In recent years there has been a growing interest in the study of sparse representation of signals. Using an overcomplete dictionary that contains prototype signal-atoms, signals are described by sparse linear combinations of these atoms. Applications that use sparse representation are many and include compression, regularization in inverse problems, feature extraction, and more. Recent activity in this field has concentrated mainly on the study of pursuit algorithms that decompose signals with respect to a given dictionary. Designing dictionaries to better fit the above model can be done by either selecting one from a prespecified set of linear transforms or adapting the dictionary to a set of training signals. Both of these techniques have been considered, but this topic is largely still open. In this paper we propose a novel algorithm for adapting dictionaries in order to achieve sparse signal representations. Given a set of training signals, we seek the dictionary that leads to the best representation for each member in this set, under strict sparsity constraints. We present a new method-the K-SVD algorithm-generalizing the K-means clustering process. K-SVD is an iterative method that alternates between sparse coding of the examples based on the current dictionary and a process of updating the dictionary atoms to better fit the data. The update of the dictionary columns is combined with an update of the sparse representations, thereby accelerating convergence. The K-SVD algorithm is flexible and can work with any pursuit method (e.g., basis pursuit, FOCUSS, or matching pursuit). We analyze this algorithm and demonstrate its results both on synthetic tests and in applications on real image data

https://users.ece.utexas.edu/~sanghavi/courses/papers/KSVD.pdf

$rm K$ -SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation

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.

Stable recovery of sparse overcomplete representations in the presence of noise

Edited by the people who were forerunners in creating the field, together with contributions from 34 leading international experts, this handbook provides the definitive reference on Blind Source Separation, giving a broad and comprehensive description of all the core principles and methods, numerical algorithms and major applications in the fields of telecommunications, biomedical engineering and audio, acoustic and speech processing. Going beyond a machine learning perspective, the book reflects recent results in signal processing and numerical analysis, and includes topics such as optimization criteria, mathematical tools, the design of numerical algorithms, convolutive mixtures, and time frequency approaches. This Handbook is an ideal reference for university researchers, RD algebraic identification of under-determined mixtures, time-frequency methods, Bayesian approaches, blind identification under non negativity approaches, semi-blind methods for communicationsShows the applications of the methods to key application areas such as telecommunications, biomedical engineering, speech, acoustic, audio and music processing, while also giving a general method for developing applications

https://ie.technion.ac.il/~mcib/Gribonval_Zibulevsky_SCA_chapter.pdf

Handbook of Blind Source Separation: Independent Component Analysis and Applications

The minimum lscr1-norm solution to an underdetermined system of linear equations y=Ax is often, remarkably, also the sparsest solution to that system. This sparsity-seeking property is of interest in signal processing and information transmission. However, general-purpose optimizers are much too slow for lscr1 minimization in many large-scale applications.In this paper, the Homotopy method, originally proposed by Osborne et al. and Efron et al., is applied to the underdetermined lscr1-minimization problem min parxpar1 subject to y=Ax. Homotopy is shown to run much more rapidly than general-purpose LP solvers when sufficient sparsity is present. Indeed, the method often has the following k-step solution property: if the underlying solution has only k nonzeros, the Homotopy method reaches that solution in only k iterative steps. This k-step solution property is demonstrated for several ensembles of matrices, including incoherent matrices, uniform spherical matrices, and partial orthogonal matrices. These results imply that Homotopy may be used to rapidly decode error-correcting codes in a stylized communication system with a computational budget constraint. The approach also sheds light on the evident parallelism in results on lscr1 minimization and orthogonal matching pursuit (OMP), and aids in explaining the inherent relations between Homotopy, least angle regression (LARS), OMP, and polytope faces pursuit.

/pdf/fast-solution-of-ell-1-norm-minimization-problems-when-the-36p60aek8y.pdf

Fast Solution of $\ell _{1}$ -Norm Minimization Problems When the Solution May Be Sparse

This paper explores various aspects of the image decomposition problem using modern variational techniques. We aim at splitting an original image f into two components u and ?, where u holds the geometrical information and ? holds the textural information. The focus of this paper is to study different energy terms and functional spaces that suit various types of textures. Our modeling uses the total-variation energy for extracting the structural part and one of four of the following norms for the textural part: L2, G, L1 and a new tunable norm, suggested here for the first time, based on Gabor functions. Apart from the broad perspective and our suggestions when each model should be used, the paper contains three specific novelties: first we show that the correlation graph between u and ? may serve as an efficient tool to select the splitting parameter, second we propose a new fast algorithm to solve the TV ? L1 minimization problem, and third we introduce the theory and design tools for the TV-Gabor model.

https://www.math.u-bordeaux.fr/~jaujol/HDR/A7.pdf

Structure-Texture Image Decomposition--Modeling, Algorithms, and Parameter Selection

This paper presents a novel method for separating images into
texture and piecewise smooth parts. The proposed approach is based
on a combination of the Basis Pursuit Denoising (BPDN) algorithm
and the Total-Variation (TV) regularization scheme. The basic idea
promoted in this paper is the use of two appropriate dictionaries,
one for the representation of textures, and the other for the
natural scene parts. Each dictionary is designed for sparse
representation of a particular type of image-content (either
texture or piecewise smooth). The use of BPDN with the two
augmented dictionaries leads to the desired separation, along with
noise removal as a by-product. As the need to choose a proper
dictionary for natural scene is very hard, a TV regularization is
employed to better direct the separation process. Experimental
results validate the algorithm's performance.

Mikael Elad

Papers

Image decomposition: separation of texture from piecewise smooth content