Distorted Born Iterative Method Research Articles

Accelerating the T‐matrix approach to seismic full‐waveform inversion by domain decomposition

ABSTRACTA seismic variant of the distorted Born iterative inversion method, which is commonly used in electromagnetic and acoustic (medical) imaging, has been recently developed on the basis of the T‐matrix approach of multiple scattering theory. The distorted Born iterative method is consistent with the Gauss–Newton method, but its implementation is different, and there are potentially significant computational advantages of using the T‐matrix approach in this context. It has been shown that the computational cost associated with the updating of the background medium Green functions after each iteration can be reduced via the use of various linearisation or quasi‐linearisation techniques. However, these techniques for reducing the computational cost may not work well in the presence of strong contrasts. To deal with this, we have now developed a domain decomposition method, which allows one to decompose the seismic velocity model into an arbitrary number of heterogeneous domains that can be treated separately and in parallel. The new domain decomposition method is based on the concept of a scattering‐path matrix, which is well known in solid‐state physics. If the seismic model consists of different domains that are well separated (e.g., different reservoirs within a sedimentary basin), then the scattering‐path matrix formulation can be used to derive approximations that are sufficiently accurate but far more speedy and much less memory demanding because they ignore the interaction between different domains. However, we show here that one can also use the scattering‐path matrix formulation to calculate the overall T‐matrix for a large model exactly without any approximations at a computational cost that is significantly smaller than the cost associated with an exact formal matrix inversion solution. This is because we have derived exact analytical results for the special case of two interacting domains and combined them with Strassen's formulas for fast recursive matrix inversion. To illustrate the fact that we have accelerated the T‐matrix approach to full‐waveform inversion by domain decomposition, we perform a series of numerical experiments based on synthetic data associated with a complex salt model and a simpler two‐dimensional model that can be naturally decomposed into separate upper and lower domains. If the domain decomposition method is combined with an additional layer of multi‐scale regularisation (based on spatial smoothing of the sensitivity matrix and the data residual vector along the receiver line) beyond standard sequential frequency inversion, then one apparently can also obtain stable inversion results in the absence of ultra‐low frequencies and reduced computation times.

A New Inversion Method Based on Distorted Born Iterative Method for Grounded Electrical Source Airborne Transient Electromagnetics

A new iterative inversion algorithm is proposed to reconstruct the electrical conductivity profile in a stratified underground medium for the grounded electrical source airborne transient electromagnetic (GREATEM) system. In forward modeling, we simplify the mathematical expressions of the magnetic fields generated by a finite line source in the layered ground to semianalytical forms in order to save the computation time. The Frechet derivative is derived for the electromagnetic response at the receivers due to a small perturbation of the conductivity in a certain layer underground. The initial expression of the Frechet derivative has an expensive triple integral and contains the Bessel function in the integrand. It is simplified by partially eliminating the integration along the source line and deriving the analytical expression for the integration in the vertical direction inside the perturbed layer. In the inverse solution, we use the distorted Born iterative method (DBIM). This is the first time that the DBIM is applied to data measured by the GREATEM system. Besides, the forward and inverse procedures are carried out in the frequency domain and based on the Frechet derivative of a line source. We demonstrate the validity of our forward model, Frechet derivative, inverse model, and the precision as well as robustness of the inversion algorithm through numerical computation and comparisons. Finally, we apply the inversion algorithm to the measured data and compare the retrieved conductivity to the actual drilling data.

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

The electromagnetic inverse scattering problem can be effectively handled by means of the so-called distorted-Born iterative method (DBIM). A new method, denoted as the subspace-based DBIM (S-DBIM), is proposed. It updates the Green’s function for the inhomogeneous background at each step of the iterative procedure, like DBIM. By linearly retrieving the deterministic subspace of the induced current, the S-DBIM estimates the total electric field more accurately than the DBIM does and thus exhibits a faster convergence speed. To avoid the computationally demanding and arguably hard choice of the optimal Tikhonov regularization term involved in the procedure, a second version of S-DBIM, which is more robust against noise and has faster convergence, is also proposed. Thorough numerical simulations show that both versions of S-DBIM achieve super-resolved retrievals of the benchmark “Austria” profile.

Deterministic compressive sampling for high-quality image reconstruction of ultrasound tomography

BackgroundA well-known diagnostic imaging modality, termed ultrasound tomography, was quickly developed for the detection of very small tumors whose sizes are smaller than the wavelength of the incident pressure wave without ionizing radiation, compared to the current gold-standard X-ray mammography. Based on inverse scattering technique, ultrasound tomography uses some material properties such as sound contrast or attenuation to detect small targets. The Distorted Born Iterative Method (DBIM) based on first-order Born approximation is an efficient diffraction tomography approach. One of the challenges for a high quality reconstruction is to obtain many measurements from the number of transmitters and receivers. Given the fact that biomedical images are often sparse, the compressed sensing (CS) technique could be therefore effectively applied to ultrasound tomography by reducing the number of transmitters and receivers, while maintaining a high quality of image reconstruction.MethodsThere are currently several work on CS that dispose randomly distributed locations for the measurement system. However, this random configuration is relatively difficult to implement in practice. Instead of it, we should adopt a methodology that helps determine the locations of measurement devices in a deterministic way. For this, we develop the novel DCS-DBIM algorithm that is highly applicable in practice. Inspired of the exploitation of the deterministic compressed sensing technique (DCS) introduced by the authors few years ago with the image reconstruction process implemented using l1 regularization.ResultsSimulation results of the proposed approach have demonstrated its high performance, with the normalized error approximately 90% reduced, compared to the conventional approach, this new approach can save half of number of measurements and only uses two iterations. Universal image quality index is also evaluated in order to prove the efficiency of the proposed approach.ConclusionsNumerical simulation results indicate that CS and DCS techniques offer equivalent image reconstruction quality with simpler practical implementation. It would be a very promising approach in practical applications of modern biomedical imaging technology.

Real-Time Three-Dimensional Microwave Monitoring of Interstitial Thermal Therapy.

We report a method for real-time three-dimensional monitoring of thermal therapy through the use of noncontact microwave imaging. This method is predicated on using microwaves to image changes in the dielectric properties of tissue with changing temperature. Instead of the precomputed linear Born approximation that was used in prior work to speed up the frame-to-frame inversions, here we use the nonlinear distorted Born iterative method (DBIM) to solve the electric volume integral equation (VIE) to image the temperature change. This is made possible by using a recently developed graphic processing unit accelerated conformal finite difference time domain method to solve the forward problem and update the electric field in the monitored region in each DBIM iteration. Compared to our previous work, this approach provides a far superior approximation of the electric field within the VIE, and thus yields a more accurate reconstruction of tissue temperature change. The proposed method is validated using a realistic numerical model of interstitial thermal therapy for a deep-seated brain lesion. With the new DBIM, we reduced the average estimation error of the mean temperature within the region of interest from 2.5 to 1.0 for the noise-free case, and from 2.9 to 1.7 for the 2% background noise case.

Multiple-Frequency DBIM-TwIST Algorithm for Microwave Breast Imaging

A novel distorted Born iterative method (DBIM) algorithm is proposed for microwave breast imaging based on the two-step iterative shrinkage/thresholding method. We show that this implementation is more flexible and robust than using traditional Krylov subspace methods such as the CGLS as solvers of the ill-posed linear problem. This paper presents several strategies to increase the algorithm’s robustness: a hybrid multifrequency approach to achieve an optimal tradeoff between imaging accuracy and reconstruction stability; a new approach to estimate the average breast tissues properties, based on sampling along their range of possible values and running a few DBIM iterations to find the minimum error; and finally, a new regularization strategy for the DBIM method based on the $L^{1}$ norm and the Pareto curve. We present reconstruction examples which illustrate the benefits of these optimization strategies, which have resulted in a DBIM algorithm that outperforms our previous implementations for microwave breast imaging.

Microwave Imaging via Distorted Iterated Virtual Experiments

The linearity of the scattering phenomenon with respect to primary sources allows to recombine a posteriori the available experiments and build, in a synthetic fashion, new “virtual” experiments. Starting from this circumstance, an iterative procedure is proposed as an effective approach to tackle nonlinear inverse scattering problems. In this procedure, the virtual experiments, the Green's function, and the corresponding physical inspired field approximations are updated at each iteration. The structure and the complexity of the approach are comparable with those of the widely adopted distorted Born iterative method, but its performances are remarkably better, thanks to extended validity of the exploited field approximation. The overall approach also takes advantage of a compressive sensing inspired regularization scheme to promote sparsity in the search of piecewise constant dielectric profiles and further improve the accuracy of the imaging results. Examples with numerical and experimental data are given to assess the method.

Reconstruction of sound speed distributions from pulse-echo data

Ultrasonic imaging is perhaps the most ubiquitous form of biomedical imaging currently employed in the clinical setting due its safety, affordability, and real-time imaging capabilities. However, the poor image quality of standard brightness mode (B-Mode) images has inspired research into alternative acoustic imaging modalities, such as sound speed imaging. Historically, sound speed tomography has been employed in transmission-mode because of the limited k-space coverage available in backscattered data. However, such imaging techniques require specialized hardware configurations, and are often only limited to tissue such as the breast where full angular coverage is available. A recent approach [Jaeger et al., UMB, 41: 235-250, 2015] has demonstrated the feasibility of sound speed imaging using only pulse-echo data, which only requires a single array probe, commonly available in a clinical setting. This approach allows for complementary k-space coverage by utilizing relative temporal shifts between successive steered plane wave transmissions. Motivated by these successes, we present simulated and experimental pulse-echo sound speed reconstructions using three different methods: a modified Jaeger-like approach, the Born iterative method (BIM), and the distorted Born iterative method (DBIM).

Modification of enhanced distorted Born iterative method for the 2D inverse problem

This study presents a modified method based on the enhanced distorted Born iterative method for solving the 2D inverse problem. The proposed method overcomes two shortcomings of the enhanced distorted Born iterative method: (i) the regularisation parameter has to be determined artificially beforehand; (ii) The effects of the two regularisation processes in the enhanced distorted Born iterative method are redundant. The synthetic numerical reconstruction results, as well as the experimental reconstruction results, show that the proposed algorithm is capable and efficient for solving the 2D inverse scattering problem.

L2-Regularized Iterative Weighted Algorithm for Inverse Scattering

We propose a new inverse scattering technique based on sparsity for the application of microwave imaging. The underdetermined inverse problem appeared in the distorted born iterative method (DBIM) technique is solved using the suggested $L_{2}$ -regularized iterative weighted algorithm ( $L_{2}$ -IWA). The $L_{2}$ -regularizer has been introduced to stabilize the algorithm against nonlinear approximations, and the sparsity is enforced with the aid of another reweighted $L_{2}$ -norm regularizer to address the ill-posedness of the inverse problem. The derived algorithm is a three-step iterative technique which solves the underdetermined set of equations at each DBIM iteration. Moreover, the convergence of the $L_{2}$ -IWA technique is proved, analytically. The suggested method outperforms its other counterparts in various scenarios of homogeneous and heterogeneous breast models. Besides improving the resolution of the breast tumors, the $L_{2}$ -IWA technique is shown to be robust against additive noise.

Influence of dual-frequency combination on the quality improvement of ultrasound tomography

The most useful feature of ultrasound tomography founded on the inverse scattering theory is that it can detect small structures below the wavelength of the pressure wave. A popular method introduced in ultrasound tomography is the Distorted Born Iterative Method (DBIM). Recently, the dual-frequency combination technique has been utilized to improve the reconstruction quality and increase the convergence rate of the DBIM. This method uses two frequencies, f1 (low) and f2 (high), to estimate the sound contrast in [Formula: see text] and [Formula: see text] iterations, respectively. However, the influence of these iteration parameters on the overall performance of the system is not yet known. In this paper, it is shown by using the simulation technique that if we do not pay attention to the choice of these parameters, the reconstruction quality might be worse than that using a single frequency. Furthermore, we focus on the best way to select the parameters in order to improve the reconstruction quality of ultrasound tomography. Given a fixed sum [Formula: see text] of [Formula: see text] and [Formula: see text], simulation results show that the best value of [Formula: see text] is [Formula: see text]/2; this choice of parameters offers a normalized error that reduces by 67.6%, compared to the conventional DBIM using a single frequency.

Sensitivity of the Distorted Born Iterative Method to the Initial Guess in Microwave Breast Imaging

The distorted Born iterative method (DBIM) has been explored recently for microwave breast imaging. DBIM is an iterative method; thus, it requires an initial guess of the dielectric properties of the breast. In this paper, we study the sensitivity of DBIM imaging accuracy and convergence speed to the properties of a homogeneous initial guess. We conduct this investigation for a multifrequency formulation of DBIM, wherein the dispersive breast tissue properties are described with a Debye model. The parameters of the Debye model for the initial guess are linearly linked to reflect typical breast tissue dielectric properties. The static permittivity characterizing the properties is swept over an appropriate range of initial guesses and DBIM is used to obtain imaging results for each guess. Image quality and the number of iterations required for convergence is evaluated using three-dimensional (3-D) anatomically realistic numerical breast phantoms. This investigation not only vividly illustrates the sensitivity of DBIM to the initial guess but also definitively demonstrates that the use of initial values close to the average properties of the breast yields near-optimal performance. Finally, we present and evaluate a practical algorithm for estimating the average properties to be used as the initial guess.

Full waveform inversion in the frequency domain using direct iterative T-matrix methods

We present two direct iterative solutions to the nonlinear seismic waveform inversion problem that are based on volume integral equation methods for seismic forward modelling in the acoustic approximation. The solutions are presented in the frequency domain, where accurate inversion results can often be obtained using a relatively low number of frequency components. Our inverse scattering approach effectively replaces an ill-posed nonlinear inverse problem with a series of linear ill-posed inverse problems, for which there already exist efficient (regularized) solution methods. Both these solutions update the wavefield within the scattering domain after each iteration. The main difference is that the background medium Green functions are kept fixed in the first solution, but updated after each iteration in the second solution. This means that our solutions are very similar to the Born iterative (BI) and the distorted Born iterative (DBI) methods that are commonly used in acoustic and electromagnetic inverse scattering. However, we have eliminated the need to perform a full forward simulation (or to invert a huge matrix) at each iteration via the use of an iterative T-matrix method for fixed background media for the BI method and a variational T-matrix method for dynamic background media for the DBI method. The T-matrix (variation) is linearly related with the seismic wavefield data (residuals), but related with the unknown scattering potential model parameter (updates) in a non-linear manner, which is independent of the source-receiver configuration. This mathematical structure, which allows one to peel off the effects of the source-receiver configuration, is very attractive when dealing with multiple (simultaneous) sources, and is also compatible with the (future) use of renormalization methods for dealing with local minima problems. To illustrate the performance and potential of the two direct iterative methods for FWI, we performed a series of numerical experiments on synthetic seismic waveform data associated with a simple 2D model and the more complicated Marmousi model. The results of these numerical experiments suggest that the use of a fixed (e.g. smooth and ray-tracing friendly) background medium may be adequate for some applications with moderately large velocity contrasts, but the solution based on a dynamic (non-smooth and constantly updated) background medium will normally provide superiour inversion results; also in the case of low signal-to-noise ratios.

Wavelet-based regularization for robust microwave imaging in medical applications.

Microwave imaging (MWI) is an emerging tool for medical diagnostics, potentially offering unique advantages such as the capability of providing quantitative images of the inspected tissues. This involves, however, solving a challenging nonlinear and ill-posed electromagnetic inverse scattering problem. This paper presents a robust method for quantitative MWI in medical applications where very little, if any, a priori information on the imaging scenario is available. This is accomplished by employing a distorted Born iterative method and a regularization by projection technique, which reconstructs the tissue parameters using a wavelet basis expansion to represent the unknown contrast. This approach is suited for any microwave medical imaging application where the requirement for increased resolution dictates the use of higher frequency data and, consequently, a robust regularization strategy. To demonstrate the robustness of the proposed approach, this paper presents reconstructions of highly heterogeneous anatomically realistic numerical breast phantoms in a canonical 2-D configuration.

Fast Microwave Medical Imaging Based on Iterative Smoothed Adaptive Thresholding

This letter presents a fast microwave imaging technique based on the concept of smoothed minimization and adaptive thresholding. The distorted Born iterative method (DBIM) is used to solve the electromagnetic (EM) inverse scattering problem. We propose to solve the set of underdetermined equations at each iteration of the DBIM algorithm using an L2 regularized iterative smoothed adaptive thresholding ( L2-ISATCS) technique. Our simulation results confirm that this technique can reduce considerably the required reconstruction times for the DBIM method relative to previously suggested compressed sensing (CS)-based approaches .

Microwave medical imaging based on sparsity and an iterative method with adaptive thresholding.

We propose a new image recovery method to improve the resolution in microwave imaging applications. Scattered field data obtained from a simplified breast model with closely located targets is used to formulate an electromagnetic inverse scattering problem, which is then solved using the Distorted Born Iterative Method (DBIM). At each iteration of the DBIM method, an underdetermined set of linear equations is solved using our proposed sparse recovery algorithm, IMATCS. Our results demonstrate the ability of the proposed method to recover small targets in cases where traditional DBIM approaches fail. Furthermore, in order to regularize the sparse recovery algorithm, we propose a novel L(2) -based approach and prove its convergence. The simulation results indicate that the L(2)-regularized method improves the robustness of the algorithm against the ill-posed conditions of the EM inverse scattering problem. Finally, we demonstrate that the regularized IMATCS-DBIM approach leads to fast, accurate and stable reconstructions of highly dense breast compositions.

Beamforming-Enhanced Inverse Scattering for Microwave Breast Imaging.

We present a focal-beamforming-enhanced formulation of the distorted Born iterative method (DBIM) for microwave breast imaging. Incorporating beamforming into the imaging algorithm has the potential to mitigate the effect of noise on the image reconstruction. We apply the focal-beamforming-enhanced DBIM algorithm to simulated array measurements from two MRI-derived, anatomically realistic numerical breast phantoms and compare its performance to that of the DBIM formulated with two non-focal schemes. The first scheme simply averages scattered field data from reciprocal antenna pairs while the second scheme discards reciprocal pairs. Images of the dielectric properties are reconstructed for signal-to-noise ratios (SNR) ranging from 35 dB down to 0 dB. We show that, for low SNR, the focal beamforming algorithm creates reconstructions that are of higher fidelity with respect to the exact dielectric profiles of the phantoms as compared to reconstructions created using the non-focal schemes. At high SNR, the focal and non-focal reconstructions are of comparable quality.

Non-linear inverse scattering: High resolution quantitative breast tissue tomography

Recent published results in inverse scattering generally show the difficulty in dealing with moderate to high contrast inhomogeneities when employing linearized or iteratively linearized algorithms (e.g., distorted Born iterative method). This paper presents a fully nonlinear algorithm utilizing full wave field data, that results in ultrasound computed tomographic images from a laboratory breast scanner, and shows several such unique images from volunteer subjects. The forward problem, data collection process and inverse scattering algorithm used are discussed. A functional that represents the "best fit" between predicted and measured data is minimized, and therefore requires a very fast forward problem solver, Jacobian calculation, and gradient estimation, all of which are described. The data collection device is described. The algorithm and device yield quantitative estimates of human breast tissue in vivo. Several high resolution images, measuring ∼150 by 150 wavelengths, obtained from the 2D inverse scattering algorithms, using data collected from a first prototype, are shown and discussed. The quantitative values are compared with previous published work.

Quantitative Microwave Imaging of Realistic Numerical Breast Phantoms Using an Enclosed Array of Multiband, Miniaturized Patch Antennas.

We present a 3-D microwave breast imaging study in which we reconstruct the dielectric profiles of MRI-derived numerical breast phantoms from simulated array measurements using an enclosed array of multiband, miniaturized patch antennas. The array is designed to overcome challenges relating to the ill-posed nature of the inverse scattering system. We use a multifrequency formulation of the distorted Born iterative method to image four normal-tissue breast phantoms, each corresponding to a different density class. The reconstructed fibroglandular distributions are very faithful to the true distributions in location and basic shape. These results establish the feasibility of using an enclosed array of miniaturized, multiband patch antennas for quantitative microwave breast imaging.

Density imaging using a multiple-frequency DBIM approach

Current inverse scattering methods for quantitative density imaging have limitations that keep them from practical experimental implementations. In this work, an improved approach, termed the multiple-frequency distorted Born iterative method (MF-DBIM) algorithm, was developed for imaging density variations. The MF-DBIM approach consists of inverting the wave equation by solving for a single function that depends on both sound speed and density variations at multiple frequencies. Density information was isolated by using a linear combination of the reconstructed single-frequency profiles. Reconstructions of targets using MF-DBIM from simulated data were compared with reconstructions using methods currently available in the literature, i.e., the dual-frequency DBIM (DF-DBIM) and T-matrix approaches. Useful density reconstructions, i.e., root mean square errors (RMSEs) less than 30%, were obtained with MF-DBIM even with 2% Gaussian noise in the simulated data and using frequency ranges spanning less than an order of magnitude. Therefore, the MFDBIM approach outperformed both the DF-DBIM method (which has problems converging with noise even an order of magnitude smaller) and the T-matrix method (which requires a ka factor close to unity to achieve convergence). However, the convergence of all the density imaging algorithms was compromised when imaging targets with object functions exhibiting high spatial frequency content.