Distorted Born Iterative Method Research Articles

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.

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.

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.

Calibrated Frequency-Division Distorted Born Iterative Tomography for Real-Life Head Imaging.

The clinical use of microwave tomography (MT) requires addressing the significant mismatch between simulated environment, which is used in the forward solver, and real-life system. To alleviate this mismatch, a calibrated tomography, which uses two homogeneous calibration phantoms and a modified distorted Born iterative method (DBIM), is presented. The two phantoms are used to derive a linear model that matches the forward solver to real-life measurements. Moreover, experimental observations indicate that signal quality at different frequencies varies between different antennas due to inevitably inconsistent manufacturing tolerance and variances in radio-frequency chains. An optimum frequency, at which the simulated and measured signals of the antenna present maximum similarity when irradiating the calibrated phantoms, is thus calculated for each antenna. A frequency-division DBIM (FD-DBIM), in which different antennas in the array transmit their corresponding optimum frequencies, is subsequently developed. A clinical brain scanner is then used to assess performance of the algorithm in lab and healthy volunteers' tests. The linear calibration model is first used to calibrate the measured data. After that FD-DBIM is used to solve the problem and map the dielectric properties of the imaged domain. The simulated and experimental results confirm validity of the presented approach and its superiority to other tomographic method.

Distorted Born iterative method with back-propagation improves permittivity reconstruction

AbstractThe distorted Born iterative method (DBIM) is a widely used quantitative reconstruction algorithm to solve the inverse scattering problems of microwave imaging. The major mathematical challenges in solving such problems are non-uniqueness, non-linearity, and ill-posedness. Due to these issues, the optimization algorithm converges to a local minimum. This drawback can be overcome by selecting the correct initial guess solution, which helps to escape local minima and thus guides the inversion algorithm to a satisfactory result. This study uses a back-propagation algorithm to calculate the initial estimate, which significantly accelerates the rate of convergence and improves the accuracy of the standard DBIM approach. The results of this method are compared with zero initialization and Born-approximated initialization. For comparison, weak as well as strong scattering profiles of synthetic and experimental dataset are considered. The results suggest that the proposed method provides a significant improvement in terms of computing cost and efficiency. Furthermore, the proposed technique has the potential to successfully push the limits of reconstructible contrast.

Fractional Regularized Distorted Born Iterative Method for Permittivity Reconstruction

Fractional Regularized Distorted Born Iterative Method for Permittivity Reconstruction

Adaptive Clustering Distorted Born Iterative Method for Microwave Brain Tomography With Stroke Detection and Classification.

A modified distorted Born iterative method (DBIM), which includes clustering of reconstructed electrical properties (EPs) after certain iterations, is presented for brain imaging aiming at stroke detection and classification. For this approach to work, a rough estimation of number of different materials (or bio-tissues) in the imaged domain and their corresponding rough dielectric properties (permittivity and conductivity) are needed as a prior information. The proposed adaptive clustering DBIM (AC-DBIM) is compared with three conventional methods (DBIM, multiplicative regularized contrast source inversion (MR-CSI), and CSI for shape and location reconstruction (SL-CSI)) in two-dimensional scenario on a head phantom and numerical head model with different strokes. Three-dimensional simulations are also conducted to indicate the suitability of AC-DBIM in real-life brain imaging. Lastly, the proposed algorithm is assessed using a clinical electromagnetic head scanner developed on phantoms. The simulation and experimental results show superiority of AC-DBIM compared to conventional methods. AC-DBIM achieves significant improvement in the size and shape reconstruction and reduction in errors and standard deviation of the reconstructed εr and σ at clinical scenarios compared with conventional DBIM.

Three-Dimensional Microwave Head Imaging with GPU-Based FDTD and the DBIM Method.

We present a preliminary study of microwave head imaging using a three-dimensional (3-D) implementation of the distorted Born iterative method (DBIM). Our aim is to examine the benefits of using the more computationally intensive 3-D implementation in scenarios where limited prior information is available, or when the target occupies an area that is not covered by the imaging array’s transverse planes. We show that, in some cases, the 3-D implementation outperforms its two-dimensional (2-D) counterpart despite the increased number of unknowns for the linear problem at each DBIM iteration. We also discuss how the 3-D algorithm can be implemented efficiently using graphic processing units (GPUs) and validate this implementation with experimental data from a simplified brain phantom. In this work, we have implemented a non-linear microwave imaging approach using DBIM with GPU-accelerated FDTD. Moreover, the paper offers a direct comparison of 2-D and 3-D microwave tomography implementations for head imaging and stroke detection in inhomogenous anatomically complex numerical head phantoms.

Experimental Validation of Microwave Imaging Prototype and DBIM-IMATCS Algorithm for Bone Health Monitoring

The evaluation of the microwave imaging (MWI) prototype and imaging algorithms on experimental bone phantoms is a precursor step before clinical testing for measuring <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in vivo</i> dielectric properties of human bones. To this end, this paper presents microwave tomographic image reconstruction of experimental phantoms of normal and diseased human calcaneus bone using an MWI prototype and distorted Born iterative method (DBIM) algorithm for bone health monitoring application. A two-layered simplified cylindrical-shaped 3-D printed phantom was used to mimic the human calcaneus bone. The external and internal layers of the bone phantom mimic the cortical bone and trabecular bone, respectively. Liquid tissue-mimicking mixtures (TMM) for normal bone, osteoporotic bone, and osteoarthritis bone were prepared. The phantoms were placed in the imaging prototype and the electromagnetic inverse scattering problem was solved using the DBIM to create the complex permittivity images. An <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L_{2}$ </tex-math></inline-formula> -based regularization approach was adopted along with the iterative method with adaptive thresholding for compressed sensing (IMATCS) to overcome the ill-posedness and to solve the underdetermined set of linear equations at each DBIM iteration. The reconstruction of dielectric properties of bone phantoms have shown that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L_{2}$ </tex-math></inline-formula> -IMATCS approach provides a robust reconstruction of diverse bone phantoms with acceptable accuracy. Moreover, the osteoporotic and osteoarthritis bone phantoms were distinguished based on reconstructed dielectric properties with an average percentage difference of 26% at 3 GHz. This paper has made the first attempt to validate an MWI prototype for bone imaging application. A DBIM-based iterative method has been employed to classify normal and diseased bone phantoms.

Experimental Validation of the DBIM-TwIST Algorithm for Brain Stroke Detection and Differentiation Using a Multi-Layered Anatomically Complex Head Phantom

We present an experimental validation of the distorted Born iterative method with the two-step iterative shrinkage thresholding (DBIM-TwIST) algorithm for the problem of brain stroke detection and differentiation, using an anatomically accurate, multi-layer head phantom. To this end, we have developed a gelatine-based, anatomically complex head phantom which mimics various brain tissues and also includes a target mimicking hemorrhagic or ischemic stroke. We simulated the model and setup using CST Microwave Studio and then used our experimental imaging setup to collect numerical and measured data, respectively. We then used our DBIM-TwIST algorithm to reconstruct the dielectric properties of the imaging domain for both simulated and measured data. Results from our CST simulations showed that we are able to locate and reconstruct the permittivity of different stroke targets using an approximate initial guess. Our experimental results demonstrated the potential and challenges for successful detection and differentiation of the stroke targets.

Resolution Enhanced Distorted Born Iterative Method Using ROI Limiting Scheme for Microwave Breast Imaging

This paper presents a high-resolution inverse scattering approach using the distorted Born iterative method (DBIM) by utilizing a region-of-interest limitation scheme that assumes microwave breast cancer imaging modality. The DBIM-based tomography approach provides a quantitative dielectric profile, contributing to reliable cancer recognition, especially for dense breast media. However, if the inversion cell is small, the original DBIM will be inaccurate because the unknowns will far exceed the number of data samples. Thus, high-resolution reconstruction is a challenging issue and is not unique only to the DBIM. To address this difficulty, this paper introduces a region-of-interest (ROI)-limited scheme by developing low-resolution DBIM images. Numerical tests based on 2D FDTD demonstrate that the proposed approach provides high-resolution and accurate dielectric profiles for highly heterogeneous realistic phantoms.

An Enhanced Multi-Frequency Distorted Born Iterative Method for Ultrasound Tomography Based on Fundamental Tone and Overtones

Conventional Distorted Born Iterative Method (DBIM) using single frequency has low resolution and is prone to creating images with high-contrast subjects. We propose a productive frequency combination method to better result in tomographic ultrasound imaging based on the multi-frequency technique. This study uses the natural mechanism of emitting oscillators' frequencies and uses these frequencies for imaging in iterations. We use a fundamental tone (i.e., the starting frequency f0) for the first iteration in DBIM, then consecutively use its overtones for the next ones. The digital simulation scenarios are tested with other multi-frequency approaches to prove our method's feasibility. We performed 57 different simulation scenarios on the use of multi-frequency information for the DBIM method. As a result, the proposed method for the smallest normalization error (RRE = 0.757). The proposed method's imaging time is not significantly longer than the way of using single frequency information.

Influence of the multi-resolution technique on tomographic reconstruction in ultrasound tomography

The greatest advantage of scattering theory-based ultrasound tomography (UT) is its ability to investigate small structures. DBIM is the Distorted Born Iterative Method. The nearest neighbour interpolation method is used to enhance the reconstruction performance and reduce the reconstruction time. The raw (N 1 × N 1) and dense (N 2 × N 2) meshed integration areas are reconstructed in NN 1 and NN 2 iterations, respectively. However, choosing the best value of NN 1 to get the highest performance was not mentioned in previous works. If it is not well chosen, the reconstruction quality is even worse than that when using no interpolation. This study proposes a method to enhance the UT reconstruction by using the nearest neighbour interpolation (MR-DBIM). The corresponding algorithms are specified by the graphical concurrent programming language of Sleptsov nets. Some significant results are (1) the MR-DBIM is only meaningful when (i.e. sparse scattering domain); (2) the best performance is obtained in the DBIM when Nt = Nr , but in the MR-DBIM when Nr = 2Nt ; (3) the well-investigated value of NN 1 is 2 when and is 3 when .