-Wave Nonlinear Inverse Scattering Problems

The phase information's role in deep neural networks (DNNs) to solve the electromagnetic inverse scattering problems is investigated. The feedforward neutral network model with complex-valued (CV) data stream and the corresponding CV backpropagation training algorithm are proposed to realize CV convolutional neural networks. Numerical examples are carried out to demonstrate the phase information's role in DNNs in terms of generalization capability as well as the convergence speed in the training stage.

Low-Frequency Data Prediction With Iterative Learning for Highly Nonlinear Inverse Scattering Problems

In this work, we present a deep-learning-based low-frequency (LF) data prediction scheme to solve the highly nonlinear inverse scattering problem (ISP) with strong scatterers. The nonlinearity of ISP is alleviated by introducing the LF components in full-wave inversion. In this scheme, a deep neural network (DNN) is trained to predict the absent LF scattered field data from the measured high-frequency (HF) data. Then, a frequency-hopping technique is applied to invert the predicted LF data and measured HF data, where the inverted LF model is served as an initial guess for the HF data inversion. In this way, the risk of HF data inversion getting trapped in local minima is largely reduced. Furthermore, an iterative training method is employed to continuously update the DNN based on the previous inverted model to predict more accurate LF data, thereby improving the reconstruction result. Both synthetic and experimental results are performed to verify the efficacy of this scheme.

Accelerating Uncertainty Quantification for Nonlinear Inverse Scattering Problems With High Contrast Media by Direct Envelope Methods and Krylov Subspace Iterative Integral Equation Solvers

Uncertainty quantification related to nonlinear inverse scattering problems often involves the posterior covariance matrix that cannot be effectively estimated but still need to be considered as an important part of the nonlinear inverse scattering problem. The objective of this work is to show how to take advantage of a Bayesian framework to estimate the uncertainty of velocity reconstruction in the nonlinear inverse scattering imaging. The key ingredients are twofold. On the one hand, we rely on a Kalman Filter method, equipped with an optimization scheme, to solve the inverse problem. We use the distorted Born iterative method to formulate the sensitivity kernel. It directly uses an explicit representation of the data sensitivity function in terms of Green functions, rather than the indirect optimization approach based on the adjoint state method, in which the Green’s functions are based on integral equations. On the other hand, we use the direct envelope methods to provide the initial guess (large-scale smooth model) to overcome the nonlinearity in the inverse scattering problems. We first investigate the uncertainty of velocity imaging in high contrast media. Then we show by means of Lippmann-Schwinger-type equations that the uncertainty of multiparameter inverse scattering problems can be addressed. Numerical results dealing with the uncertainty of nonlinear inverse scattering problems in the case of both isotropic and anisotropic media highlight the important role played by uncertainty quantification.

Investigation of the effect of super-resolution in nonlinear inverse scattering

The idea that a solution to a nonlinear inverse scattering problem (ISP) can contain information about the target on a subwavelength scale and thus allow one to achieve super-resolution (spatial resolution beyond the diffraction limit) has been around since the 1990s. However, a solid mathematical theory of super-resolution in nonlinear image reconstruction is still lacking. In this paper, we investigate the effect of super-resolution in nonlinear ISPs (both analytically and numerically) by analyzing several inverse problems in which the limit of spatial resolution can be defined precisely. The conclusions we obtain are not optimistic. Although it is possible to create examples of exactly solvable models in which account of nonlinearity in the ISP results in additional mathematically independent equations (one such example is shown herein), our results indicate that super-resolution is not achievable in any practical sense. Rather, we find that the linear subspace of possible solutions to a band-limited linearized ISP is transformed into a more general curved manifold due to the effects of nonlinearity. In the one-dimensional problem with realistic interaction that we have considered, the manifold can have a slightly smaller dimensionality that the subspace of solutions to the linearized problem but it does not contract to a point and the effect is practically insignificant.

Wavelet-Based Subspace Regularization for Solving Highly Nonlinear Inverse Scattering Problems with Contraction Integral Equation

A wavelet transform twofold subspace-based optimization method (WT-TSOM) is proposed to solve the highly nonlinear inverse scattering problems with contraction integral equation for inversion (CIE-I). While the CIE-I is able to suppress the multiple scattering effects within inversion (without compromising the accuracy of the physics), proper regularization is needed. In this paper, we investigate a new type subspace regularization technique based on wavelet expansions for the induced currents. We found that the bior3.5 wavelet is a good choice to stabilize the inversions with the CIE-I model and in the meanwhile it also can rectify the contrast profile. Numerical tests against both synthetic and experimental data show that WT-TSOM is a promising regularization technique for inversion with CIE-I.

Integral Equation Methods With Multiple Scattering and Gaussian Beams in Inhomogeneous Background Media for Solving Nonlinear Inverse Scattering Problems

I develop inverse scattering methods for velocity reconstruction in the subsurface, based on multiple scattering theory, Gaussian beams and nonlinear Born approximation. This method is based on a new version of the distorted Born iterative inverse scattering method. It directly uses an explicit representation of the data sensitivity function in terms of Green functions, rather than the indirect optimization approach based on the adjoint state method. I propose three direct scattering methods for forward modeling, namely, Gaussian beam integral equation propagators, nonlinear Born approximation, and multiple scattering. First, I apply a sensitivity kernel incorporating the Gaussian beam summation-based integral equation method and compute the background medium Green's function in an inhomogeneous medium. Then I extend an approximate solution of the Lippmann-Schwinger equation, referred to as nonlinear Born approximation to inverse scattering problem. I apply the nonlinear Born approximation to forward modeling at each iteration. Furthermore, I combine the Gaussian beams with the multiple scattering theory for nonlinear scattering problems. I obtain the inversion results using the proposed approaches, which shows the capability for inverse scattering imaging. I have carried out several numerical experiments that involve reconstructing the resampled Marmousi and Society of Exploration Geophysicists (SEG)/European Association of Geoscientists and Engineers (EAGE) salt models from its smoothed version.

A Multiresolution Contraction Integral Equation Method for Solving Highly Nonlinear Inverse Scattering Problems

The solution of highly nonlinear inverse scattering problems is addressed. An innovative multiresolution microwave imaging technique, which leverages on the profitable integration of the contraction integral equation for inversion (CIE-I) and the iterative multiscaling approach (IMSA), is proposed. Thanks to the joint exploitation of the features of the CIE-I formulation and the IMSA, the arising IMSA-CIE-I inversion technique provides faithful reconstructions of complex-shaped strong scatterers with enhanced resolution and computational efficiency with respect to the state-of-the-art competitive alternatives. Numerical and experimental test cases verify the effectiveness and limitations of the proposed method.

Solving Phaseless Highly Nonlinear Inverse Scattering Problems With Contraction Integral Equation for Inversion

At high frequencies, phase measurement is difficult to meet the accuracy required for imaging, and it is more susceptible to noise pollution. Consequently, it is important to develop inversion models and algorithms for electromagnetic inverse scattering problems with phaseless data (PD-ISPs). Compared to the ISPs with full-data (FD-ISPs), the PD-ISPs are more nonlinear due to the lack of the phase information. Aiming at solving highly nonlinear PD-ISPs, in this article we present a contraction integral equation for inversion (CIE-I) regularized with Fourier subspace via contrast source inversion method, denoted as PD-CSI-CIE-I. Under the model of CIE-I, the nonlinearity of PD-ISPs can be effectively alleviated. In order to stabilize CIE-I, a nested optimization scheme with multi-round procedure is adopted by choosing the proper number of low-frequency components from unknown induced current spanned in the Fourier subspace. By the Fourier bases expansion of induced current, the computational cost can be further reduced. Numerical and experimental examples validate the efficiency of the proposed inversion method. Futher, it is shown that the proposed PD-CSI-CIE-I has a stronger inversion capability than the one with the Lippmann-Schwinger integral equation (LSIE) when tackling high contrast scatterers with phaseless data.

Deep-Learning Schemes for Full-Wave Nonlinear Inverse Scattering Problems

This paper is devoted to solving a full-wave inverse scattering problem (ISP), which is aimed at retrieving permittivities of dielectric scatterers from the knowledge of measured scattering data. ISPs are highly nonlinear due to multiple scattering, and iterative algorithms with regularizations are often used to solve such problems. However, they are associated with heavy computational cost, and consequently, they are often time-consuming. This paper proposes the convolutional neural network (CNN) technique to solve full-wave ISPs. We introduce and compare three training schemes based on U-Net CNN, including direct inversion, backpropagation, and dominant current schemes (DCS). Several representative tests are carried out, including both synthetic and experimental data, to evaluate the performances of the proposed methods. It is demonstrated that the proposed DCS outperforms the other two schemes in terms of accuracy and is able to solve typical ISPs quickly within 1 s. The proposed deep-learning inversion scheme is promising in providing quantitative images in real time.

A Hybrid Regularization Technique for Solving Highly Nonlinear Inverse Scattering Problems

Solving inverse scattering problems (ISPs) for targets with high contrasts and/or large dimensions suffer from severe ill-posedness and strong nonlinearity. Recently, a family of new integral equations (NIE) has been proposed to tackle such problems, in which the multiple scattering effects in estimating contrasts during inversions are suppressed by the local wave effects. This effectively reduces the nonlinearity of ISPs by transforming the problems into a new form. As in most inversions, to achieve better (stabler and faster) inversion efficiency, proper regularization techniques are needed. This paper provides the detailed studies on the two different types of regularization techniques in the inversions with the NIE, i.e., the twofold subspace-based optimization method, directly applied in the modeling, and the total variation type multiplicative regularization, conventionally applied on the unknowns. We will show that how each regularization works with the NIE and how they work together with the NIE to obtain the better performance in terms of reducing the nonlinearity and increasing the stability of the inversions. Numerical tests against synthetic data and experimental data are provided to verify the interests.

A New Integral Equation Method to Solve Highly Nonlinear Inverse Scattering Problems

A family of new integral equations (NIE) is proposed in this paper, which are transformed from the original Lippmann–Schwinger integral equation. It can be shown that the NIE can effectively reduce the nonlinearity of inverse scattering problems by reducing the global nonlinear effects, introduced by the multiple scattering behaviors, in estimating the contrast. Equipping the previously proposed twofold subspace-based optimization method with such NIE, the new inversion method is able to solve inverse scattering problems with strong scatterers, like with high contrast and/or large dimensions (in terms of wavelength) ones. Furthermore, such a family of NIE could provide a convenient tool to appraise reconstructed results. Several representative numerical tests are carried out, using both synthetic and experimental data, to verify the efficacy of the new inversion method.

A Novel Effective Model for Solving 3-D Nonlinear Inverse Scattering Problems in Lossy Scenarios

A proper extension of the contrast source extended Born model, recently introduced in the two-dimensional scalar case, is proposed to establish a novel full-wave inversion method for nonlinear vectorial three-dimensional scattering problems. In particular, a suitable auxiliary function, embedding the parameters of the unknown targets, is introduced as fundamental unknown of the inverse problem instead of the widely used contrast function. Numerical examples confirm effectiveness and reliability of the proposed inversion approach, which also favorably compares with other methods previously known in the literature

Progress towards in vitro quantitative imaging of human femur using compound quantitative ultrasonic tomography

The objective of this study is to make cross-sectional ultrasonic quantitative tomography of the diaphysis of long bones. Ultrasonic propagation in bones is affected by the severe mismatch between the acoustic properties of this biological solid and those of the surrounding soft medium, namely, the soft tissues in vivo or water in vitro. Bone imaging is then a nonlinear inverse-scattering problem. In this paper, we showed that in vitro quantitative images of sound velocities in a human femur cross section could be reconstructed by combining ultrasonic reflection tomography (URT), which provides images of the macroscopic structure of the bone, and ultrasonic transmission tomography (UTT), which provides quantitative images of the sound velocity. For the shape, we developed an image-processing tool to extract the external and internal boundaries and cortical thickness measurements. For velocity mapping, we used a wavelet analysis tool adapted to ultrasound, which allowed us to detect precisely the time of flight from the transmitted signals. A brief review of the ultrasonic tomography that we developed using correction algorithms of the wavepaths and compensation procedures are presented. Also shown are the first results of our analyses on models and specimens of long bone using our new iterative quantitative protocol.

An Algorithm for the Fully Nonlinear Inverse Scattering Problem at Fixed Frequency

We reconstruct an object which is described by a complex valued function from the scattered waves generated by irradiating plane waves at fixed frequency. The scattering process is modeled by the Helmholtz equation and includes multiple scattering. We present numerical results from computer generated data.

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.