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Compressed data separation via dual frames based split-analysis with Weibull matrices

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Abstract

In this paper, we consider data separation problem, where the original signal is composed of two distinct subcomponents, via dual frames based Split-analysis approach. We show that the two distinct subcomponents, which are sparse in two different general frames respectively, can be exactly recovered with high probability, when the measurement matrix is a Weibull random matrix (not Gaussian) and the two frames satisfy a mutual coherence property. Our result may be significant for analysing Split-analysis model for data separation.

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References

  1. R. Baraniuk, M. Davenport, R. Decore and M. Wakin. A simple proof of the restricted isometry property for random matrices, Constr Approx, 2008, 28: 253–263.

    Article  MATH  MathSciNet  Google Scholar 

  2. J. Cai, S. Osher and Z. Shen. Split Bregman methods and frame based image restoration, Multiscale Model Simul, 2009, 8: 337–369.

    Article  MathSciNet  Google Scholar 

  3. T. Cai, L. Wang and G. Xu. New bounds for restricted isometry constants, IEEE Trans Inform Theory, 2010, 56: 4388–4394.

    Article  MathSciNet  Google Scholar 

  4. T. Cai and A. Zhang. Sharp RIP bound for sparse signal and low-rank matrix recovery, Appl Comput Harmon Anal, 2012, 31: 59–73.

    MathSciNet  Google Scholar 

  5. T. Cai and A. Zhang. Sparse representation of a polytope and recovery of sparse signals and lowrank matrices, manuscript, available at http://arxiv.org/abs/1306.1154.pdf.

  6. T. Cai, G. Xu and J. Zhang. On recovery of sparse signals via l 1 minimization, IEEE Trans Inform Theory, 2009, 55: 3388–3397.

    Article  MathSciNet  Google Scholar 

  7. E. J. Candès. The restricted isometry property and its implications for compressed sensing, C R Acad Sci Paris Sér I Math, 2008, 346: 589–592.

    Article  MATH  Google Scholar 

  8. E. J. Candès, J. Romberg and T. Tao. Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Trans Inform Theory, 2006, 52: 489–509.

    Article  MATH  MathSciNet  Google Scholar 

  9. E. J. Candès, J. Romberg and T. Tao. Stable signal recovery from incomplete and inaccurate measurements, Comm Pure Appl Math, 2006, 59: 1207–1223.

    Article  MATH  MathSciNet  Google Scholar 

  10. E. J. Candès, Y. C. Eldar, D. Needell and P. Randall. Compressed sensing with coherent and redundant dictionaries, Appl Comput Harmon Anal, 2011, 31: 59–73.

    Article  MATH  MathSciNet  Google Scholar 

  11. E. J. Candès and D. L. Donoho. New tight frames of curvelets and optimal representations of objects with piecewise C 2 singularities, Comm Pure Appl Math, 2004, 57: 489–509.

    Google Scholar 

  12. O. Christensen. An Introduction to Frames and Riesz Bases, Birkauser, 2002.

    Google Scholar 

  13. D. Donoho. Compressed sensing, IEEE Trans Inform Theory, 2006, 52: 1289–1306.

    Article  MathSciNet  Google Scholar 

  14. M. Elad, J. L. Starck, P. Querre and D. L. Donoho. Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA), Appl Comput Harmon Anal, 2005, 19: 340–358.

    Article  MATH  MathSciNet  Google Scholar 

  15. Y. C. Eldar and G. Kutyniok. Compressed Sensing: Theory and Applications, Cambridge University Press, New York, 2012.

    Book  Google Scholar 

  16. S. Foucart. Stability and robustness of l 1 -minimization with Weibull matrices and redundant dictionaries, Linear Algebra Appl, 2012. [Online]_Available: http://dx.doi.org/10.1016/j.laa.2012.10.003.

    Google Scholar 

  17. S. Foucart and M. Lai. Sparse recovery with pre-Gaussian random matrices, Studia Math, 2010, 200: 91–102.

    Article  MATH  MathSciNet  Google Scholar 

  18. S. Li and J. Lin. Compressed sensing with coherent tight frame via lq minimization, Inverse Probl Imaging, accepted, 2013.

    Google Scholar 

  19. J. Lin, S. Li and Y. Shen. Compressed data separation with redundant dictionaries, IEEE Trans Inform Theory, 2013, 39: 4309–4315.

    Article  MathSciNet  Google Scholar 

  20. J. Lin, S. Li and Y. Shen. New bounds for restricted isometry constants with coherent tight frames, IEEE Trans Signal Process, 2013, 61: 611–621.

    Article  MathSciNet  Google Scholar 

  21. Y. Liu, T. Mi and S. Li. Compressed sensing with general frames via optimal-dual-based l 1-analysis, IEEE Trans Inform Theory, 2012, 58: 4201–4214.

    Article  MathSciNet  Google Scholar 

  22. S. Mallat. A Wavelet Tour of Signal Processing, 2nd ed, Academic Press, London, 1999.

    MATH  Google Scholar 

  23. Q. Mo and S. Li. New bounds on the restricted isometry constant δ 2 k, Appl Comput Harmon Anal, 2011, 31: 460–468.

    Article  MATH  MathSciNet  Google Scholar 

  24. H. Rauhut, K. Schnass and P. Vandergheynst. Compressed sensing and redundant dictionaries, IEEE Trans Inform Theory, 2008, 54: 2210–2219.

    Article  MathSciNet  Google Scholar 

  25. M. Rudelson and R. Vershynin. On sparse reconstruction from Fourier and Gaussian measurements, Comm Pure Appl Math, 2008, 61: 1025–1045.

    Article  MATH  MathSciNet  Google Scholar 

  26. R. Zhang and S. Li. Optimal D-RIP bound for compressed sensing, preprint.

  27. X. Zhang and S. Li. Compressed sensing via dual frame based l 1 -analysis with Weibull matrices, IEEE Signal Process Letters, 2012. [Online] Available: http://dx.doi.org/10.1109/LSP.2013.2242060.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

    Yun Cai & Song Li

Authors
  1. Yun Cai
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  2. Song Li
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Correspondence to Song Li.

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Dedicated to Professor Jiangong Chen on the occasion of his 120th birthday

Supported by the National Natural Science Foundation of China (11171299 and 91130009).

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Cai, Y., Li, S. Compressed data separation via dual frames based split-analysis with Weibull matrices. Appl. Math. J. Chin. Univ. 28, 427–437 (2013). https://doi.org/10.1007/s11766-013-3201-z

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  • DOI: https://doi.org/10.1007/s11766-013-3201-z

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