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38
Asymptotic analysis of complex LASSO via complex approximate message passing
 IEEE Trans. Inf. Theory
, 2011
"... Recovering a sparse signal from an undersampled set of random linear measurements is the main problem of interest in compressed sensing. In this paper, we consider the case where both the signal and the measurements are complexvalued. We study the popular reconstruction method of ℓ1regularized lea ..."
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Cited by 10 (3 self)
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Recovering a sparse signal from an undersampled set of random linear measurements is the main problem of interest in compressed sensing. In this paper, we consider the case where both the signal and the measurements are complexvalued. We study the popular reconstruction method of ℓ1regularized least squares or LASSO. While several studies have shown that the LASSO algorithm offers desirable solutions under certain conditions, the precise asymptotic performance of this algorithm in the complex setting is not yet known. In this paper, we extend the approximate message passing (AMP) algorithm to the complexvalued signals and measurements to obtain the complex approximate message passing algorithm (CAMP). We then generalize the state evolution framework recently introduced for the analysis of AMP, to the complex setting. Using the state evolution, we derive accurate formulas for the phase transition and noise sensitivity of both LASSO and CAMP. Our results are theoretically proved for the case of i.i.d. Gaussian sensing matrices. But we confirm through simulations that our results hold for larger class of random matrices. 1
Robust learning of discriminative projection for multicategory classification on the Stiefel manifold
 in Proc. IEEE CS Conf. Comput. Vis. Pattern Recognit
, 2008
"... Learning a robust projection with a small number of training samples is still a challenging problem in face recognition, especially when the unseen faces have extreme variation in pose, illumination, and facial expression. To address this problem, we propose a framework formulated under statistical ..."
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Cited by 7 (0 self)
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Learning a robust projection with a small number of training samples is still a challenging problem in face recognition, especially when the unseen faces have extreme variation in pose, illumination, and facial expression. To address this problem, we propose a framework formulated under statistical learning theory that facilitates robust learning of a discriminative projection. Dimensionality reduction using the projection matrix is combined with a linear classifier in the regularized framework of lasso regression. The projection matrix in conjunction with the classifier parameters are then found by solving an optimization problem over the Stiefel manifold. The experimental results on standard face databases suggest that the proposed method outperforms some recent regularized techniques when the number of training samples is small. 1.
Optimal Estimation of Deterioration from Diagnostic Image Sequence
 IEEE TRANSACTIONS ON SIGNAL PROCESSING, SUBMITTED MAY 2007
, 2007
"... This paper considers estimation of pixelwise monotonic increasing (or decreasing) data from a time series of noisy blurred images. The motivation comes from estimation of mechanical structure damage that accumulates irreversibly over time. We formulate a Maximum A posteriory Probablity (MAP) estima ..."
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Cited by 5 (4 self)
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This paper considers estimation of pixelwise monotonic increasing (or decreasing) data from a time series of noisy blurred images. The motivation comes from estimation of mechanical structure damage that accumulates irreversibly over time. We formulate a Maximum A posteriory Probablity (MAP) estimation problem and find a solution by direct numerical optimization of a loglikelihood index. Spatial continuity of the damage is modeled using a Markov Random Field (MRF). The MRF prior includes the temporal monotonicity constraints. We tune the MRF prior, using a spatial frequency domain loopshaping technique to achieve a tradeoff between noise rejection and signal restoration properties of the estimate. The MAP optimization is a largescale Quadratic Programming (QP) problem that could have more than a million of decision variables and constraints. We describe and implement an efficient interiorpoint method for solving such optimization problem. The method uses a preconditioned conjugate gradient method to compute the search step. The developed QP solver relies on the special structure of the problem and can solve the problems of this size in a few tens of minutes, on a PC. The application example in the paper describes structural damage images obtained using a Structural Health Monitoring (SHM) system. The damage signal is distorted by environmental temperature that varies for each acquired image in the series. The solution for the experimental data is demonstrated to provide an excellent estimate of the damage accumulation trend while rejecting the spatial and temporal noise.
Implementing Algorithms for Signal and Image Reconstruction on Graphical Processing Units
"... Several highly effective algorithms that have been proposed recently for compressed sensing and image processing applications can be implemented efficiently on commodity graphical processing units (GPUs). The properties of algorithms and application that make for efficient GPU implementation are d ..."
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Cited by 5 (2 self)
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Several highly effective algorithms that have been proposed recently for compressed sensing and image processing applications can be implemented efficiently on commodity graphical processing units (GPUs). The properties of algorithms and application that make for efficient GPU implementation are discussed, and computational results for several algorithms are presented that show large speedups over CPU implementations.
Sparse Representation of Cast Shadows via ℓ1Regularized Least Squares
"... Scenes with cast shadows can produce complex sets of images. These images cannot be well approximated by lowdimensional linear subspaces. However, in this paper we show that the set of images produced by a Lambertian scene with cast shadows can be efficiently represented by a sparse set of images ge ..."
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Cited by 4 (3 self)
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Scenes with cast shadows can produce complex sets of images. These images cannot be well approximated by lowdimensional linear subspaces. However, in this paper we show that the set of images produced by a Lambertian scene with cast shadows can be efficiently represented by a sparse set of images generated by directional light sources. We first model an image with cast shadows as composed of a diffusive part (without cast shadows) and a residual part that captures cast shadows. Then, we express the problem in an ℓ1regularized least squares formulation, with nonnegativity constraints. This sparse representation enjoys an effective and fast solution, thanks to recent advances in compressive sensing. In experiments on both synthetic and real data, our approach performs favorably in comparison to several previously proposed methods. 1.
Accelerating gradient projection methods for ℓ1constrained signal recovery by steplength selection rules
, 2009
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CONVERGENCE OF FIXEDPOINT CONTINUATION ALGORITHMS FOR MATRIX RANK MINIMIZATION
, 2009
"... Abstract. The matrix rank minimization problem has applications in many fields such as system identification, optimal control, lowdimensional embedding, etc. As this problem is NPhard in general, its convex relaxation, the nuclear norm minimization problem, is often solved instead. Recently, Ma, G ..."
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Abstract. The matrix rank minimization problem has applications in many fields such as system identification, optimal control, lowdimensional embedding, etc. As this problem is NPhard in general, its convex relaxation, the nuclear norm minimization problem, is often solved instead. Recently, Ma, Goldfarb and Chen proposed a fixedpoint continuation algorithm for solving the nuclear norm minimization problem [33]. By incorporating an approximate singular value decomposition technique in this algorithm, the solution to the matrix rank minimization problem is usually obtained. In this paper, we study the convergence/recoverability properties of the fixedpoint continuation algorithm and its variants for matrix rank minimization. Heuristics for determining the rank of the matrix when its true rank is not known are also proposed. Some of these algorithms are closely related to greedy algorithms in compressed sensing. Numerical results for these algorithms for solving affinely constrained matrix rank minimization problems are reported.
Minimum Error Bounded Efficient ℓ1 Tracker with Occlusion Detection
"... Recently, sparse representation has been applied to visual tracking to find the target with the minimum reconstruction error from the target template subspace. Though effective, these L1 trackers require high computational costs due to numerous calculations for ℓ1 minimization. In addition, the inhe ..."
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Cited by 2 (2 self)
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Recently, sparse representation has been applied to visual tracking to find the target with the minimum reconstruction error from the target template subspace. Though effective, these L1 trackers require high computational costs due to numerous calculations for ℓ1 minimization. In addition, the inherent occlusion insensitivity of the ℓ1 minimization has not been fully utilized. In this paper, we propose an efficient L1 tracker with minimum error bound and occlusion detection which we call Bounded Particle Resampling (BPR)L1 tracker. First, the minimum error bound is quickly calculated from a linear least squares equation, and serves as a guide for particle resampling in a particle filter framework. Without loss of precision during resampling, most insignificant samples are removed before solving the computationally expensive ℓ1 minimization function. The BPR technique enables us to speed up the L1 tracker without sacrificing accuracy. Second, we perform occlusion detection by investigating the trivial coefficients in the ℓ1 minimization. These coefficients, by design, contain rich information about image corruptions including occlusion. Detected occlusions enhance the template updates to effectively reduce the drifting problem. The proposed method shows good performance as compared with several stateoftheart trackers on challenging benchmark sequences. 1.
A Unified Approach to Sparse Signal Processing
, 2009
"... A unified view of the area of sparse signal processing is presented in tutorial form by bringing together various fields in which the property of sparsity has been successfully exploited. For each of these fields, various algorithms and techniques, which have been developed to leverage sparsity, ar ..."
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Cited by 2 (1 self)
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A unified view of the area of sparse signal processing is presented in tutorial form by bringing together various fields in which the property of sparsity has been successfully exploited. For each of these fields, various algorithms and techniques, which have been developed to leverage sparsity, are described succinctly. The common potential benefits of significant reduction in sampling rate and processing manipulations through sparse signal processing are revealed. The key application domains of sparse signal processing are sampling, coding, spectral estimation, array processing, component analysis, and multipath channel estimation. In terms of the sampling process and reconstruction algorithms, linkages are made with random sampling, compressed sensing and rate of innovation. The redundancy introduced by channel coding in finite and real Galois fields is then related to oversampling with similar reconstruction algorithms. The methods of Prony, Pisarenko, and MUltiple SIgnal Classification (MUSIC) are next shown to be targeted at analyzing signals with sparse frequency domain representations. Specifically, the relations of the approach of Prony to an annihilating filter in rate of innovation and Error Locator Polynomials in coding are emphasized; the Pisarenko and MUSIC methods are further improvements of the Prony method. Such narrowband spectral estimation is then related to multisource location and direction of arrival estimation in array processing. The notions of sparse array beamforming and sparse sensor networks are also introduced. Sparsity in unobservable source signals is also shown to facilitate source separation in Sparse Component Analysis (SCA); the algorithms developed in this area are also widely used in compressed sensing. Finally, the nature of the multipath channel estimation problem is shown to have a sparse formulation; algorithms similar to sampling and coding are used to estimate typical multicarrier communication channels.