Robust Minimum Variance Beamforming Research Articles

A low-complexity and robust minimum variance beamformer for ultrasound imaging systems using beamspace dominant mode rejection

Minimum variance beamformer (MVB) has a high computational complexity that is mainly due to the inversion of an L×L covariance matrix involved during weight vector estimation, where L is the length of the subarray. In this work an attempt is made to reduce the computational complexity as well as increase the robustness against signal mismatch. The computational complexity is reduced by projecting the element-space data on to beamspace domain and then using dominant mode rejection on the beamspace covariance matrix (BCM). This reduces the dimension of covariance matrix and also eliminates the matrix inversion thereby reducing the computational complexity. Further, a closeness factor is introduced to determine the interference components that have to be suppressed, leading to increased robustness against signal mismatch. Performance of the proposed method has been evaluated on both simulated and experimental datasets. Results indicate that the proposed beamformer has a lateral resolution of 0.07 mm and a contrast resolution of 0.80, which are comparable to that of MVB which has a lateral and contrast resolution of 0.10 mm and 0.78, that too with a 12-fold reduction in computational complexity. The robustness of the peak magnitude estimate and spatial resolution of the beamformer with respect to error in estimation of sound velocity in the medium have also been evaluated. The variation in lateral resolution of proposed beamformer and MVB is approximately 1.21 mm and 1 mm. Further, the proposed beamformer has a maximum deviation in peak magnitude estimate of 0.7 dB whereas that of MVB is 2.5 dB, thus indicating the increased robustness of proposed method in peak magnitude estimate. Overall, the proposed beamformer has a 12-fold lower computational complexity compared to MVB with additional flexibility to increase the robustness against error in sound velocity.

A Sparsity-constraint Capon Beamforming Algorithm

In the presence of the steering vector of the signal of interest mismatches, nulls are formed in the direction of the signal of interest, which affects the performance of the beamforming. In this paper, a sparsity-constraint Capon beamforming based on iterative robust minimum variance beamforming is proposed. Using iteractive robust minimum variance beamforming, the mismatched steering vector is corrected. To achieve sidelobe level and suppress iterference, sparse constraints on beam response is added in the angel of sidelobe domain. Computer simulations have been conducted to verify validity and superiority of the proposed algorithm. It is shown that in the presence of mismatched steering vector, the proposed algorithm is more robust and has lower sidelobe level than the existing methods.

Estimating neural sources using a worst-case robust adaptive beamforming approach

Recently, brain source localization and beamforming methods have played an important role in enhancing the utility of the electroencephalograph (EEG) and/or the magnetoencephalograph (MEG). Source localization methods are in general very sensitive to parameter values used to describe the underlying lead–field matrix, such as head shape, electrode positions, conductivity of various tissues of the head, etc. Errors in these parameter values can cause significant degradation in performance of these algorithms.In this paper, we develop a robust minimum variance beamformer (RMVB) specifically for EEG applications that can deal with an arbitrary mismatch between the assumed and true lead field matrix. The approach optimizes the worst-case uncertainty performance, sacrificing a distortionless response in exchange for more robust performance. The performance of the RMVB is compared with the classic minimum variance beamformer (MVB) as well as its regularized and eigenspace-based variations. The simulation scenarios highlight the superior ability of the RMVB to cope with arbitrary mismatches.

Electromagnetic Brain Source Imaging by Means of a Robust Minimum Variance Beamformer.

Adaptive beamformer methods that have been extensively used for functional brain imaging using EEG/MEG (magnetoencephalography) signals are sensitive to model mismatches. We propose a robust minimum variance beamformer (RMVB) technique, which explicitly incorporates the uncertainty of the lead field matrix into the estimation of spatial-filter weights that are subsequently used to perform the imaging. The uncertainty of the lead field is modeled by ellipsoids in the RMVB method; these hyperellipsoids (ellipsoids in higher dimensions) define regions of uncertainty for a given nominal lead field vector. These ellipsoids are estimated empirically by sampling lead field vectors surrounding each point of the source space, or more generally by building several forward models for the source space. Once these uncertainty regions (ellipsoids) are estimated, they are used to perform the source-imaging task. Computer simulations are conducted to evaluate the performance of the proposed RMVB technique. Our results show that robust beamformers can outperform conventional beamformers in terms of localization error, recovering source dynamics, and estimation of the underlying source extents when uncertainty in the lead field matrix is properly determined and modeled. The RMVB can be substituted for conventional beamformers, especially in applications where source imaging is performed off-line, and computational speed and complexity are not of major concern. A high-quality source imaging can be utilized in various applications, such as determining the epileptogenic zone in medically intractable epilepsy patients or estimating the time course of activity, which is a required step for computing the functional connectivity of brain networks.

A Distributionally Robust Minimum Variance Beamformer Design

This letter is concerned with a robust minimum variance beamformer design. To hedge the mismatch between the true and the assumed steering vectors, a distributionally robust beamformer (DR-beamformer) is proposed. The tractable reformulation of this beamformer is developed. Compared with the existing robust beamformers (e.g., worst-case robust beamformer and Gaussian robust beamformer), the proposed robust beamformer does not assume full knowledge of the channel mismatch. Therefore, it is more flexible in practice and more general in formulation. In addition, the relationships of the proposed robust beamformer to the existing ones are investigated. The performance gain of the DR-beamformer over the other robust beamformers is highlighted through numerical simulations.

Effects of Phase Aberration and Phase Aberration Correction on the Minimum Variance Beamformer.

The minimum variance (MV) beamformer has the potential to enhance the resolution and contrast of ultrasound images but is sensitive to steering vector errors. Robust MV beamformers have been proposed but mainly evaluated in the presence of gross sound speed mismatches, and the impact of phase aberration correction (PAC) methods in mitigating the effects of phase aberration in MV beamformed images has not been explored. In this study, an analysis of the effects of aberration on conventional MV and eigenspace MV (ESMV) beamformers is carried out. In addition, the impact of three PAC algorithms on the performance of MV beamforming is analyzed. The different beamformers were tested on simulated data and on experimental data corrupted with electronic and tissue-based aberration. It is shown that all gains in performance of the MV beamformer with respect to delay-and-sum (DAS) are lost at high aberration strengths. For instance, with an electronic aberration of 60 ns, the lateral resolution of DAS degrades by 17% while MV degrades by 73% with respect to the images with no aberration. Moreover, although ESMV shows robustness at low aberration levels, its degradation at higher aberrations is approximately the same as that of regular MV. It is also shown that basic PAC methods improve the aberrated MV beamformer. For example, in the case of electronic aberration, multi-lag reduces degradation in lateral resolution from 73% to 28% and contrast loss from 85% to 25%. These enhancements allow the combination of MV and PAC to outperform DAS and PAC and ESMV in moderate and strong aberrations. We conclude that the effect of aberration on the MV beamformer is stronger than previously reported in the literature and that PAC is needed to improve its clinical potential.

Robust Minimum Variance Beamforming Based on Signal Power Maximization

Conventional beamformers are very sensitive to steering vector mismatches and their performance degrades in the presence of array mismatches. This paper proposes a robust minimum variance beamforming algorithm based on signal power maximization. In the proposed algorithm, the beamforming vector is determined by maximizing the signal power while minimizing the output power of the beamformer, with the signal distortionless response constrained to be unity. Numerical results show that the proposed beamformer is robust to the steering vector mismatch, and achieves better SINR performance than other robust beamforming algorithms when the mismatch in the signal look direction is large.

Robust adaptive beamforming with random steering vector mismatch

In this paper, random steering vector mismatches in sensor arrays are considered and probability constraints are imposed for designing a robust minimum variance beamformer (RMVB). To solve the resultant design problem, a Bernstein-type inequality for stochastic processes of quadratic forms of Gaussian variables is employed to transform the probabilistic constraint to a deterministic form. With the use of convex optimization techniques, the deterministic problem is reformulated to a semidefinite programming (SDP) problem which can be efficiently solved. In order to overcome the degradation caused by the presence of the signal-of-interest (SOI) in the training snapshots, two methods with different application conditions to interference-plus-noise covariance matrix (INCM) construction are also introduced. Additionally, the uncertainty of the sample covariance matrix is taken into account to improve the robustness when the INCM-based approaches are not feasible. Numerical examples are presented to demonstrate the performances of the proposed robust beamformers in different scenarios.

MAGNITUDE CONSTRAINT MINIMUM VARIANCE BEAMFORMER WITH CONJUGATE SYMMETRIC CONSTRAINT AND NORM CONSTRAINT

In this paper, an improved robust minimum variance beamformer against direction of arrival (DOA) mismatch and flnite sample efiect is proposed. Multiple inequality magnitude constraints are imposed to broaden the main lobe of beampattern. The conjugate symmetric structure of the optimal weight is utilized to transform the non-convex inequality magnitude constraints into convex ones. A quadratic constraint on the norm of weight is introducing to make further improvement on robustness against DOA mismatch and flnite sample efiect. The proposed beamforming problem can be reformulated in the form of the second order cone programming and solved e-ciently by interior point method. Simulation results show that the proposed beamformer outperforms several other adaptive beamformers.

Robust minimum variance multiple‐input multiple‐output radar beamformer

A robust minimum variance (MV) beamforming approach is proposed for improving the robustness of multiple-input multiple-output (MIMO) radar against the mismatches of the steering vector and the finite sample effects. In contrast to existing robust MV beamformers (RMVBs), the proposed RMVB utilises a specific structured model of virtual steering vector (also named transmit–receive steering vector) of MIMO radar rather than the commonly used unstructured model in phased-array radar. The basic idea of the proposed RMVB is to estimate the desired transmit and receive steering vectors under two quadratic constraints. To solve this problem, an iterative algorithm is developed. Simulations are provided to confirm the effectiveness of the proposed method.

Iterative Robust Minimum Variance Beamforming

Based on worst-case performance optimization, the recently developed adaptive beamformers utilize the uncertainty set of the desired array steering vector to achieve robustness against steering vector mismatches. In the presence of large steering vector mismatches, the uncertainty set has to expand to accommodate the increased error. This degrades the output signal-to-interference-plus-noise ratios (SINRs) of these beamformers since their interference-plus-noise suppression abilities are weakened. In this paper, an iterative robust minimum variance beamformer (IRMVB) is proposed which uses a small uncertainty sphere (and a small flat ellipsoid) to search for the desired array steering vector iteratively. This preserves the interference-plus-noise suppression ability of the proposed beamformer and results in a higher output SINR. Theoretical analysis and simulation results are presented to show the effectiveness of the proposed beamformer.

Unifying Experiment Design and Convex Regularization Techniques for Enhanced Imaging With Uncertain Remote Sensing Data—Part I: Theory

This paper considers the problem of high-resolution remote sensing (RS) of the environment formalized in the terms of a nonlinear ill-posed inverse problem of estimation of the power spatial spectrum pattern (SSP) of the wavefield scattered from an extended remotely sensed scene via processing the discrete measurements of a finite number of independent realizations of the observed degraded data signals [single realization of the trajectory signal in the case of synthetic aperture radar (SAR)]. We address a new descriptive experiment design regularization (DEDR) approach to treat the SSP reconstruction problem in the uncertain RS environment that unifies the paradigms of maximum likelihood nonparametric spectral estimation, descriptive experiment design, and worst case statistical performance optimization-based regularization. Pursuing such an approach, we establish a family of the DEDR-related SSP estimators that encompass a manifold of algorithms ranging from the traditional matched filter to the modified robust adaptive spatial filtering and minimum variance beamforming methods. The theoretical study is resumed with the development of a fixed-point iterative DEDR technique that incorporates the regularizing projections onto convex solution sets into the SSP reconstruction procedures to enforce the robustness and convergence. For the imaging SAR application, the proposed DEDR approach is aimed at performing, in a single optimized processing, adaptive SAR focusing, speckle reduction and RS scene image enhancement, and accounts for the possible presence of uncertain trajectory deviations.

On the Relationship Between Robust Minimum Variance Beamformers With Probabilistic and Worst-Case Distortionless Response Constraints

An interesting relationship between the probability-constrained and worst-case optimization based robust minimum variance (MV) beamformers has been discovered. It is shown that both in the cases of circularly symmetric Gaussian and worst-case distributions of the steering vector mismatch, the probability-constrained robust MV beamforming problem can be tightly approximated as a convex second-order cone programming (SOCP) problem. The latter problem is mathematically equivalent to that resulting from the deterministic worst-case approach and, therefore, probability-constrained beamformers can be interpreted and implemented using their deterministic worst-case counterparts. However, an important advantage of the developed probability-constrained MV beamformers with respect to their standard worst-case counterparts is that the former approaches enable to explicitly quantify the parameters of the uncertainty region in terms of the beamformer outage probability.

Image-based refocusing of ultrasound arrays in the presence of strongly scattering objects.

Ultrasound phased arrays are currently being investigated for a dual-mode (imaging/therapy) operation in targeting deep-seated abdominal tumors for selective destruction noninvasively. In some cases, e.g., liver tumors, the target is partially obstructed by the ribcage, which limits the array gain at the target. In addition, the intensity at the ribs may be high enough to cause treatment-limiting pain and/or collateral damage to the tissues surrounding the ribs in the path of the therapeutic beam. Using the imaging capabilities of these dual-mode arrays, we have formulated and experimentally verified a robust minimum variance beamforming algorithm for adaptive refocusing in the presence of the ribs. The algorithm utilizes image data of the treatment region formed using conventional beamforming in estimating the array steering vector at the target and the weighting matrix that maximizes the array gain at the target. In this paper, we will describe the mathematical formulation and present experimental data to demonstrate its robustness. We will also discuss the implications of the method, the use of ultrasound phased arrays for imaging, and therapy of trans-thoracic targets such as the liver and the heart.

Removal of the Eye-Blink Artifacts From EEGs via STF-TS Modeling and Robust Minimum Variance Beamforming

A novel scheme for the removal of eye-blink (EB) artifacts from electroencephalogram (EEG) signals based on a novel space-time-frequency (STF) model of EEGs and robust minimum variance beamformer (RMVB) is proposed. In this method, in order to remove the artifact, the RMVB is provided with a priori information, namely, an estimation of the steering vector corresponding to the point source EB artifact. The artifact-removed EEGs are subsequently reconstructed by deflation. The a priori knowledge, the vector corresponding to the spatial distribution of the EB factor, is identified using the STF model of EEGs, provided by the parallel factor analysis (PARAFAC) method. In order to reduce the computational complexity present in the estimation of the STF model using the three-way PARAFAC, the time domain is subdivided into a number of segments, and a four-way array is then set to estimate the STF-time/segment (TS) model of the data using the four-way PARAFAC. The correct number of the factors of the STF model is effectively estimated by using a novel core consistency diagnostic- (CORCONDIA-) based measure. Subsequently, the STF-TS model is shown to closely approximate the classic STF model, with significantly lower computational cost. The results confirm that the proposed algorithm effectively identifies and removes the EB artifact from raw EEG measurements.

A robust minimum variance beamformer with new constraint on uncertainty of steering vector

A robust minimum variance beamformer, also called robust Capon beamformer (RCB), with a new constraint on the uncertainty of nominal array steering vector (ASV) is proposed in this paper. Unlike the uncertainty constraints used in literature, the proposed constraint is constructed by replacing the nominal ASV with the projected one onto the signal-plus-interference subspace. With this compact uncertainty constraint, the proposed RCB achieves higher output signal-to-noise-plus-interference ratio (SINR) compared with the conventional RCBs. Theoretical analysis and simulation results show the effectiveness of the proposed RCB.

COMPUTATIONALLY EFFICIENT MONTE CARLO ESTIMATION ALGORITHMS FOR MATCHED FIELD PROCESSING IN UNCERTAIN OCEAN ENVIRONMENTS

In this paper, Monte Carlo estimation techniques are presented for computationally efficient implementation of two methods for matched field source localization in uncertain ocean channels. In the Optimal Uncertain Field Processor (OUFP), Monte Carlo integration is used to integrate out the environmental parameters and thus estimate the a posteriori distribution of the source location parameters. In the Minimum Variance Beamformer with Environmental Perturbation Constraints (MV-EPC), Monte Carlo estimation of the signal correlation matrix averaged over the ensemble of environmental realizations is used to estimate the beamformer constraints. Using the OUFP, detection performance bounds are evaluated which indicate that source position uncertainty affects performance much more than environmental uncertainty. An upper bound on source localization performance is also obtained indicating that for short observation times a threshold signal-to-noise ratio (SNR) exists, dependent upon environmental uncertainty, below which source localization performance rapidly degrades. Among robust minimum variance beamforming methods, the MV-EPC method demonstrated superior probability of correct localization (PCL), both in single source scenarios and in the presence of interference. The OUFP at high SNR and the MV-EPC at large observation times both achieved near perfect source localization performance, although for large environmental uncertainty the OUFP provides an upper bound on P CL .