Ali Pezeshki

Sensitivity to Basis Mismatch in Compressed Sensing

The theory of compressed sensing suggests that successful inversion of an image of the physical world (broadly defined to include speech signals, radar/sonar returns, vibration records, sensor array snapshot vectors, 2-D images, and so on) for its source modes and amplitudes can be achieved at measurement dimensions far lower than what might be expected from the classical theories of spectrum or modal analysis, provided that the image is sparse in an apriori known basis. For imaging problems in spectrum analysis, and passive and active radar/sonar, this basis is usually taken to be a DFT basis. However, in reality no physical field is sparse in the DFT basis or in any apriori known basis. No matter how finely we grid the parameter space the sources may not lie in the center of the grid cells and consequently there is mismatch between the assumed and the actual bases for sparsity. In this paper, we study the sensitivity of compressed sensing to mismatch between the assumed and the actual sparsity bases. We start by analyzing the effect of basis mismatch on the best k-term approximation error, which is central to providing exact sparse recovery guarantees. We establish achievable bounds for the l1 error of the best k -term approximation and show that these bounds grow linearly with the image (or grid) dimension and the mismatch level between the assumed and actual bases for sparsity. We then derive bounds, with similar growth behavior, for the basis pursuit l1 recovery error, indicating that the sparse recovery may suffer large errors in the presence of basis mismatch. Although, we present our results in the context of basis pursuit, our analysis applies to any sparse recovery principle that relies on the accuracy of best k-term approximations for its performance guarantees. We particularly highlight the problematic nature of basis mismatch in Fourier imaging, where spillage from off-grid DFT components turns a sparse representation into an incompressible one. We substantiate our mathematical analysis by numerical examples that demonstrate a considerable performance degradation for image inversion from compressed sensing measurements in the presence of basis mismatch, for problem sizes common to radar and sonar.

Doppler Resilient Golay Complementary Waveforms

We describe a method of constructing a sequence (pulse train) of phase-coded waveforms, for which the ambiguity function is free of range sidelobes along modest Doppler shifts. The constituent waveforms are Golay complementary waveforms which have ideal ambiguity along the zero Doppler axis but are sensitive to nonzero Doppler shifts. We extend this construction to multiple dimensions, in particular to radar polarimetry, where the two dimensions are realized by orthogonal polarizations. Here we determine a sequence of two-by-two Alamouti matrices where the entries involve Golay pairs and for which the range sidelobes associated with a matrix-valued ambiguity function vanish at modest Doppler shifts. The Prouhet-Thue-Morse sequence plays a key role in the construction of Doppler resilient sequences of Golay complementary waveforms.

Sensitivity to basis mismatch in compressed sensing

The theory of compressed sensing suggests that successful inversion of an image of the physical world (broadly defined to include speech signals, radar/sonar returns, vibration records, sensor array snapshot vectors, 2-D images, and so on) for its source modes and amplitudes can be achieved at measurement dimensions far lower than what might be expected from the classical theories of spectrum or modal analysis, provided that the image is sparse in an apriori known basis. For imaging problems in spectrum analysis, and passive and active radar/sonar, this basis is usually taken to be a DFT basis. However, in reality no physical field is sparse in the DFT basis or in any apriori known basis. No matter how finely we grid the parameter space the sources may not lie in the center of the grid cells and consequently there is mismatch between the assumed and the actual bases for sparsity. In this paper, we study the sensitivity of compressed sensing to mismatch between the assumed and the actual sparsity bases. We start by analyzing the effect of basis mismatch on the best k-term approximation error, which is central to providing exact sparse recovery guarantees. We establish achievable bounds for the l1 error of the best k -term approximation and show that these bounds grow linearly with the image (or grid) dimension and the mismatch level between the assumed and actual bases for sparsity. We then derive bounds, with similar growth behavior, for the basis pursuit l1 recovery error, indicating that the sparse recovery may suffer large errors in the presence of basis mismatch. Although, we present our results in the context of basis pursuit, our analysis applies to any sparse recovery principle that relies on the accuracy of best k-term approximations for its performance guarantees. We particularly highlight the problematic nature of basis mismatch in Fourier imaging, where spillage from off-grid DFT components turns a sparse representation into an incompressible one. We substantiate our mathematical analysis by numerical examples that demonstrate a considerable performance degradation for image inversion from compressed sensing measurements in the presence of basis mismatch, for problem sizes common to radar and sonar.

Eigenvalue Beamforming Using a Multirank MVDR Beamformer and Subspace Selection

We derive eigenvalue beamformers to resolve an unknown signal of interest whose spatial signature lies in a known subspace, but whose orientation in that subspace is otherwise unknown. The unknown orientation may be fixed, in which case the signal covariance is rank-1, or it may be random, in which case the signal covariance is multirank. We present a systematic treatment of such signal models and explain their relevance for modeling signal uncertainties. We then present a multirank generalization of the MVDR beamformer. The idea is to minimize the power at the output of a matrix beamformer, while enforcing a data dependent distortionless constraint in the signal subspace, which we design based on the type of signal we wish to resolve. We show that the eigenvalues of an error covariance matrix are fundamental for resolving signals of interest. Signals with rank-1 covariances are resolved by the largest eigenvalues of the error covariance, while signals with multirank covariances are resolved by the smallest eigenvalues. Thus, the beamformers we design are eigenvalue beamformers, which extract signal information from eigen-modes of an error covariance. We address the tradeoff between angular resolution of eigenvalue beamformers and the fraction of the signal power they capture.

Undersea Target Classification Using Canonical Correlation Analysis

Canonical correlation analysis is employed as a multiaspect feature extraction method for underwater target classification. The method exploits linear dependence or coherence between two consecutive sonar returns, at different aspect angles. This is accomplished by extracting the dominant canonical correlations between the two sonar returns and using them as features for classifying mine-like objects from nonmine-like objects. The experimental results on a wideband acoustic backscattered data set, which contains sonar returns from several mine-like and nonmine-like objects in two different environmental conditions, show the promise of canonical correlation features for mine-like versus nonmine-like discrimination. The results also reveal that in a fixed bottom condition, canonical correlation features are relatively invariant to changes in aspect angle.

Wideband DOA estimation algorithms for multiple moving sources using unattended acoustic sensors

The problem of direction of arrival (DOA) estimation for multiple wideband sources using unattended passive acoustic sensors is considered. Several existing methods for narrowband DOA estimation are extended to resolve multiple closely spaced sources in presence of interference and wind noise. New wideband Capon beamforming methods are developed that use various algorithms for combining the power spectra at different frequency bins. A robust wideband Capon method is also studied to account for the inherent uncertainties in the array steering vector. Finally, to improve the resolution within an angular sector of interest and to provide robustness to sensor data loss, the beamspace method is extended and applied to the wideband problems. These methods are then tested and benchmarked on two real acoustic signature data sets that contain multiple ground vehicles moving in various formations.

Two-channel constrained least squares problems: solutions using power methods and connections with canonical coordinates

The problem of two-channel constrained least squares (CLS) filtering under various sets of constraints is considered, and a general set of solutions is derived. For each set of constraints, the solution is determined by a coupled (asymmetric) generalized eigenvalue problem. This eigenvalue problem establishes a connection between two-channel CLS filtering and transform methods for resolving channel measurements into canonical or half-canonical coordinates. Based on this connection, a unified framework for reduced-rank Wiener filtering is presented. Then, various representations of reduced-rank Wiener filters in canonical and half-canonical coordinates are introduced. An alternating power method is proposed to recursively compute the canonical coordinate and half-canonical coordinate mappings. A deflation process is introduced to extract the mappings associated with the dominant coordinates. The correctness of the alternating power method is demonstrated on a synthesized data set, and conclusions are drawn.

Wideband DOA estimation algorithms for multiple target detection and tracking using unattended acoustic sensors

The problem of detection, tracking and localization of multiple wideband sources (ground vehicles) using unattended passive acoustic sensors is considered in this paper. Existing methods typically fail to detect, resolve and track multiple closely spaced sources in tight formations, especially in the presence of clutter and wind noise. In this paper, several existing wideband direction of arrival (DOA) estimation algorithms are extended and applied to this problem. A modified version of the Steered Covariance Matrix (STCM) algorithm is presented that uses a two-step search process. To overcome the problems of existing DOA estimation methods, new wideband versions of the narrowband Capon beamforming method are proposed that use various algorithms for combining power spectra from different frequency bins. These methods are then implemented and benchmarked on a real acoustic signature data set that contains multiple ground targets moving in tight formations.

Empirical canonical correlation analysis in subspaces

This paper addresses canonical correlation analysis of two-channel data, when channel covariances are estimated from a limited number of samples, and are not necessarily full-rank. We show that empirical canonical correlations measure the cosines of the principal angles between the row spaces of the data matrices for the two channels. When the number of samples is smaller than the sum of the ranks of the two data matrices, some of the empirical canonical correlations become one, regardless of the two-channel model that generates the samples. In such cases, the empirical canonical correlations may not be used as estimates of correlation between random variables.

Canonical Coordinates are the Right Coordinates for Low-Rank Gauss–Gauss Detection and Estimation

In this correspondence, our aim is to establish a connection between low-rank detection, low-rank estimation, and canonical coordinates. The key to this connection is the observation that Gauss-Gauss detectors and estimators share canonical coordinates in the case where the underlying model is a signal-plus-noise model. We show that in Gauss-Gauss detection J-divergence is a function of squared canonical correlations, and hence is invariant to nonsingular transformations of the data channels. Further, we show that J-divergence has a special decomposition in canonical coordinates, impelling their use for rank-reduction. Canonical coordinates have been found earlier to be fundamental for low-rank estimation, as they decompose three important performance measures, namely the relative volume of error concentration ellipse, processing gain, and information rate. This correspondence shows that canonical coordinates are also fundamental for low-rank detection, making them more useful in signal processing and communication problems when low-rank modeling is required to achieve computational efficiency or robustness against noise and model uncertainties