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Publication# Compressive Sampling Strategies for Multichannel Signals

Abstract

Over the past decade researches in applied mathematics, signal processing and communications have introduced compressive sampling (CS) as an alternative to the Shannon sampling theorem. The two key observations making CS theory widely applicable to numerous areas of signal processing are: i) due to their structural properties, natural signals typically have sparse representations in properly chosen orthogonal bases, ii) the number of linear non-adaptive measurements required to acquire high-dimensional data with CS is proportional to the signal’s sparsity level in the chosen basis. In multichannel signal applications the data of different channels is often highly correlated and therefore the unstructured sparsity hypothesis deployed by the classical CS theory results in suboptimal measurement rates. Meanwhile, the wide range of applications of multichannel signals and the extremely large and increasing flow of data in those applications motivates the development of more comprehensive models incorporating both inter and intra-channel data structures in order to achieve more efficient dimensionality reduction. The main focus of this thesis is on studying two new models for efficient multichannel signal compressed sensing. Our first approach proposes a simultaneous low-rank and joint-sparse matrix model for multichannel signals. As a result, we introduce a novel CS recovery scheme based on Nuclear-l2/l1 norm convex minimization for low-rank and joint-sparse matrix approximation. Our theoretical analysis indicates that using this approach can achieve significantly lower sampling rates for a robust multichannel data CS acquisition than state-of-the-art methods. More remarkably, our analysis confirms the near-optimality of this approach: the number of CS measurements are nearly proportional to the few degrees of freedom of such structured data. Our second approach introduces a stronger model for multichannel data synthesized by a linear mixture model. Here we assume that the mixture parameters are given as side information. As a result, multichannel data CS recovery turns into a compressive source separation problem, where we propose a novel decorrelating scheme to exploit the knowledge of mixture parameters for a robust and numerically efficient source identification. Our theoretical guarantees explain the fundamental limits of this approach in terms of the number of CS measurements, the sparsity level of sources, the sampling noise, and the conditioning of the mixture parameters. We apply these two approaches to compressive hyperspectral image recovery and source separation, and compare the efficiency of our methods to state-of-the-art approaches for several challenging real-world hyperspectral datasets. Note that applications of these methods are not limited to hyperspectral imagery and it can have a broad impact on numerous multichannel signal applications. As an example, for sensor network applications deploying compressive sampling schemes, our results indicate a tight tradeoff between the number of available sensors (channels) and the complexity/cost of each sensor. Finally, in a different multichannel signal application, we deal with a simple but very important problem in computer vision namely the detection and localization of people given their multi-view silhouettes captured by networks of cameras. The main challenge in many existing solutions is the tradeoff between robustness and numerical efficiency. We model this problem by a boolean (non-linear) inverse problem where, by penalizing the sparsity of the solution, we achieve accurate results comparable to state-of-the-art methods. More remarkably, using boolean arithmetics enables us to propose a real-time and memory efficient approximation algorithm that is mainly rooted in the classical literature of group testing and set cover.

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Related concepts (3)

Compressed sensing

Compressed sensing (also known as compressive sensing, compressive sampling, or sparse sampling) is a signal processing technique for efficiently acquiring and reconstructing a signal, by finding solutions to underdetermined linear systems. This is based on the principle that, through optimization, the sparsity of a signal can be exploited to recover it from far fewer samples than required by the Nyquist–Shannon sampling theorem. There are two conditions under which recovery is possible.

Measurement

Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared to a basic reference quantity of the same kind. The scope and application of measurement are dependent on the context and discipline. In natural sciences and engineering, measurements do not apply to nominal properties of objects or events, which is consistent with the guidelines of the International vocabulary of metrology published by the International Bureau of Weights and Measures.

Mathematics

Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them.