Frames for compressed sensing using coherence

Subject Areas : Functional analysis

L. Gavruta ¹ , G. Zamani Eskandani ² , P. Gavruta ³

1 - Politehnica University of Timisoara, Department of Mathematics, Piata Victoriei no.2, 300006 Timisoara, Romania
2 - Faculty of Sciences, Department of Mathematics, University of Tabriz, Tabriz, Iran
3 - Politehnica University of Timisoara, Department of Mathematics, Piata Victoriei no.2, 300006 Timisoara, Romania

Received: 2014-12-10 Accepted : 2015-03-20 Published : 2015-04-01

Keywords: coherence, compressed sensing, frames,

Abstract :

We give some new results on sparse signal recovery in the presence of noise, forweighted spaces. Traditionally, were used dictionaries that have the norm equal to 1, but, forrandom dictionaries this condition is rarely satisfied. Moreover, we give better estimationsthen the ones given recently by Cai, Wang and Xu.

References:

[1] R. Baraniuk, P. Steeghs, Compressive radar imaging, IEEE Radar Conference, Waltham, Massachusetts, April 2007.

[2] T. Cai, L. Wang, and G. Xu, Stable Recovery of Sparse Signals and an Oracle Inequality, IEEE Trans. Inf. Theory, 56(2010) 3516–3522.

[3] E. Candes, The restricted isometry property and its implications for compressed sensing, C. R. Acad. Sci. Paris, Ser. I 346 (2008) 589–592.

[4] E. Candes, M. Wakin, An introduction to Compressive Sampling, IEEE Signal Processing Magazine, 25(2)(2008) 21–30.

[5] E. Candes, T. Tao, Decoding by Linear Programming, IEEE Trans. Inform. Theory, 51(12)(2005) 4203–4215.

[6] E. Candes, J. Romberg, T. Tao, Stable signal recovery from incomplete and inaccurate measurements, Comm. Pure Appl. Math. 59(2006) 1207–1223.

[7] S.S. Chen, D.L. Donoho, M.A. Saunders, Atomic decomposition by basis pursuit, SIAM J. Sci. Comput., 20(1)(1998) 33-61.

[8] O. Christensen, An Introduction to Frames and Riesz bases, Applied and Numerical Harmonic Analysis, Birkhauser, Boston, 2003.

[9] D. L. Donoho, Compressed sensing, IEEE Trans. Inform. Theory, 52(4)(2006) 1289-1306.

[10] D. L. Donoho and M. Elad, Optimally Sparse Representation in General (nonorthogonal) Dictionaries via L1 Minimization, the Proc. Nat. Aca. Sci., 100(2003) 2197–2202.

[11] M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, R. Baraniuk, Single-pixel imaging via compressive sampling, IEEE Signal Processing Magazine, 25(2)(2008) 83–91.

[12] M. Elad, Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing, Springer, 2010.

[13] M. Lustig, D.L. Donoho, J.M. Pauly, Sparse MRI: The application of compressed sensing for rapid MR imaging, Magnetic Resonance in Medicine, 58(6)(2007) 1182–1195.

[14] M. Lustig, D.L. Donoho, J.M. Santos, J.M. Pauly, Compressed sensing MRI, IEEE Signal Processing Magazine, 25(2)(2008) 72–82.

[15] S.G. Mallat, Z. Zhang, Matching pursuits with time-frequency dictionaries, IEEE Trans. Signal Proc., 41(12)(1993) 3397-3415.

[16] L. Potter, P. Schniter, J. Ziniel, Sparse reconstruction for RADAR, SPIE Algorithms for Synthetic Aperture Radar Imagery XV, 2008.

[17] J.A. Tropp, Greed is good: Alogorithmic results for sparse approximation, IEEE Trans. Inform. Theory, 50(10)(2004) 2231–2242.

[18] L.R. Welch, Lower Bounds on the Maximum Cross Correlation of Signals, IEEE Trans. Inform. Theory, 20(1974) 397–399.