[Show abstract][Hide abstract] ABSTRACT: We extend the generalized approximate message passing (G-AMP) approach,
originally proposed for high-dimensional generalized-linear regression in the
context of compressive sensing, to the generalized-bilinear case, which enables
its application to matrix completion, robust PCA, dictionary learning, and
related matrix-factorization problems. In the first part of the paper, we
derive our Bilinear G-AMP (BiG-AMP) algorithm as an approximation of the
sum-product belief propagation algorithm in the high-dimensional limit, where
central-limit theorem arguments and Taylor-series approximations apply, and
under the assumption of statistically independent matrix entries with known
priors. In addition, we propose an adaptive damping mechanism that aids
convergence under finite problem sizes, an expectation-maximization (EM)-based
method to automatically tune the parameters of the assumed priors, and two
rank-selection strategies. In the second part of the paper, we discuss the
specializations of EM-BiG-AMP to the problems of matrix completion, robust PCA,
and dictionary learning, and present the results of an extensive empirical
study comparing EM-BiG-AMP to state-of-the-art algorithms on each problem. Our
numerical results, using both synthetic and real-world datasets, demonstrate
that EM-BiG-AMP yields excellent reconstruction accuracy (often best in class)
while maintaining competitive runtimes and avoiding the need to tune
IEEE Transactions on Signal Processing 10/2013; 62(22). DOI:10.1109/TSP.2014.2357776 · 3.20 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Shape- and motion-reconstruction is inherently ill-conditioned such that estimates rapidly degrade in the presence of noise, outliers, and missing data. For moving-target radar imaging applications, methods which infer the underlying geometric invariance within back-scattered data are the only known way to recover completely arbitrary target motion. We previously demonstrated algorithms that recover the target motion and shape, even with very high data drop-out (e.g., greater than 75%), which can happen due to self-shadowing, scintillation, and destructive-interference effects. We did this by combining our previous results, that a set of rigid scattering centers forms an elliptical manifold, with new methods to estimate low-rank subspaces via convex optimization routines. This result is especially significant because it will enable us to utilize more data, ultimately improving the stability of the motion-reconstruction process.
Since then, we developed a feature- based shape- and motion-estimation scheme based on newly developed object-image relations (OIRs) for moving targets collected in bistatic measurement geometries. In addition to generalizing the previous OIR-based radar imaging techniques from monostatic to bistatic geometries, our formulation allows us to image multiple closely-spaced moving targets, each of which is allowed to exhibit missing data due to target self-shadowing as well as extreme outliers (scattering centers that are inconsistent with the assumed physical or geometric models). The new method is based on exploiting the underlying structure of the model equations, that is, far-field radar data matrices can be decomposed into multiple low-rank subspaces while simultaneously locating sparse outliers.
[Show abstract][Hide abstract] ABSTRACT: Image acquisition systems such as synthetic aperture radar (SAR) and magnetic resonance imaging often measure irregularly spaced Fourier samples of the desired image. In this paper we show the relationship between sample locations, their associated backprojection weights, and image resolution as characterized by the resulting point spread function (PSF). Two new methods for computing data weights, based on different optimization criteria, are proposed. The first method, which solves a maximal-eigenvector problem, optimizes a PSF-derived resolution metric which is shown to be equivalent to the volume of the Cramer–Rao (positional) error ellipsoid in the uniform-weight case. The second approach utilizes as its performance metric the Frobenius error between the PSF operator and the ideal delta function, and is an extension of a previously reported algorithm. Our proposed extension appropriately regularizes the weight estimates in the presence of noisy data and eliminates the superfluous issue of image discretization in the choice of data weights. The Frobenius-error approach results in a Tikhonov-regularized inverse problem whose Tikhonov weights are dependent on the locations of the Fourier data as well as the noise variance. The two new methods are compared against several state-of-the-art weighting strategies for synthetic multistatic point-scatterer data, as well as an 'interrupted SAR' dataset representative of in-band interference commonly encountered in very high frequency radar applications.
[Show abstract][Hide abstract] ABSTRACT: The linear sampling method (LSM) offers a qualitative image reconstruction approach, which is known as a viable alternative for obstacle support identification to the well-studied filtered backprojection (FBP), which depends on a linearized forward scattering model. Of practical interest is the imaging of obstacles from sparse aperture far-field data under a fixed single frequency mode of operation. Under this scenario, the Tikhonov regularization typically applied to LSM produces poor images that fail to capture the obstacle boundary. In this paper, we employ an alternative regularization strategy based on constraining the sparsity of the solution's spatial gradient. Two regularization approaches based on the spatial gradient are developed. A numerical comparison to the FBP demonstrates that the new method's ability to account for aspect-dependent scattering permits more accurate reconstruction of concave obstacles, whereas a comparison to Tikhonov-regularized LSM demonstrates that the proposed approach significantly improves obstacle recovery with sparse-aperture data.
[Show abstract][Hide abstract] ABSTRACT: In this work, we consider a general form of noisy compressive sensing (CS) when there is uncertainty in the measurement matrix as well as in the measurements. Matrix uncertainty is motivated by practical cases in which there are imperfections or unknown calibration parameters in the signal acquisition hardware. While previous work has focused on analyzing and extending classical CS algorithms like the LASSO and Dantzig selector for this problem setting, we propose a new algorithm whose goal is either minimization of mean-squared error or maximization of posterior probability in the presence of these uncertainties. In particular, we extend the Approximate Message Passing (AMP) approach originally proposed by Donoho, Maleki, and Montanari, and recently generalized by Rangan, to the case of probabilistic uncertainties in the elements of the measurement matrix. Empirically, we show that our approach performs near oracle bounds. We then show that our matrix-uncertain AMP can be applied in an alternating fashion to learn both the unknown measurement matrix and signal vector. We also present a simple analysis showing that, for suitably large systems, it suffices to treat uniform matrix uncertainty as additive white Gaussian noise.
Circuits, Systems and Computers, 1977. Conference Record. 1977 11th Asilomar Conference on 01/2011; DOI:10.1109/ACSSC.2011.6190118
[Show abstract][Hide abstract] ABSTRACT: Traditional high-value monostatic imaging systems employ frequency-diverse pulses to form images from small synthetic apertures. In contrast, RF tomography utilizes a network of spatially diverse sensors to trade geometric diversity for bandwidth, permitting images to be formed with narrowband waveforms. Such a system could use inexpensive sensors with minimal ADC requirements, provide multiple viewpoints into urban canyons and other obscured environments, and offer graceful performance degradation under sensor attrition. However, numerous challenges must be overcome to field and operate such a system, including multistatic autofocus, precision timing requirements, and the development of appropriate image formation algorithms for large, sparsely populated synthetic apertures with anisotropic targets. AFRL has recently constructed an outdoor testing facility to explore these challenges with measured data. Preliminary experimental results are provided for this system, along with a description of remaining challenges and future research directions.