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Sparse and redundant representations. From theory to applications in signal and image processing. (English) Zbl 1211.94001
New York, NY: Springer (ISBN 978-1-4419-7010-7/hbk; 978-1-4419-7011-4/ebook). xx, 376 p. (2010).
The concept of sparse representations for signals and images is explored in the book under review. Although sparse representation is a poorly defined problem and a computationally impractical goal in general, the authors give a long series of recent mathematical results to show that under certain conditions it is possible to obtain results of a positive nature, guaranteeing uniqueness, stability, and computational practicality. The author discusses how these can be used to form an interesting and strong source model for signals and images. Several potential applications are shown that use this model of sparse representation in real image processing settings, and cases where this leads to state-of-the-art results. More specific problems such as image denoising, image deblurring, facial image compression, image inpainting, and image scale-up, all benefit from this model. The book offers an important and organized view of this field, setting the foundations of future research.
The book is divided into two parts. The theoretical and numerical foundations of sparse and redundant representations are considered in the first part. It covers such topics as: uncertainty principle and solutions, uniqueness analysis for the general case, pursuit algorithms practice, approximate solutions, iterative-shrinkage algorithms, average performance analysis and Dantzig-Selector algorithm. In the second part of the book are considered: the sparsity-seeking methods in signal processing, image deblurring, the applications of maximum aposteriory probability and minimum-mean squared error estimators, dictionary-learning algorithms and training structured dictionaries, image compression and denoising and other applications.
The book is written to serve as the material for an advanced one-semester graduate course for engineering students. It will be of interest for all specialists working in the area of sparse and redundant representations application in signal and image processing.

MSC:
94-02 Research exposition (monographs, survey articles) pertaining to information and communication theory
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
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