The ℓ1 analysis approach by sparse dual frames for sparse signal recovery represented by frames

A sparse-dual-frame based ℓ1-analysis approach for compressed sensing (CS) is proposed. The sparse dual frame is a notion of optimal dual frames of a non-exact frame. It is motivated in the study of compressed sensing problems where signals are sparse with respect to redundant dictionaries (frames). An alternating iterative algorithm is proposed. An error bound ensuring the correct signal recovery is obtained. Empirical studies over generally difficult CS problems demonstrate that the new sparse-dual-based approach provides satisfactory solutions, whereas other existing means may not.

[1]  Yulong Liu,et al.  Compressed Sensing With General Frames via Optimal-Dual-Based $\ell _{1}$-Analysis , 2012, IEEE Transactions on Information Theory.

[2]  Michael Elad,et al.  Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ1 minimization , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[3]  Jean-Jacques Fuchs,et al.  On sparse representations in arbitrary redundant bases , 2004, IEEE Transactions on Information Theory.

[4]  Michael Elad,et al.  Analysis versus synthesis in signal priors , 2006, 2006 14th European Signal Processing Conference.

[5]  Emmanuel J. Candès,et al.  Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information , 2004, IEEE Transactions on Information Theory.

[6]  Xiaoming Huo,et al.  Uncertainty principles and ideal atomic decomposition , 2001, IEEE Trans. Inf. Theory.

[7]  Yulong Liu,et al.  Performance analysis of ℓ1-synthesis with coherent frames , 2012, 2012 IEEE International Symposium on Information Theory Proceedings.

[8]  Yonina C. Eldar,et al.  Compressed Sensing with Coherent and Redundant Dictionaries , 2010, ArXiv.

[9]  E.J. Candes.  Compressive Sampling , 2022 .

[10]  Rémi Gribonval,et al.  Sparse representations in unions of bases , 2003, IEEE Trans. Inf. Theory.

[11]  Michael A. Saunders,et al.  Atomic Decomposition by Basis Pursuit , 1998, SIAM J. Sci. Comput..

[12]  Michael Elad,et al.  A generalized uncertainty principle and sparse representation in pairs of bases , 2002, IEEE Trans. Inf. Theory.

[13]  Yulong Liu,et al.  Fast thresholding algorithms with feedbacks for sparse signal recovery , 2012, ArXiv.

[14]  Pierre Vandergheynst,et al.  Compressed Sensing and Redundant Dictionaries , 2007, IEEE Transactions on Information Theory.

[15]  Shidong Li,et al.  On general frame decompositions , 1995 .