Deterministic Compressed Sensing Using Binary Measurement Matrices
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Deterministic Compressed Sensing Using Binary Measurement Matrices
The fundamental objective of compressed sensing is to recover a high dimensional but low complexity vector or matrix from only a few linear measurements. In most of the initial publications in the field of compressed sensing, the emphasis is on using random matrices such as Gaussian, Bernoulli, etc. The limitations of this framework include high memory requirements, and CPU time. This dissertation mainly focuses on binary measurement matrices and highlights the superiority of binary matrices in terms of CPU time, complexity, and the required memory compared to the random measurement matrices. In the first part of the dissertation (Chapter 2), we present a new recovery algorithm for compressed sensing that makes use of binary measurement matrices and achieves exact recovery of ultra sparse vectors, in a single pass and without any iterations. Due to its non-iterative nature, our algorithm is hundreds of times faster than iterative algorithms such as l 1-norm minimization, and methods based on expander graphs. Our algorithm can accommodate nearly sparse vectors, in which case it recovers the index set of the largest components, and can also accommodate shot noise measurements. In the second part of the dissertation (Chapter 3), we focus on using binary measurement matrices to solve the problem of compressed sensing while l1-norm minimization (basis pursuit) is considered as the recovery algorithm. We derive new upper and lower bounds on the number of measurements to achieve robust sparse recovery with binary matrices. We prove sufficient conditions for a column-regular binary matrix to satisfy the robust null space property (RNSP) and derive universal lower bounds on the number of measurements that any binary matrix needs to have in order to satisfy the weaker sufficient condition based on the RNSP. Then we show that binary matrices that are bi-adjacency matrices of bipartite graphs of girth six are optimal. Two classes of binary matrices, namely parity check matrices of array codes, and the matrices based on Euler Squares, have girth six and are nearly optimal in the sense of almost satisfying the lower bound. In principle randomly generated Gaussian measurement matrices are "order-optimal"; however in practice, Gaussian matrices require more measurements than binary matrices when n 106
Compressive Sensing in Healthcare
Compressive Sensing in Healthcare, part of the Advances in Ubiquitous Sensing Applications for Healthcare series gives a review on compressive sensing techniques in a practical way, also presenting deterministic compressive sensing techniques that can be used in the field. The focus of the book is on healthcare applications for this technology. It is intended for both the creators of this technology and the end users of these products. The content includes the use of EEG and ECG, plus hardware and software requirements for building projects. Body area networks and body sensor networks are explored. - Provides a toolbox for compressive sensing in health, presenting both mathematical and coding information - Presents an intuitive introduction to compressive sensing, including MATLAB tutorials - Covers applications of compressive sensing in health care
An Introduction to Compressed Sensing
Compressed sensing is a relatively recent area of research that refers to the recovery of high-dimensional but low-complexity objects from a limited number of measurements. The topic has applications to signal/image processing and computer algorithms, and it draws from a variety of mathematical techniques such as graph theory, probability theory, linear algebra, and optimization. The author presents significant concepts never before discussed as well as new advances in the theory, providing an in-depth initiation to the field of compressed sensing. An Introduction to Compressed Sensing contains substantial material on graph theory and the design of binary measurement matrices, which is missing in recent texts despite being poised to play a key role in the future of compressed sensing theory. It also covers several new developments in the field and is the only book to thoroughly study the problem of matrix recovery. The book supplies relevant results alongside their proofs in a compact and streamlined presentation that is easy to navigate. The core audience for this book is engineers, computer scientists, and statisticians who are interested in compressed sensing. Professionals working in image processing, speech processing, or seismic signal processing will also find the book of interest.