Sparse approximation

About: Sparse approximation is a research topic. Over the lifetime, 18037 publications have been published within this topic receiving 497739 citations. The topic is also known as: Sparse approximation.

An Analysis and Implementation of the BM3D Image Denoising Method

Abstract: BM3D is a recent denoising method based on the fact that an image has a locally sparse representation in transform domain. This sparsity is enhanced by grouping similar 2D image patches into 3D groups. In this paper we propose an open-source implementation of the method. We discuss the choice of all parameter methods and confirm their actual optimality. The description of the method is rewritten with a new notation. We hope this new notation is more transparent than in the original paper. A final index gives nonetheless the correspondence between the new notation and the original notation.

Exactly Sparse Delayed-State Filters for View-Based SLAM

Abstract: This paper reports the novel insight that the simultaneous localization and mapping (SLAM) information matrix is exactly sparse in a delayed-state framework. Such a framework is used in view-based representations of the environment that rely upon scan-matching raw sensor data to obtain virtual observations of robot motion with respect to a place it has previously been. The exact sparseness of the delayed-state information matrix is in contrast to other recent feature-based SLAM information algorithms, such as sparse extended information filter or thin junction-tree filter, since these methods have to make approximations in order to force the feature-based SLAM information matrix to be sparse. The benefit of the exact sparsity of the delayed-state framework is that it allows one to take advantage of the information space parameterization without incurring any sparse approximation error. Therefore, it can produce equivalent results to the full-covariance solution. The approach is validated experimentally using monocular imagery for two datasets: a test-tank experiment with ground truth, and a remotely operated vehicle survey of the RMS Titanic

Fast $\ell_{1}$ -Minimization Algorithms for Robust Face Recognition

Abstract: l 1-minimization refers to finding the minimum l1-norm solution to an underdetermined linear system \mbib=A\mbix. Under certain conditions as described in compressive sensing theory, the minimum l1-norm solution is also the sparsest solution. In this paper, we study the speed and scalability of its algorithms. In particular, we focus on the numerical implementation of a sparsity-based classification framework in robust face recognition, where sparse representation is sought to recover human identities from high-dimensional facial images that may be corrupted by illumination, facial disguise, and pose variation. Although the underlying numerical problem is a linear program, traditional algorithms are known to suffer poor scalability for large-scale applications. We investigate a new solution based on a classical convex optimization framework, known as augmented Lagrangian methods. We conduct extensive experiments to validate and compare its performance against several popular l1-minimization solvers, including interior-point method, Homotopy, FISTA, SESOP-PCD, approximate message passing, and TFOCS. To aid peer evaluation, the code for all the algorithms has been made publicly available.

Sparse approximation using least squares support vector machines

Abstract: In least squares support vector machines (LS-SVMs) for function estimation Vapnik's /spl epsiv/-insensitive loss function has been replaced by a cost function which corresponds to a form of ridge regression. In this way nonlinear function estimation is done by solving a linear set of equations instead of solving a quadratic programming problem. The LS-SVM formulation also involves less tuning parameters. However, a drawback is that sparseness is lost in the LS-SVM case. In this paper we investigate imposing sparseness by pruning support values from the sorted support value spectrum which results from the solution to the linear system.

Hyperspectral Image Classification Based on Structured Sparse Logistic Regression and Three-Dimensional Wavelet Texture Features

Abstract: Hyperspectral remote sensing imagery contains rich information on spectral and spatial distributions of distinct surface materials. Owing to its numerous and continuous spectral bands, hyperspectral data enable more accurate and reliable material classification than using panchromatic or multispectral imagery. However, high-dimensional spectral features and limited number of available training samples have caused some difficulties in the classification, such as overfitting in learning, noise sensitiveness, overloaded computation, and lack of meaningful physical interpretability. In this paper, we propose a hyperspectral feature extraction and pixel classification method based on structured sparse logistic regression and 3-D discrete wavelet transform (3D-DWT) texture features. The 3D-DWT decomposes a hyperspectral data cube at different scales, frequencies, and orientations, during which the hyperspectral data cube is considered as a whole tensor instead of adapting the data to a vector or matrix. This allows the capture of geometrical and statistical spectral-spatial structures. After the feature extraction step, sparse representation/modeling is applied for data analysis and processing via sparse regularized optimization, which selects a small subset of the original feature variables to model the data for regression and classification purpose. A linear structured sparse logistic regression model is proposed to simultaneously select the discriminant features from the pool of 3D-DWT texture features and learn the coefficients of the linear classifier, in which the prior knowledge about feature structure can be mapped into the various sparsity-inducing norms such as lasso, group, and sparse group lasso. Furthermore, to overcome the limitation of linear models, we extended the linear sparse model to nonlinear classification by partitioning the feature space into subspaces of linearly separable samples. The advantages of our methods are validated on the real hyperspectral remote sensing data sets.