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Compressive sampling
, 2006
"... Conventional wisdom and common practice in acquisition and reconstruction of images from frequency data follow the basic principle of the Nyquist density sampling theory. This principle states that to reconstruct an image, the number of Fourier samples we need to acquire must match the desired res ..."
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Cited by 1026 (15 self)
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Conventional wisdom and common practice in acquisition and reconstruction of images from frequency data follow the basic principle of the Nyquist density sampling theory. This principle states that to reconstruct an image, the number of Fourier samples we need to acquire must match the desired resolution of the image, i.e. the number of pixels in the image. This paper surveys an emerging theory which goes by the name of “compressive sampling” or “compressed sensing,” and which says that this conventional wisdom is inaccurate. Perhaps surprisingly, it is possible to reconstruct images or signals of scientific interest accurately and sometimes even exactly from a number of samples which is far smaller than the desired resolution of the image/signal, e.g. the number of pixels in the image. It is believed that compressive sampling has far reaching implications. For example, it suggests the possibility of new data acquisition protocols that translate analog information into digital form with fewer sensors than what was considered necessary. This new sampling theory may come to underlie procedures for sampling and compressing data simultaneously. In this short survey, we provide some of the key mathematical insights underlying this new theory, and explain some of the interactions between compressive sampling and other fields such as statistics, information theory, coding theory, and theoretical computer science.
Explicit constructions for compressed sensing of sparse signals
 In Proceedings of the 19th Annual ACMSIAM Symposium on Discrete Algorithms
, 2008
"... Over the recent years, a new approach for obtaining a succinct approximate representation of ndimensional ..."
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Cited by 45 (3 self)
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Over the recent years, a new approach for obtaining a succinct approximate representation of ndimensional
A remark on compressed sensing
, 2007
"... Abstract—Recently, a new direction in signal processing – “Compressed Sensing " is being actively developed. A number of authors have pointed out a connection between the Compressed Sensing problem and the problem of estimating the Kolmogorov widths, studied in the seventies and eighties of the ..."
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Cited by 23 (0 self)
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Abstract—Recently, a new direction in signal processing – “Compressed Sensing " is being actively developed. A number of authors have pointed out a connection between the Compressed Sensing problem and the problem of estimating the Kolmogorov widths, studied in the seventies and eighties of the last century. In this paper we make the above mentioned connection more precise. DOI: 10.1134/S0001434607110193
Sparse representations and approximation theory
, 2011
"... This paper is an attempt to both expound and expand upon, from an approximation theorist’s point of view, some of the theoretical results that have been obtained in the sparse representation (compressed sensing) literature. In particular, we consider in detail ℓm 1approximation, which is fundamenta ..."
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Cited by 1 (0 self)
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This paper is an attempt to both expound and expand upon, from an approximation theorist’s point of view, some of the theoretical results that have been obtained in the sparse representation (compressed sensing) literature. In particular, we consider in detail ℓm 1approximation, which is fundamental in the theory of sparse representations, and the connection between the theory of sparse representations and certain nwidth concepts. We try to illustrate how the theory of sparse representation leads to new and interesting problems in approximation theory, while the results and techniques of approximation theory can further add to the theory of sparse representations.
A REVIEW OF NUMERICAL METHODS FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
"... To HeinzOtto Kreiss with friendship and appreciation Abstract. Numerical methods were first put into use as an effective tool for solving partial differential equations (PDEs) by John von Neumann in the mid1940s. In a 1949 letter von Neumann wrote “the entire computing machine is merely one compon ..."
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To HeinzOtto Kreiss with friendship and appreciation Abstract. Numerical methods were first put into use as an effective tool for solving partial differential equations (PDEs) by John von Neumann in the mid1940s. In a 1949 letter von Neumann wrote “the entire computing machine is merely one component of a greater whole, namely, of the unity formed by the computing machine, the mathematical problems that go with it, and the type of planning which is called by both. ” The “greater whole ” is viewed today as scientific computation: over the past sixty years, scientific computation has emerged as the most versatile tool to complement theory and experiments, and numerical methods for solving PDEs are at the heart of many of today’s advanced scientific computations. Numerical solutions found their way from financial models on Wall Street to traffic models on Main Street. Here we provide a bird’s eye view on the development of these numerical methods with
Derivative Compressive Sampling with Application to Inverse Problems and Imaging
"... c ○ Mahdi S. Hosseini 2010I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ..."
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c ○ Mahdi S. Hosseini 2010I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public.
On the size of incoherent systems ⋆
"... This paper concerns special redundant systems, namely, incoherent systems or systems with small coherence parameter. Simple greedytype algorithms perform well on these systems. For example, known results show that the smaller the coherence parameter, the better the performance of the Orthogonal Gre ..."
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This paper concerns special redundant systems, namely, incoherent systems or systems with small coherence parameter. Simple greedytype algorithms perform well on these systems. For example, known results show that the smaller the coherence parameter, the better the performance of the Orthogonal Greedy Algorithm. These systems with small coherence parameter are also useful in the construction of compressed sensing matrices. Therefore, it is very desirable to build dictionaries with small coherence parameter. We discuss the following problem for both R n and C n: How large can a system with coherence parameter not exceeding a fixed number µ be? We obtain upper and lower bounds for the maximal cardinality of such systems. Although the results herein are either known or simple corollaries of known results, our objective is to demonstrate how fundamental results from different areas of mathematics–linear algebra, probability, and number theory–collaborate on this important approximation theoretic problem.
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, 804
"... On the reconstruction of blocksparse signals with an optimal number of measurements ..."
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On the reconstruction of blocksparse signals with an optimal number of measurements