Sparse Representation Problem Research Articles

Robust visual tracking with structured sparse representation appearance model

In this paper, we present a structured sparse representation appearance model for tracking an object in a video system. The mechanism behind our method is to model the appearance of an object as a sparse linear combination of structured union of subspaces in a basis library, which consists of a learned Eigen template set and a partitioned occlusion template set. We address this structured sparse representation framework that preferably matches the practical visual tracking problem by taking the contiguous spatial distribution of occlusion into account. To achieve a sparse solution and reduce the computational cost, Block Orthogonal Matching Pursuit (BOMP) is adopted to solve the structured sparse representation problem. Furthermore, aiming to update the Eigen templates over time, the incremental Principal Component Analysis (PCA) based learning scheme is applied to adapt the varying appearance of the target online. Then we build a probabilistic observation model based on the approximation error between the recovered image and the observed sample. Finally, this observation model is integrated with a stochastic affine motion model to form a particle filter framework for visual tracking. Experiments on some publicly available benchmark video sequences demonstrate the advantages of the proposed algorithm over other state-of-the-art approaches. ► We model the appearance as a sparse linear combination of structured subspaces. ► Block orthogonal matching pursuit is used to reduce the computational complexity. ► The incremental PCA is applied to adapt the appearance variations online. ► The proposed method outperforms the reference trackers on benchmark datasets.

Hand posture recognition via joint feature sparse representation

In this study, we cast hand posture recognition as a sparse representation problem, and propose a novel approach called joint feature sparse representation classifier for efficient and accurate sparse representation based on multiple features. By integrating different features for sparse representation, including gray-level, texture, and shape feature, the proposed method can fuse benefits of each feature and hence is robust to partial occlusion and varying illumination. Additionally, a new database optimization method is introduced to improve computational speed. Experimental results, based on public and self-build databases, show that our method performs well compared to the state-of-the-art methods for hand posture recognition.

Feature-Based Sparse Representation for Image Similarity Assessment

Assessment of image similarity is fundamentally important to numerous multimedia applications. The goal of similarity assessment is to automatically assess the similarities among images in a perceptually consistent manner. In this paper, we interpret the image similarity assessment problem as an information fidelity problem. More specifically, we propose a feature-based approach to quantify the information that is present in a reference image and how much of this information can be extracted from a test image to assess the similarity between the two images. Here, we extract the feature points and their descriptors from an image, followed by learning the dictionary/basis for the descriptors in order to interpret the information present in this image. Then, we formulate the problem of the image similarity assessment in terms of sparse representation. To evaluate the applicability of the proposed feature-based sparse representation for image similarity assessment (FSRISA) technique, we apply FSRISA to three popular applications, namely, image copy detection, retrieval, and recognition by properly formulating them to sparse representation problems. Promising results have been obtained through simulations conducted on several public datasets, including the Stirmark benchmark, Corel-1000, COIL-20, COIL-100, and Caltech-101 datasets.

Kernel sparse representation based classification

Sparse representation has attracted great attention in the past few years. Sparse representation based classification (SRC) algorithm was developed and successfully used for classification. In this paper, a kernel sparse representation based classification (KSRC) algorithm is proposed. Samples are mapped into a high dimensional feature space first and then SRC is performed in this new feature space by utilizing kernel trick. Since samples in the high dimensional feature space are unknown, we cannot perform KSRC directly. In order to overcome this difficulty, we give the method to solve the problem of sparse representation in the high dimensional feature space. If an appropriate kernel is selected, in the high dimensional feature space, a test sample is probably represented as the linear combination of training samples of the same class more accurately. Therefore, KSRC has more powerful classification ability than SRC. Experiments of face recognition, palmprint recognition and finger-knuckle-print recognition demonstrate the effectiveness of KSRC.

Time delay and Doppler estimation for wideband acoustic signals in multipath environments

Estimation of the parameters of a multipath underwater acoustic channel is of great interest for a variety of applications. This paper proposes a high-resolution method for jointly estimating the multipath time delays, Doppler scales, and attenuation amplitudes of a time-varying acoustical channel. The proposed method formulates the estimation of channel parameters into a sparse representation problem. With the [script-l](1)-norm as the measure of sparsity, the proposed method makes use of the basis pursuit (BP) criterion to find the sparse solution. The ill-conditioning can be effectively reduced by the [script-l](1)-norm regularization. Unlike many existing methods that are only applicable to narrowband signals, the proposed method can handle both narrowband and wideband signals. Simulation results are provided to verify the performance and effectiveness of the proposed algorithm, indicating that it has a super-resolution in both delay and Doppler domain, and it is robust to noise.

Direction of arrival estimation of near‐field sources using sparse representation framework.

This paper uses the sparse representation framework to investigate localization of near‐field sources (e.g., underwater bottom or buried targets) from the data captured using two uniform linear sensor (hydrophone) subarrays. The connection between the two array steering transformation matrices of the near‐field sources corresponding to the two subarrays is first analyzed using the second Taylor expansion. This connection allows the construction of a new equivalent far‐field steering matrix for each near‐field source, hence converting the near‐field source localization problem to a more convenient far‐field one. The relationship between the signals observed by the two subarrays and the new constructed far‐field directional matrix is investigated indicating that the DoA estimates of the sources can be cast into finding a solution to a sparse representation problem. The resolution of the sparse equation will also be discussed. Finally, simulation results are presented to demonstrate the effects of different noise levels on the accuracy of the DoA estimation for the cases of single and multiple sources.

Sparse representation-based synthetic aperture radar imaging

There is increasing interest in using synthetic aperture radar (SAR) images in automated target recognition and decision-making tasks. The success of such tasks depends on how well the reconstructed SAR images exhibit certain features of the underlying scene. Based on the observation that typical underlying scenes usually exhibit sparsity in terms of such features, this paper presents an image formation method that formulates the SAR imaging problem as a sparse signal representation problem. For problems of complex-valued nature, such as SAR, a key challenge is how to choose the dictionary and the representation scheme for effective sparse representation. Since features of the SAR reflectivity magnitude are usually of interest, the approach is designed to sparsely represent the magnitude of the complex-valued scattered field. This turns the image reconstruction problem into a joint optimisation problem over the representation of magnitude and phase of the underlying field reflectivities. The authors develop the mathematical framework for this method and propose an iterative solution for the corresponding joint optimisation problem. The experimental results demonstrate the superiority of this method over previous approaches in terms of both producing high-quality SAR images and exhibiting robustness to uncertain or limited data.

Greedy approximation of characteristic functions

We discuss the problem of sparse representation of domains in ℝd. We demonstrate how the recently developed general theory of greedy approximation in Banach spaces can be used in this problem. The use of greedy approximation has two important advantages: (1) it works for an arbitrary dictionary of sets used for sparse representation and (2) the method of approximation does not depend on smoothness properties of the domains and automatically provides a near optimal rate of approximation for domains with different smoothness properties. We also give some lower estimates of the approximation error and discuss a specific greedy algorithm for approximation of convex domains in ℝ2.

Improved FOCUSS Method With Conjugate Gradient Iterations

Focal Underdetermined System Solver (FOCUSS) is a powerful tool for sparse representation and underdetermined inverse problems. In this correspondence, we strengthen the FOCUSS method with the following main contributions: 1) we give a more rigorous derivation of the FOCUSS for the sparsity parameter 0 < p < 1 by a nonlinear transform and 2) we develop the CG-FOCUSS by incorporating the conjugate gradient (CG) method to the FOCUSS, which significantly reduces a computational cost with respect to the standard FOCUSS and extends its availability for large scale problems. We justify the CG-FOCUSS based on a probability theory. Furthermore, the high performance of the CG-FOCUSS is demonstrated with experiments.

A Frame Construction and a Universal Distortion Bound for Sparse Representations

We consider approximations of signals by the elements of a frame in a complex vector space of dimension N and formulate both the noiseless and the noisy sparse representation problems. The noiseless representation problem is to find sparse representations of a signal r given that such representations exist. In this case, we explicitly construct a frame, referred to as the Vandermonde frame, for which the noiseless sparse representation problem can be solved uniquely using O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) operations, as long as the number of non-zero coefficients in the sparse representation of r is isinN for some 0 les isin les 0.5. It is known that isin les 0.5 cannot be relaxed without violating uniqueness. The noisy sparse representation problem is to find sparse representations of a signal r satisfying a distortion criterion. In this case, we establish a lower bound on the tradeoff between the sparsity of the representation, the underlying distortion and the redundancy of any given frame.

Performance of high‐resolution sensor array processing algorithms in the localization of acoustic sources

The localization of noise sources from a specified direction may often be accomplished with an array of sensors. One commonly used processor consists of delay and add networks: a conventional beamformer, however its spectrum suffers from the Rayleigh resolution and its performance is highly degraded, specially in lower frequency range. In the communication, the performance of some typical high‐resolution sensor array processing algorithms: Minimum Variance, MUSIC, Mini‐Norm algorithms are investigated for wideband source location. Their performances are compared with a new source localization algorithm which is based on a sparse representation of sensor measurements with an overcomplete basis composed of samples from the array manifold. The key of the method is the use of the SVD for data reduction and the formulation of a joint multiple‐sample sparse representation problem in the signal subspace domain. Increased resolution and improved robustness to noise is obtained with this algorithm applied to vario...

A bilinear algorithm for sparse representations

We consider the following sparse representation problem: represent a given matrix X?? m×N as a multiplication X=AS of two matrices A?? m×n (m?n<N) and S?? n×N , under requirements that all m×m submatrices of A are nonsingular, and S is sparse in sense that each column of S has at least n?m+1 zero elements. It is known that under some mild additional assumptions, such representation is unique, up to scaling and permutation of the rows of S. We show that finding A (which is the most difficult part of such representation) can be reduced to a hyperplane clustering problem. We present a bilinear algorithm for such clustering, which is robust to outliers. A computer simulation example is presented showing the robustness of our algorithm.

An Empirical Bayesian Strategy for Solving the Simultaneous Sparse Approximation Problem

Given a large overcomplete dictionary of basis vectors, the goal is to simultaneously represent L>1 signal vectors using coefficient expansions marked by a common sparsity profile. This generalizes the standard sparse representation problem to the case where multiple responses exist that were putatively generated by the same small subset of features. Ideally, the associated sparse generating weights should be recovered, which can have physical significance in many applications (e.g., source localization). The generic solution to this problem is intractable and, therefore, approximate procedures are sought. Based on the concept of automatic relevance determination, this paper uses an empirical Bayesian prior to estimate a convenient posterior distribution over candidate basis vectors. This particular approximation enforces a common sparsity profile and consistently places its prominent posterior mass on the appropriate region of weight-space necessary for simultaneous sparse recovery. The resultant algorithm is then compared with multiple response extensions of matching pursuit, basis pursuit, FOCUSS, and Jeffreys prior-based Bayesian methods, finding that it often outperforms the others. Additional motivation for this particular choice of cost function is also provided, including the analysis of global and local minima and a variational derivation that highlights the similarities and differences between the proposed algorithm and previous approaches.

Equivalence of Reconstruction From the Absolute Value of the Frame Coefficients to a Sparse Representation Problem

The purpose of this letter is to prove, for real frames, that signal reconstruction from the absolute value of the frame coefficients is equivalent to solution of a sparse signal optimization problem, namely a minimum lscrp (quasi)norm over a linear constraint. This linear constraint reflects the coefficients relationship within the range of the analysis operator

Stable recovery of sparse overcomplete representations in the presence of noise

Overcomplete representations are attracting interest in signal processing theory, particularly due to their potential to generate sparse representations of signals. However, in general, the problem of finding sparse representations must be unstable in the presence of noise. This paper establishes the possibility of stable recovery under a combination of sufficient sparsity and favorable structure of the overcomplete system. Considering an ideal underlying signal that has a sufficiently sparse representation, it is assumed that only a noisy version of it can be observed. Assuming further that the overcomplete system is incoherent, it is shown that the optimally sparse approximation to the noisy data differs from the optimally sparse decomposition of the ideal noiseless signal by at most a constant multiple of the noise level. As this optimal-sparsity method requires heavy (combinatorial) computational effort, approximation algorithms are considered. It is shown that similar stability is also available using the basis and the matching pursuit algorithms. Furthermore, it is shown that these methods result in sparse approximation of the noisy data that contains only terms also appearing in the unique sparsest representation of the ideal noiseless sparse signal.

On Polar Polytopes and the Recovery of Sparse Representations

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Suppose we have a signal <emphasis><formula formulatype="inline"><tex>${\mmb y}$</tex></formula></emphasis> which we wish to represent using a linear combination of a number of basis atoms <formula><tex>$\break {\mmb a}\!\!\!$ </tex></formula><emphasis><formula> <tex>$_i,{\mmb y}$</tex></formula></emphasis><emphasis><formula formulatype="inline"> <tex>$=\!\!\!\sum_i x_i {\mmb a}_i ={\mmb A}{\mmb x}$</tex></formula></emphasis>. The problem of finding the minimum <emphasis><formula formulatype="inline"> <tex>$\ell_0$</tex></formula></emphasis> norm representation for <formula formulatype="inline"><tex>${\mmb y}$</tex></formula> is a hard problem. The Basis Pursuit (BP) approach proposes to find the minimum <emphasis><formula formulatype="inline"><tex>$\ell_1$</tex></formula></emphasis> norm representation instead, which corresponds to a linear program (LP) that can be solved using modern LP techniques, and several recent authors have given conditions for the BP (minimum <emphasis><formula formulatype="inline"><tex>$\ell_1$</tex> </formula></emphasis> norm) and sparse (minimum <emphasis><formula formulatype="inline"> <tex>$\ell_0$</tex></formula></emphasis> norm) representations to be identical. In this paper, we explore this <emphasis emphasistype="boldital">sparse representation problem</emphasis> using the geometry of convex polytopes, as recently introduced into the field by Donoho. By considering the dual LP we find that the so-called polar polytope <emphasis><formula formulatype="inline"><tex>$P^*$</tex></formula></emphasis> of the centrally symmetric polytope <emphasis><formula formulatype="inline"> <tex>$P$</tex></formula></emphasis> whose vertices are the atom pairs <formula formulatype="inline"><tex>${\pm}{\mmb a}$</tex></formula><emphasis><formula> <tex>$_i$</tex></formula></emphasis> is particularly helpful in providing us with geometrical insight into optimality conditions given by Fuchs and Tropp for non-unit-norm atom sets. In exploring this geome</para>