Born Iterative Method Research Articles

Deep Learning-Based Electric Field Enhancement Imaging Method for Brain Stroke

In clinical settings, computed tomography (CT), magnetic resonance imaging (MRI), or positron emission tomography (PET) are commonly employed in brain imaging to assist clinicians in determining the type of stroke in patients. However, these modalities are associated with potential hazards or limitations. In contrast, microwave imaging emerges as a promising technique, offering advantages such as non-ionizing radiation, low cost, lightweight, and portability. The primary challenges faced by microwave tomography include the severe ill-posedness of the electromagnetic inverse scattering problem and the time-consuming nature and unsatisfactory resolution of iterative quantitative algorithms. This paper proposes a learning electric field enhancement imaging method (LEFEIM) to achieve quantitative brain imaging based on a microwave tomography system. LEFEIM comprises two cascaded networks. The first, based on a convolutional neural network, utilizes the electric field from the receiving antenna to predict the electric field distribution within the imaging domain. The second network employs the electric field distribution as input to learn the dielectric constant distribution, thereby realizing quantitative brain imaging. Compared to the Born Iterative Method (BIM), LEFEIM significantly improves imaging time, while enhancing imaging quality and goodness-of-fit to a certain extent. Simultaneously, LEFEIM exhibits anti-noise capabilities.

Frequency-hopping along with resolution-turning for fast and enhanced reconstruction in ultrasound tomography

The distorted Born iterative (DBI) method is considered to obtain images with high-contrast and resolution. Besides satisfying the Born approximation condition, the frequency-hopping (FH) technique is necessary to gradually update the sound contrast from the first iteration and progress to the actual sound contrast of the imaged object in subsequent iterations. Inspired by the fact that the higher the frequency, the higher the resolution. Because low-frequency allows for low-resolution object imaging, hence for high-resolution imaging requirements, using low-frequency to possess a high-resolution image from the first iteration will be less efficient. For an effective reconstruction, the object’s resolution at low frequencies should be small. And similarly, with high frequencies, the object resolution should be larger. Therefore, in this paper, the FH, and the resolution-turning (RT) technique are proposed to obtain object images with high-contrast and -resolution. The convergence speed in the initial iterations is rapidly achieved by utilizing low frequency in the frequency-turning technique and low image resolution in the resolution-turning technique. It is crucial to ensure accurate object reconstruction for subsequent iterations. The desired spatial resolution is attained by employing high frequency and large image resolution. The resolution-turning distorted Born iterative (RT-DBI) and frequency-hopping distorted Born iterative (FH-DBI) solutions are thoroughly investigated to exploit their best performance. This makes sense because if it is not good to choose the number of iterations for the frequency f1 in FH-DBI and for the resolution of N1 × N1 in RT-DBI, then these solutions give even worse quality than traditional DBI. After that, the RT-FH-DBI integration was investigated in two sub-solutions. We found that the lower frequency f1 used both before and after the RT would get the best performance. Consequently, compared to the traditional DBI approaches, the normalized error and total runtime for the reconstruction process were dramatically decreased, at 83.6% and 18.6%, respectively. Besides fast and quality imaging, the proposed solution RT-FH-DBI is promised to produce high-contrast and high-resolution object images, aiming at object reconstruction at the biological tissue. The development of 3D imaging and experimental verification will be studied further.

A Hybrid Born Iterative Method for Diagnosis of Planar Arrays With Amplitude-Only Far-Field Measurements

A Hybrid Born Iterative Method for Diagnosis of Planar Arrays With Amplitude-Only Far-Field Measurements

Deep learning based distorted Born iterative method for improving microwave imaging

Abstract The distorted Born iterative method (DBIM) is a popular quantitative reconstruction algorithm for solving electromagnetic inverse scattering problems. These problems are non-linear and ill-posed. As a result, the efficiency of the method is limited by local minima. To overcome this, a correct initial guess solution is needed to obtain a satisfactory result. The U-Net based Convolutional Neural Network (CNN) is used in this study to make a good initial guess for the DBIM technique. The permittivity estimate produced at the output of U-Net is then refined using an existing iterative optimization process. This method’s findings are compared with the conventional DBIM approach. Strong scattering profiles of synthetic and experimental datasets with homogeneous and heterogeneous scatterers are investigated to validate the efficiency of the proposed technique. The results suggest that the use of the deep learning technique for an initial guess of DBIM improves accuracy and convergence rate significantly.

A new microwave nondestructive testing method of relative permittivity and conductivity for stratified media based on dyadic Green's function I: Theory

Stratified composite materials are employed in many industrial scenes, including bridge construction, vehicle, ship, aviation, etc. However, minor imperfections in composite materials can have significant effects and even result in disasters when used. In this work, a multi-parameter inversion method that can simultaneously reconstruct the conductivity and permittivity parameters of layered media is devised and presented for microwave nondestructive testing (NDT) of the minor imperfections in stratified composite materials. Firstly, the calculations for the electric field in the layered medium model with the emission source as the point source are provided. Secondly, the spatial electric field variations (the Fréchet derivative) caused by changes in conductivity and permittivity of arbitrary layers are calculated. Then the permittivity and conductivity are reconstructed using the Distorted Born iterative method (DBIM) technique. The numerical examples at microwave frequencies show that the inversion technique has a strong capacity to reconstruct for a single parameter (relative permittivity or conductivity) and multi-parameter (relative permittivity and conductivity). Our suggested multi-parameter inversion technique in this research has excellent potential for use in the identification of multilayer composites and subsurface inclusions.

Microwave Imaging of 3-D Dielectric–Magnetic Penetrable Objects Based on Integral Equation Method

Some objects like mineral substances may be both dielectric and magnetic and their high-resolution inner imaging is very desirable when detecting and analyzing their ingredients. An integral equation method is developed for reconstructing or imaging such objects by using microwave illumination. Since the objects are both dielectric and magnetic, full volume integral equations (VIEs) are needed to describe the problem. The objects can be reconstructed by alternatively solving the forward scattering VIEs (FSVIE) and the inverse scattering VIEs (ISVIEs) under the Born iterative method (BIM) or distorted BIM (DBIM). The Nyström method is used to solve the FSVIEs while a Gauss-Newton minimization method (GNMM) with a multiplicative regularization scheme (MRS) is employed to solve the ISVIE. The Nyström method is more suitable for solving the FSVIEs of inverse problems because it is a point-matching method. To accelerate the solution, the multilevel fast multipole algorithm (MLFMA) is incorporated with the Nyström method which is new for solving inverse problems. Also, using the MRS-based GNMM to solve the ISVIEs can facilitate the selection of regularization parameter and reduce the requirements on measured data. Numerical examples are provided to demonstrate the approach and good results have been achieved.

Experimental Assessment of Axillary Lymph Node Microwave Tomography Using Anthropomorphic Phantoms

We assess the application of microwave tomography (MWT) for the detection of axillary lymph nodes (ALNs) in breast cancer patients. We numerically study the effects of limiting angular view in axillary MWT, as probes can only be placed on a limited arc around the axillary region. We also numerically study the possibility of increasing the amount of retrievable information by acquiring data in two consecutive steps, with a single antenna set in two different angular positions. We finally experimentally test axillary MWT on anthropomorphic phantoms with different levels of anatomical fidelity, and different ALN positions. Our MWT system (0.8–2.5 GHz) employs six monopole antennas placed on a single transverse plane, facing the axillary region. The reconstruction algorithm implements the distorted Born iterative method, combined with the two-step iterative shrinkage/thresholding for the inversion (DBIM-TwIST). Our numerical results i) highlight the challenges associated with the limited angular view, and ii) show that performing two-step angular measurements enhances imaging results, suggesting that rotating the antenna set between consecutive measurements is an effective means to increase the retrievable information in ALN MWT. Our experimental results show that our MWT system can detect an ALN in different positions. To the best of our knowledge this is the first paper to assess ALN MWT in a realistic 3-D experimental scenario.

Bayesian inverse scattering theory for multiparameter full-waveform inversion in transversely isotropic media

Full-waveform inversion (FWI) is an effective method that uses kinematic and dynamic information to provide an accurate image of the subsurface target. By applying the Bayesian approaches to FWI workflow, the traditional deterministic solution is replaced with a maximum a posteriori solution. This provides not only a reliable subsurface model for the nonlinear inverse scattering problem but also an uncertainty quantification. The high computational cost of the Bayesian FWI and the complexity of a large-scale anisotropy inversion, however, restrict its application in industrial settings. We have developed an FWI approach for tilted transversely isotropic (TTI) media. The strain field and the particle displacement of the anisotropic media are expressed by an integral method based on the scattering theory and an explicit multiscattering modified Green’s function. To reduce the computational cost, the wavefield is calculated by using the preconditioned generalized minimal residual iterative solver, which approximates the solution in the Krylov subspace. The Fréchet derivatives are expressed by using the distorted Born iterative method. An iterated extended Kalman filter is applied to solve the inverse problem and the posterior covariance matrix frequency-by-frequency. Numerical experiments indicate that the presented methodology can reasonably reconstruct the elastic moduli of the TTI medium. Furthermore, the numerical experiments demonstrate the accuracy and feasibility of our methodology.

A matrix‐free variant of the distorted Born iterative method for seismic full‐waveform inversion

AbstractThe distorted Born iterative method reduces a nonlinear inverse scattering problem to a sequence of (ill‐posed) linear inverse scattering problems that can be solved using a regularized least‐squares formulation. This method was originally applied to two‐dimensional electromagnetic problems but has been implemented to solve acoustic and electromagnetic problems in three dimensions. It has also been applied to seismic problems but only for moderately large two‐dimensional models. Previous applications of the distorted Born iterative method to seismic inverse scattering were based on a matrix representation of the relevant integral operators. The matrix‐based implementation is simple and transparent but not very suitable for large‐scale computations since the memory requirement and computational cost scales like N2 and N3, where N is the number of grid blocks. In this paper, we introduce a matrix‐free variant of the distorted Born iterative method, which is much more suitable for large‐scale problems, since the memory requirements and computational cost have been reduced to N and Nlog(N), respectively. Our matrix‐free implementation utilizes a fast‐Fourier‐transform‐accelerated iterative method to solve the linear system that results after discretization. In the computation of Fréchet derivatives, we avoid the direct computation of Green's functions for heterogeneous media, by solving an equivalent direct scattering problem. The adjoint of the Fréchet derivative operator is also computed in an efficient matrix‐free manner, by exploiting the physical interpretation and reciprocity of the Green's function. We illustrate the potential of the matrix‐free variant of the distorted Born iterative method using synthetic waveform data for large two‐ and three‐dimensional models. The main conclusion is that the distorted Born iterative method has been transformed into a more practical tool for seismic as well as electromagnetic and acoustic nonlinear inverse scattering.

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.

Neural Born Iterative Method for Solving Inverse Scattering Problems: 2D Cases

In this article, we propose the neural Born iterative method (NeuralBIM) for solving 2-D inverse scattering problems (ISPs) by drawing on the scheme of the physics-informed supervised residual learning (PhiSRL) to emulate the computing process of the traditional Born iterative method (TBIM). NeuralBIM uses independent convolutional neural networks (CNNs) to learn the alternate update rules of two different candidate solutions regarding the residuals. Two different schemes are presented in this article, including the supervised and unsupervised learning schemes. With the dataset generated by the method of moments (MoM), supervised NeuralBIM is trained with the knowledge of the total fields and contrasts. Unsupervised NeuralBIM is guided by the physics-embedded objective function founding on the governing equations of ISPs, which results in no requirement of the total fields and contrasts for training. Numerical and experimental results further validate the efficacy of NeuralBIM.

A BIM-Based Algorithm for Quantitative Monitoring of Temperature Distribution During Breast Hyperthermia Treatments

Microwave hyperthermia (MH) treatment for breast cancer is a research interest due to its capability to initiate cell necrosis in malignant tumor or to enhance the effect of other treatment modalities such as chemotherapy. The goal of MH treatment is to increase temperature of malignant tumor up to 45°C based on the treatment plan; however, microwave energy focusing is a challenging problem and may cause unwanted hotspots on healthy tissues; therefore, there is a need to monitor the temperature. In this paper, an iterative differential microwave imaging algorithm for temperature monitoring is presented. The algorithm is based on Born iterative method (BIM) and Tikhonov regularization. Feasibility of the algorithm is shown by a large computational study using realistic digital breast phantoms via TM <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</sub> polarized 2-D scattered fields. Also, some results are given for calibrated scattering parameters, which are obtained from both a 3-D electromagnetic simulation program and a simple measurement setup. An approach for selection of matching medium in hyperthermia monitoring applications is also presented. The reconstructions are performed with scattered field data collected at 11 discrete frequency points uniformly taken from the 0.5-1.5 GHz range. For a specific heating scenario in 2-D problem, reconstruction error is lower than 0.3% with ±10% noise on reference dielectric property distribution and 40 dB signal to noise ratio (SNR). The results show that the proposed approach provides up to 3°C and 0.1°C resolution in temperature estimation with ±10% noise on reference dielectric property distribution for 30 dB and 60 dB SNR values, respectively.

Born Iterative Method Based on Convolutional Dictionary Learning for Inverse Scattering Problems

In this letter, a Born iterative method based on convolutional dictionary learning is proposed to solve inverse scattering problems. The method is inspired by compressed sensing, which indicates that the contrast can be represented by nonzero expansion coefficients with respect to expansion bases. The inverse scattering problem is changed from solving the contrast to solving the nonzero expansion coefficients under the expansion bases. In our method, the bases are obtained by convolutional dictionary learning from a set of training data, which learn and characterize the intrinsic laws of the local contrast. The performance of the proposed method is verified using simulated data. We compare it with the Born iterative method and the contrast source inversion method, and the results show that the proposed method can get better results.

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.

A Trained Iterative Shrinkage Approach Based on Born Iterative Method for Electromagnetic Imaging

A trained-based Born iterative method (TBIM) is developed for electromagnetic imaging (EMI) applications. The proposed TBIM consists of a nested loop; the outer loop executes TBIM iteration steps, while the inner loop executes a trained iterative shrinkage thresholding algorithm (TISTA). The applied TISTA runs linear Landweber iterations implemented with a trained regularization network designed based on U-net architecture. A normalization process was imposed in TISTA that made TISTA training applicable within the proposed TBIM. The iterative utilization of the regularization network in TISTA is a bottleneck that demands high memory allocation through the training process. Therefore TISTA within each TBIM step was trained separately. The TISTA regularization network in each TBIM step was initialized using the weights from the previous TBIM step. The above approach achieved high-quality image restoration after running few TBIM steps while maintained low memory allocation through the training process. The proposed framework can be extended to Newton or quasi-Newton schemes, where within each Newton iteration, a linear ill-posed problem is optimized that differs from one example to another. The numerical results illustrated in this work show the superiority of the proposed TBIM compared to the conventional sparse-based Born iterative method (SBIM).

High Sound-Contrast Inverse Scattering by MR-MF-DBIM Scheme

In ultrasound tomography, cross-sectional images represent the spatial distribution of the physical parameters of a target of interest, which can be obtained based on scattered ultrasound measurements. These measurements can be obtained from dense datasets collected at different transmitter and receiver locations, and using multiple frequencies. The Born approximation method, which provides a simple linear relationship between the objective function and the scattering field, has been adopted to resolve the inverse scattering problem. The distorted Born iterative method (DBIM), which utilizes the first-order Born approximation, is a productive diffraction tomography scheme. In this article, the range of interpolation applications is extended at the multilayer level, taking into account the advantages of integrating this multilayer level with multiple frequencies for the DBIM. Specifically, we consider: (a) a multi-resolution technique, i.e., a multi-step interpolation for the DBIM: MR-DBIM, with the advantage that the normalized absolute error is reduced by 18.67% and 37.21% in comparison with one-step interpolation DBIM and typical DBIM, respectively; (b) the integration of multi-resolution and multi-frequency techniques with the DBIM: MR-MF-DBIM, which is applied to image targets with high sound contrast in a strongly scattering medium. Relative to MR-DBIM, this integration offers a 44.01% reduction in the normalized absolute error.

A Hybrid Born Iterative Bayesian Inversion Method for Electromagnetic Imaging of Moderate-Contrast Scatterers With Piecewise Homogeneities

To tackle the problem of imaging moderate-contrast scatterers with piecewise homogeneities, a hybrid Born iterative (BI) Bayesian inversion method is proposed. First, we establish the nonlinear inverse scattering problem in the context of multiple measurement vector (MMV). Then, a computationally efficient BI formulation is used to account for the strong nonlinear relationship between the scattered field and the contrast. In each iteration, a regularization strategy that exploits the piecewise homogeneous <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$a$ </tex-math></inline-formula> priori information of the investigated domain is adopted to overcome the ill-posedness. Our proposed method can tackle the inverse scattering problem in its full nonlinearity and incorporate the available <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$a$ </tex-math></inline-formula> priori information to stabilize the inversion procedure. Several representative tests are carried out to evaluate the performance of the proposed method. It is demonstrated that the proposed method outperforms other existing state-of-the-art algorithms in terms of accuracy, convergence, robustness, and computational efficiency.

Resolution improvement using enriched Krylov subspace for microwave tomography breast imaging system

AbstractIn this work, a resolution improvement strategy has been presented for microwave tomography imaging in the dense and heterogeneous breasts. While improving the spatial resolution by reducing the size of pixels, the resultant image will be affected by spatial oscillation, and it will also lack to differentiate the dielectric values of low contrast tissue regions which are the unavoidable problems in the microwave tomography dense breast imaging. These difficulties are addressed as discrete ill‐posed and ill‐condition problems. In this paper, Enriched Conjugate Gradient Least Square (ECGLS) regularization method has been proposed and it is associated with the Distorted Born Iterative Method (DBIM) to improve the quality of high‐resolution images of heterogeneous and dense type breasts. The Proposed‐ECGLS (PECGLS) method introduces the span of enriched subspace with external penalization value to resolve the discrete ill‐posed problem. The Gram‐Schmidt factorization (or QR factorization) based step length and the external penalization value control the spatial oscillations and resolve the ill‐condition problem with quick convergence. The performance of the Proposed‐ECGLS method has been tested on heterogeneous and dense breast phantoms for synthetically embedded malignant along with differences in permittivity value of +8% to +10% in the fibrogland tissues. The results from the simulation studies depict that the Proposed‐ECGLS has achieved up to 0.8276 of structural similarity in different level discretization error and it is better than the existing methods such as ECGLS and standard CGLS. This high‐resolution microwave tomography imaging method will be helpful in malignant detection and fibroadenoma diagnosis in dense type breasts.

Solving Ultrasound Tomography's Inverse Problem: Automating Regularization Parameter Selection.

Ultrasound tomography (UT) is a noninvasive procedure that can be used to detect breast cancer. Yet, to accomplish this, reconstruction algorithms must solve an inherent nonlinear, ill-posed inverse problem. One solution is to use the distorted Born iterative (DBI) method. However, in order for successful convergence, ill-posed inverse problems must also be solved for each individual iteration. We used the Tikhonov regularization with different algorithms for choosing the regularization parameter that provides optimal balance, a solution neither overregularized nor underregularized. In this article, we propose a novel algorithm for choosing a balanced parameter based on minimizing two inversely proportional components: signal loss and scaled noise errors (SNEs). This begins with an overestimation of the noise in the measured data, which is then appropriately adjusted within each iteration of the DBI method using the discrepancy between measured and calculated data. We compared our algorithm to the L-curve method, as well as generalized cross-validation (GCV) and projection-based regularized total least-squares (PB-RTLS) methods. Four numerical simulations with varying noise levels and aperture settings showed that our algorithm provided the lowest relative error (RE) for phantom reconstruction, signifying image quality compared to the other methods.

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.