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• W2029740707 abstract "We consider the classical problem of finding the sparse representation of a signal in a pair of bases. When both bases are orthogonal, it is known that the sparse representation is unique when the sparsity K of the signal satisfies K <; 1/μ(D), where μ(D) is the mutual coherence of the dictionary. Furthermore, the sparse representation can be obtained in polynomial time by basis pursuit (BP), when K <; 0.91/μ(D). Therefore, there is a gap between the unicity condition and the one required to use the polynomial-complexity BP formulation. For the case of general dictionaries, it is also well known that finding the sparse representation under the only constraint of unicity is NP-hard. In this paper, we introduce, for the case of Fourier and canonical bases, a polynomial complexity algorithm that finds all the possible K-sparse representations of a signal under the weaker condition that K <; √2/μ(D). Consequently, when K <; 1/μ(D), the proposed algorithm solves the unique sparse representation problem for this structured dictionary in polynomial time. We further show that the same method can be extended to many other pairs of bases, one of which must have local atoms. Examples include the union of Fourier and local Fourier bases, the union of discrete cosine transform and canonical bases, and the union of random Gaussian and canonical bases." @default.
• W2029740707 created "2016-06-24" @default.
• W2029740707 creator A5038111389 @default.
• W2029740707 creator A5053203917 @default.
• W2029740707 date "2014-12-01" @default.
• W2029740707 modified "2023-10-01" @default.
• W2029740707 title "On Sparse Representation in Fourier and Local Bases" @default.
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• W2029740707 doi "https://doi.org/10.1109/tit.2014.2361858" @default.
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