Journal ArticleDOI
A generalized uncertainty principle and sparse representation in pairs of bases
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The main contribution in this paper is the improvement of an important result due to Donoho and Huo (2001) concerning the replacement of the l/sub 0/ optimization problem by a linear programming minimization when searching for the unique sparse representation.Abstract:
An elementary proof of a basic uncertainty principle concerning pairs of representations of R/sup N/ vectors in different orthonormal bases is provided. The result, slightly stronger than stated before, has a direct impact on the uniqueness property of the sparse representation of such vectors using pairs of orthonormal bases as overcomplete dictionaries. The main contribution in this paper is the improvement of an important result due to Donoho and Huo (2001) concerning the replacement of the l/sub 0/ optimization problem by a linear programming (LP) minimization when searching for the unique sparse representation.read more
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Compressed sensing
TL;DR: It is possible to design n=O(Nlog(m)) nonadaptive measurements allowing reconstruction with accuracy comparable to that attainable with direct knowledge of the N most important coefficients, and a good approximation to those N important coefficients is extracted from the n measurements by solving a linear program-Basis Pursuit in signal processing.
Journal ArticleDOI
Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information
TL;DR: In this paper, the authors considered the model problem of reconstructing an object from incomplete frequency samples and showed that with probability at least 1-O(N/sup -M/), f can be reconstructed exactly as the solution to the lscr/sub 1/ minimization problem.
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$rm K$ -SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation
TL;DR: A novel algorithm for adapting dictionaries in order to achieve sparse signal representations, the K-SVD algorithm, an iterative method that alternates between sparse coding of the examples based on the current dictionary and a process of updating the dictionary atoms to better fit the data.
Journal ArticleDOI
Decoding by linear programming
Emmanuel J. Candès,Terence Tao +1 more
TL;DR: F can be recovered exactly by solving a simple convex optimization problem (which one can recast as a linear program) and numerical experiments suggest that this recovery procedure works unreasonably well; f is recovered exactly even in situations where a significant fraction of the output is corrupted.
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Decoding by Linear Programming
Emmanuel J. Candès,Terence Tao +1 more
TL;DR: In this paper, it was shown that under suitable conditions on the coding matrix, the input vector can be recovered exactly by solving a simple convex optimization problem (which one can recast as a linear program).
References
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A wavelet tour of signal processing
TL;DR: An introduction to a Transient World and an Approximation Tour of Wavelet Packet and Local Cosine Bases.
Journal ArticleDOI
Atomic Decomposition by Basis Pursuit
TL;DR: Basis Pursuit (BP) is a principle for decomposing a signal into an "optimal" superposition of dictionary elements, where optimal means having the smallest l1 norm of coefficients among all such decompositions.
Journal ArticleDOI
Matching pursuits with time-frequency dictionaries
Stéphane Mallat,Zhifeng Zhang +1 more
TL;DR: The authors introduce an algorithm, called matching pursuit, that decomposes any signal into a linear expansion of waveforms that are selected from a redundant dictionary of functions, chosen in order to best match the signal structures.
Journal ArticleDOI
Atomic Decomposition by Basis Pursuit
TL;DR: This work gives examples exhibiting several advantages over MOF, MP, and BOB, including better sparsity and superresolution, and obtains reasonable success with a primal-dual logarithmic barrier method and conjugate-gradient solver.
Journal ArticleDOI
Uncertainty principles and ideal atomic decomposition
David L. Donoho,Xiaoming Huo +1 more
TL;DR: It is proved that if S is representable as a highly sparse superposition of atoms from this time-frequency dictionary, then there is only one such highly sparse representation of S, and it can be obtained by solving the convex optimization problem of minimizing the l/sup 1/ norm of the coefficients among all decompositions.