Variational Born Iterative Method Research Articles

Real-Time 3D Microwave Medical Imaging With Enhanced Variational Born Iterative Method.

In this paper, we present a new variational Born iterative method (VBIM) for real-time microwave imaging (MWI) applications. The S-parameter volume integral equation and waveport vector Green's function are implemented to utilize the measured signal of the MWI system. Meanwhile, the real and imaginary separation (RIS) approach is used at each iterative step to simultaneously reconstruct the dielectric permittivity and conductivity of unknown objects. Compared with the Born iterative method and distorted Born iterative method, VBIM requires less computational time to reach the convergence threshold. The graphics processing unit based acceleration technique is implemented for real-time imaging. To demonstrate the efficiency and accuracy of this VBIM-RIS method, synthetic analysis of a complex multi-layer spherical phantom is first conducted. Then, the algorithm is tested with measured data using our new MWI system prototype. Finally, a synthetic brain-tumor phantom model under a thermal therapy procedure is monitored to exemplify the real-time imaging with about 5 seconds per reconstruction frame.

A Robust Hierarchical Domain Constraint Strategy for Multiscale Contrast Objects Reconstruction

A robust hierarchical domain constraint (HDC) strategy is proposed for the multi-scale contrast objects reconstruction in this letter. The Monte Carlo method and point evaluation are adopted and modified for the classification of background and scatterers cells, the parameters of the background cells are set as the same as the actual background and removed from the inversion. The major work of the HDC strategy is to divide the inversion process into several stages, and the classification of cells is evaluated separately at each stage, making the classification reversible. And the HDC can effectively classify the inversion domain of low-contrast objects without negligence in the multi-scale contrast objects case, as well as the continuously varying inhomogeneous objects’ low-contrast parts. Another advantage of HDC is the threshold determining the classification of cells can be flexibly chosen. This robust constraint of the inversion domain compresses the solution space so that the ill-posedness of the inverse problem is mitigated, and better accuracy thus is achieved. In this letter, the HDC strategy is applied to the Variational Born Iterative Method (VBIM), numerical results validate the accuracy, robustness, and antinoise ability of the proposed VBIM-HDC.

Fast 3-D Electromagnetic Full-Wave Inversion of Dielectric Anisotropic Objects Based on ResU-Net Enhanced by Variational Born Iterative Method

In this article, a novel artificial neural network named residual U-Net (ResU-Net) is proposed to directly reconstruct 3-D dielectric anisotropic objects from scattered electromagnetic field data recorded at the receiver array. ResU-Net has the same framework as that of U-Net but the convolution kernels are replaced with residual kernels. Meanwhile, the squeeze-and-excitation (SE) operation is added to enable information interaction among different channels and further improve prediction accuracy. ResU-Net is trained by thousands of 3-D homogeneous dielectric anisotropic handwritten digits and the corresponding synthesized scattered field data. In the online prediction, ResU-Net can invert multiple anisotropic model parameters of homogeneous 3-D objects instantaneously. For an inhomogeneous object or multiple homogeneous objects, ResU-Net provides good initial profiles which are fed into the following variational Born iterative method (VBIM) full-wave inversion solver. In addition, the VBIM is implemented in a restricted domain instead of the whole 3-D inversion domain to save computational cost. Numerical experiments show that compared with the traditional iterative solver, such as VBIM, the proposed ResU-Net or the hybrid method can not only achieve higher reconstruction accuracy but also accomplish the multiparametric 3-D inversion in a much faster way.

Reconstruction of three-dimensional objects in layered composite structures from multimode orbital angular momentum.

Because of helical phase wavefront distribution, vortex electromagnetic waves are considered to carry more information and additional degrees of freedom than traditional spherical waves. Therefore, a vortex wave carrying orbital angular momentum (OAM) can improve inversion and imaging accuracy. In this work, we revisit the reconstruction of three-dimensional objects in layered composite structures extended with OAM. In forward modeling, the concentric uniform circle array is used to generate electromagnetic vortex beams. To analyze the difference of vortex beams, the electric field radiation pattern and phase pattern distribution of OAM waves with different modes are calculated. Then, the scattered field of layered media illuminated by different OAM beams is determined by the dyadic Green's function and the stabilized biconjugate gradient technique with a fast Fourier transform algorithm. In the inversion, the variational Born iterative method is used to reconstruct targets in layered composite structures, and multiple OAM modes are used to improve the reconstruction results. The numerical results prove that the permittivity of the target can be better reconstructed by using the multiple OAM modes rather than the traditional spherical wave. With the increase of OAM mode number, the reconstructed target parameters are closer to the true value. We expect that our results will provide a better understanding of the OAM and pave the way for the improvement of inversion and optical imaging technology using vortex waves.

3-D Voxel-Based Reconstruction of Multiple Objects Buried in Layered Media by VBIM Hybridized With Unsupervised Machine Learning

This article presents a novel hybrid electromagnetic inversion method. The traditional 3-D variational Born iterative method (VBIM) is combined with the unsupervised machine-learning expectation maximization (EM). In each iteration, VBIM first outputs the pseudo-randomly distributed model parameters in all discretized cells in the inversion domain. Then the EM algorithm is used to classify them and estimate the mean model parameter values of each homogeneous scatterer or subscatterer supposing that the reconstructed model parameters in all cells comply with the Gaussian mixture model (GMM). At last, partial cells in the inversion domain classified as “background” will be removed and the unknowns in the next VBIM iteration are reduced. This process is implemented iteratively until no “background” cell can be removed anymore and the data misfit between the measured scattered field and reconstructed field reaches the stop criterion. Finally, the mean value of the model parameter estimated by EM is mandatorily assigned for each homogeneous scatterer or subscatterer. Numerical examples show that the proposed hybrid method works efficiently for the reconstruction of isotropic, anisotropic, homogeneous, or inhomogeneous scatterers. It also has a certain antinoise ability.

Super-Resolution 3-D Microwave Imaging of Objects With High Contrasts by a Semijoin Extreme Learning Machine

This article proposes a semijoin extreme learning machine (SJ-ELM) for super-resolution 3-D microwave imaging of objects with high contrasts. The proposed scheme develops a shallow neural network structure with the semijoin strategy to convert the scattered field data into two output channels, namely, the permittivity and the conductivity of objects, respectively. The semijoin strategy can decrease the inner matrix dimensions to reduce the computational burden for 3-D super-resolution imaging, so it is employed to connect between the nodes of the hidden layer and the output layer. The imaging performance of the proposed SJ-ELM and the conventional variational Born iterative method (VBIM) is first compared for imaging objects with different electrical sizes and contrasts, and then, different targets of imaging resolution are designed to evaluate both solvers. The proposed SJ-ELM is also assessed for imaging objects with high contrasts and experimental data and is demonstrated to have superior super-resolution imaging capabilities for high-contrast 3-D objects.

Hybrid Electromagnetic Inversion of 3-D Irregular Scatterers Embedded in Layered Media by VBIM and MET

This communication presents an efficient hybrid nonlinear electromagnetic inversion method. The voxel-based variational Born iterative method (VBIM) is combined with the maximum entropy thresholding (MET) technique to reconstruct 3-D irregular scatterers embedded in the layered media. In each iteration, the VBIM first outputs the model parameters in discretized cells of the inversion domain, and the MET is then used to categorize them into “background” and “scatterer” types. Partial “background” cells will be removed in the next round iteration according to an adjustable threshold corresponding to the maximum entropy since they have no contribution to the measured scattered fields. Consequently, the inversion domain is gradually downsized in the successive iterations. The computation efficiency, memory cost, and the reconstruction accuracy are compared for VBIM with and without the MET. Their antinoise ability is also compared.

Fast Multiparametric Electromagnetic Full-Wave Inversion via Solving Contracting Scattering Data Equations Optimized by the 3-D MRF Model

This article presents a novel hybrid electromagnetic full-wave inversion that combines the traditional inversion method, i.e., the variational Born iterative method (VBIM), with the 3-D Markov random field (MRF) model. In each iteration, VBIM first reconstructs the model parameters of all discretized cells in the inversion domain by solving the discretized data equations. Then, MRF is adopted to classify the cells according to the reconstructed parameter values, i.e. to determine which homogenous scatterer a certain cell is belonging to. Finally, partial cells classified as “background” are removed, and partial cells classified as “scatterer” are merged. Consequently, the discretized data equations gradually contract, and the unknowns in the following VBIM iterations are reduced. Numerical experiments show that compared with the traditional VBIM, the proposed hybrid VBIM-MRF model can achieve higher reconstruction accuracy for multiple model parameters. Furthermore, the computational cost is also significantly reduced.

Simulation of 3-D Electromagnetic Scattering and Inverse Scattering by Arbitrary Anisotropic Dielectric Objects Embedded in Layered Arbitrary Anisotropic Media

This communication studies 3-D electromagnetic (EM) scattering and inverse scattering of arbitrary anisotropic dielectric objects embedded in multilayered media. Different from previous work, the scatterer is not only arbitrary anisotropic, but also the arbitrary anisotropy of the layered background media is formed by rotation of the principal axes of uniaxial media. In addition, the scatterers are allowed to be placed across layer boundaries. The forward model is formulated by the electric field integral equation (EFIE), solved by the biconjugate gradient stabilized (BCGS) method, and accelerated by the fast Fourier transform (FFT). The time complexity is analyzed for the BCGS-FFT applied to layered arbitrary anisotropic media. In the inversion model, the variational Born iterative method (VBIM) is employed to reconstruct the arbitrary anisotropic profiles of both the permittivity and conductivity. Numerical simulations are performed to verify the feasibility and antinoise ability of the forward and inverse solvers in the circumstance of layered arbitrary anisotropic background media.

Dual-Module NMM-IEM Machine Learning for Fast Electromagnetic Inversion of Inhomogeneous Scatterers With High Contrasts and Large Electrical Dimensions

A dual-module machine learning scheme is proposed to reconstruct inhomogeneous scatterers with high contrasts and large electrical dimensions. The first nonlinear mapping module (NMM) is an extreme learning machine (ELM), which is used to convert the measured scattered fields at the receiver arrays into the preliminary images of the scatterers. The second image-enhancing module (IEM) is a convolutional neural network (CNN), which is used to refine further the images from the NMM to obtain high-accuracy pixel-based model parameter distribution in the inversion domain. Compared with the traditional approximate methods such as backpropagation, the NMM-IEM machine learning can produce the preliminary image with a much higher accuracy but the unknown weight matrices of the ELM are only solved once during training. Hence, the IEM connected to the NMM has a simple architecture and can be trained at a rather low cost. The performance of the proposed dual-module NMM-IEM scheme and the conventional variational Born iterative method is compared in terms of inversion of scatterers with different electrical sizes and contrasts. Meanwhile, the NMM-IEM is also assessed for the inversion of scatterers with high contrasts and large electrical dimensions and experimental data. Finally, the NMM-IEM is compared with the CNNs used in the previous works.

Multiparametric Electromagnetic Inversion of 3-D Biaxial Anisotropic Objects Embedded in Layered Uniaxial Media Using VBIM Enhanced by Structural Consistency Constraint

This article presents the 3-D inversion of biaxial anisotropic objects embedded in layered uniaxial media by the variational Born iterative method (VBIM). Although previously only the isotropic complex permittivity profile was reconstructed, here we retrieve all the six diagonal anisotropic permittivity and conductivity parameters simultaneously. In the forward model, the stabilized biconjugate gradient fast Fourier transform (BCGS-FFT) method is used to solve the volume electric field integral equation (EFIE). In the inversion model, VBIM is employed to minimize the cost function. In order to improve the inversion results, we propose a novel structural consistency constraint (SCC) applied to VBIM. The SCC is based on the Monte Carlo method and reduces the computational domain in the Born iteration. The major new contribution of this work is to combine the deterministic inversion algorithm VBIM with the stochastic Monte Carlo method to reconstruct all the diagonal anisotropic parameters of the scatterers. Several typical numerical models are simulated, and the results validate the efficiency, antinoise ability as well as the accuracy of VBIM-SCC for reconstructing 3-D biaxial anisotropic objects embedded in multilayered uniaxial media.

Retrieval of Composite Model Parameters for 3-D Microwave Imaging of Biaxial Objects by BCGS-FFT and PSO

This article presents the 3-D quantitative microwave imaging of homogeneous biaxial anisotropic objects with their optical axes being in arbitrary directions. Two sets of model parameters, the anisotropic dielectric parameters and the angles of optical axes with respect to the user-defined coordinate system, are retrieved iteratively. In the forward model, the total fields are solved from the discretized combined field volume integral equations (CFVIEs) by the stabilized biconjugate gradient fast Fourier transform (BCGS-FFT) method. In the inversion model, the optimized solutions for two sets of model parameters are obtained by the particle swarm optimization (PSO) method. If the geometric models of the homogeneous anisotropic objects are unknown, a full-wave inversion method, the variational Born iterative method (VBIM), is combined with the multiparametric structural consistency constraint (SCC) to find the spatial boundaries of the homogeneous objects in advance. Two numerical examples are used to test the proposed method, and the results show that BCGS-FFT-PSO not only has strong adaptability for optical axes angles varying in wide ranges but also can retrieve composite model parameters of multiple homogeneous biaxial objects simultaneously even when the measured data are contaminated by noise.

Hybrid Reconstruction of Subsurface 3-D Objects Using FRTM and VBIM Enhanced by Monte Carlo Method

A hybrid method is proposed to reconstruct the subsurface 3-D objects with electromagnetic fields. The frequency-domain reverse time migration (FRTM) is first used to determine the approximate locations and sizes of the objects. Then, based on these results, the full-wave inversion, the variational Born iteration method (VBIM) is used to reconstruct both the shapes and dielectric parameters of the objects. The Monte Carlo method (MCM) is adopted to further refine the reconstructed shapes. Numerical simulations show that the proposed hybrid method can be effectively used for the subsurface imaging and detection.

Fast Electromagnetic Inversion of Inhomogeneous Scatterers Embedded in Layered Media by Born Approximation and 3-D U-Net

This letter presents a 3-D electromagnetic inversion method based on the Born approximation (BA) and a convolutional neural network (CNN), the 3-D U-Net. In the training stage, the BA is first used to obtain the preliminary 3-D images of a series of homogeneous scatterers with regular shapes that are further improved by the Monte Carlo method. Then, these images are used to train the 3-D U-Net. In the testing stage, inhomogeneous scatterers with complex shapes are reconstructed by both the trained 3-D U-Net and the traditional iterative method, variational Born iteration method (VBIM). Their performance is evaluated and compared.

3-D Electromagnetic Scattering and Inverse Scattering by Magnetodielectric Objects With Arbitrary Anisotropy in Layered Uniaxial Media

3-D electromagnetic (EM) scattering and inverse scattering of magnetodielectric objects with arbitrary anisotropy embedded in layered uniaxial media are studied. The stabilized biconjugate gradient fast Fourier transform (BCGS-FFT) method is employed to solve the forward scattering problem formulated by the combined field volume integral equation (CFVIE). In the inversion, we use the variational Born iterative method (VBIM) enhanced by the structural consistency constraint (SCC) to reconstruct eighteen unknown dielectric parameters of the scatterers simultaneously. In order to further improve the accuracy of the reconstructed permittivity, permeability, and conductivity of the scatterers, we propose the structural continuity scanning (SCS) technique. Comparisons between the BCGS-FFT solution and simulated results from finite element method (FEM) are performed to verify the reliability and accuracy of the forward solver. Meanwhile, we reconstruct the multiple anisotropic parameters of several typical structures embedded in layered uniaxial media to show the feasibility and anti-noise ability of the inversion algorithm.

Subspace-Based Variational Born Iterative Method for Solving Inverse Scattering Problems

In this letter, a subspace-based variational Born iterative method (S-VBIM) is proposed to solve the inverse scattering problem. The proposed method retrieves the deterministic subspace of the variational induced current at the beginning of the iterative computation, and thus, the process of the total electric field approximation is more accurate than the Born approximation does in VBIM. Therefore, S-VBIM achieves faster convergence speed and higher solution accuracy than VBIM. In addition, since the singular value decomposition, which is the main process of retrieving the variational induced current for radiation operator matrix, only needs to be performed once, the computational cost is reduced significantly. Finally, the numerical results validate the effectiveness of S-VBIM and comparisons with VBIM are made.

Joint Petrophysical and Structural Inversion of Electromagnetic and Seismic Data Based on Volume Integral Equation Method

A joint petrophysical and structural inversion method for electromagnetic (EM) and seismic data based on the volume integral equation (VIE) is proposed in this paper. In the forward EM problem, only the contrast of conductivity is solved by the electric field integral equation method. However, in the forward seismic problem, both the contrasts of velocity and mass density are solved by the combined field VIE method. Both forward solvers are accelerated by the fast Fourier transform. In the inversion problem, by using the petrophysical equations about the porosity and saturation and applying the chain rule, we fuse the EM and seismic data and construct the joint petrophysical inversion equations, which can be solved by the variational Born iteration method. Then, in order to further enhance the reconstructed results of the joint petrophysical inversion, we enforce the structural similarity constraint between porosity and water saturation and add the cross-gradient function to the joint petrophysical inversion cost function. Two typical geophysical models based on the remote sensing measurement are used to validate the proposed methods. One is the cross-well model, and the other is the marine surface exploration model. The advantage of the joint inversion compared with the separate inversion is evaluated based on the resolution and the data misfits of the reconstructed profiles as well as the antinoise ability.

2-D Electromagnetic Scattering and Inverse Scattering From Magnetodielectric Objects Based on Integral Equation Method

Efficient 2-D full-waveform forward and inverse electromagnetic scattering solvers are presented in this communication. The combined field volume integral equation (CFVIE) is formulated in a layered medium. Different from the previous work, the mixed-order basis function is adopted in the discretization of the 2-D CFVIE. Meanwhile, the permittivity and permeability are simultaneously reconstructed in the inversion through the variational Born iteration method, which is enhanced with a self-adaptive regularization factor and applied to the multifrequency data. The proposed forward solver is validated by comparisons with the finite-element simulations.

Electromagnetic Forward and Inverse Algorithms for 3-D Through-Casing Induction Mapping of Arbitrary Fractures

This letter extends the axisymmetric version of the efficient forward and inverse algorithms to characterize and reconstruct arbitrary 3-D fractures in a cased borehole environment. We improve our previous hybrid distorted Born approximation and stabilized biconjugate gradient fast Fourier transform method as the 3-D forward modeling algorithm that can significantly reduce the computational cost in forward modeling. The bounding constraints and model parameter transformation functions are introduced to our previous axisymmetric variational Born iterative inverse method to effectively reconstruct 3-D fractures. Numerical results show orders of magnitude higher efficiency of the forward algorithm than the finite-element method, and the effectiveness of the inverse algorithm for through-casing 3-D fracture reconstruction.

Joint Inversion of Electromagnetic and Seismic Data Based on Structural Constraints Using Variational Born Iteration Method

An efficient 2-D joint full-waveform inversion method for electromagnetic and seismic data in a layered medium background is developed. The joint inversion method based on the integral equation (IE) method is first proposed in this paper. In forward computation, the IE method is employed, which usually has smaller discretized computation domain and less cumulative error compared with the finite-difference method. In addition, fast Fourier transform is used to accelerate the convolution between Green’s functions and induced sources due to the shift invariance property of the layered Green’s functions in the horizontal direction. In the inversion model, the cross-gradient function is incorporated into the cost function of the separate inversion to enforce the structure similarity between electric conductivity and seismic-wave velocity. We use the improved variational Born iteration method and two different iteration strategies to minimize the cost function and reconstruct the contrasts. Several typical models in geophysical applications are used to validate our joint inversion method, and the numerical simulation results show that joint inversion can improve the inversion results when compared with those from the separate inversion.