Multivariate iterative hard thresholding for sparse decomposition with flexible sparsity patterns

We address the problem of decomposing several consecutive sparse signals, such as audio time frames or image patches. A typical approach is to process each signal sequentially and independently, with an arbitrary sparsity level fixed for each signal. Here, we propose to process several frames simultaneously, allowing for more flexible sparsity patterns to be considered. We propose a multivariate sparse coding approach, where sparsity is enforced on average across several frames. We propose a Multivariate Iterative Hard Thresholding to solve this problem. The usefulness of the proposed approach is demonstrated on audio coding and denoising tasks. Experiments show that the proposed approach leads to better results when the signal contains both transients and tonal components

Image Decomposition and Separation Using Sparse Representations: An Overview

This paper gives essential insights into the use of sparsity and morphological diversity in image decomposition and source separation by reviewing our recent work in this field. The idea to morphologically decompose a signal into its building blocks is an important problem in signal processing and has far-reaching applications in science and technology. Starck , proposed a novel decomposition method—morphological component analysis (MCA)—based on sparse representation of signals. MCA assumes that each (monochannel) signal is the linear mixture of several layers, the so-called morphological components, that are morphologically distinct, e.g., sines and bumps. The success of this method relies on two tenets: sparsity and morphological diversity. That is, each morphological component is sparsely represented in a specific transform domain, and the latter is highly inefficient in representing the other content in the mixture. Once such transforms are identified, MCA is an iterative thresholding algorithm that is capable of decoupling the signal content. Sparsity and morphological diversity have also been used as a novel and effective source of diversity for blind source separation (BSS), hence extending the MCA to multichannel data. Building on these ingredients, we will provide an overview the generalized MCA introduced by the authors in and as a fast and efficient BSS method. We will illustrate the application of these algorithms on several real examples. We conclude our tour by briefly describing our software toolboxes made available for download on the Internet for sparse signal and image decomposition and separation

Image decomposition and separation using sparse representations: an overview

International audienceThis paper gives essential insights into the use of sparsity and morphological diversity in image decomposition and source separation by overviewing our recent work in this field. The idea to morphologically decompose a signal into its building blocks is an important problem in signal processing and has far-reaching applications in science and technology. Starck et al. [1], [2] proposed a novel decomposition method - Morphological Component Analysis (MCA) - based on sparse representation of signals. MCA assumes that each (monochannel) signal is the linear mixture of several layers, the so-called Morphological Components, that are morphologically distinct, e.g. sines and bumps. The success of this method relies on two tenets: sparsity and morphological diversity. That is, each morphological component is sparsely represented in a specific transform domain, and the latter is highly inefficient in representing the other content in the mixture. Once such transforms are identified, MCA is an iterative thresholding algorithm that is capable of decoupling the signal content. Sparsity and morphological diversity have also been used as a novel and effective source of diversity for blind source separation (BSS), hence extending the MCA to multichannel data. Building on these ingredients, we will overview the Generalized MCA (GMCA) introduced by the authors in [3], [4] as a fast and efficient BSS method. We will illustrate the application of these algorithms on several real examples. We conclude our tour by briefly describing our software toolboxes made available for download on the Internet for sparse signal and image decomposition and separation

Sparse Modeling for Image and Vision Processing

In recent years, a large amount of multi-disciplinary research has been conducted on sparse models and their applications. In statistics and machine learning, the sparsity principle is used to perform model selection---that is, automatically selecting a simple model among a large collection of them. In signal processing, sparse coding consists of representing data with linear combinations of a few dictionary elements. Subsequently, the corresponding tools have been widely adopted by several scientific communities such as neuroscience, bioinformatics, or computer vision. The goal of this monograph is to offer a self-contained view of sparse modeling for visual recognition and image processing. More specifically, we focus on applications where the dictionary is learned and adapted to data, yielding a compact representation that has been successful in various contexts.Comment: 205 pages, to appear in Foundations and Trends in Computer Graphics and Visio

Thresholding-based Iterative Selection Procedures for Model Selection and Shrinkage

This paper discusses a class of thresholding-based iterative selection procedures (TISP) for model selection and shrinkage. People have long before noticed the weakness of the convex $l_1$-constraint (or the soft-thresholding) in wavelets and have designed many different forms of nonconvex penalties to increase model sparsity and accuracy. But for a nonorthogonal regression matrix, there is great difficulty in both investigating the performance in theory and solving the problem in computation. TISP provides a simple and efficient way to tackle this so that we successfully borrow the rich results in the orthogonal design to solve the nonconvex penalized regression for a general design matrix. Our starting point is, however, thresholding rules rather than penalty functions. Indeed, there is a universal connection between them. But a drawback of the latter is its non-unique form, and our approach greatly facilitates the computation and the analysis. In fact, we are able to build the convergence theorem and explore theoretical properties of the selection and estimation via TISP nonasymptotically. More importantly, a novel Hybrid-TISP is proposed based on hard-thresholding and ridge-thresholding. It provides a fusion between the $l_0$-penalty and the $l_2$-penalty, and adaptively achieves the right balance between shrinkage and selection in statistical modeling. In practice, Hybrid-TISP shows superior performance in test-error and is parsimonious.Comment: Submitted to the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org

A proximal iteration for deconvolving Poisson noisy images using sparse representations

We propose an image deconvolution algorithm when the data is contaminated by Poisson noise. The image to restore is assumed to be sparsely represented in a dictionary of waveforms such as the wavelet or curvelet transforms. Our key contributions are: First, we handle the Poisson noise properly by using the Anscombe variance stabilizing transform leading to a {\it non-linear} degradation equation with additive Gaussian noise. Second, the deconvolution problem is formulated as the minimization of a convex functional with a data-fidelity term reflecting the noise properties, and a non-smooth sparsity-promoting penalties over the image representation coefficients (e.g. $\ell_1$-norm). Third, a fast iterative backward-forward splitting algorithm is proposed to solve the minimization problem. We derive existence and uniqueness conditions of the solution, and establish convergence of the iterative algorithm. Finally, a GCV-based model selection procedure is proposed to objectively select the regularization parameter. Experimental results are carried out to show the striking benefits gained from taking into account the Poisson statistics of the noise. These results also suggest that using sparse-domain regularization may be tractable in many deconvolution applications with Poisson noise such as astronomy and microscopy