ETRI Journal

Electronics and Telecommunications Research Institute (한국전자통신연구원)
권호 표지
DOI QR코드

DOI QR Code

Performance Analysis of Compressed Sensing Given Insufficient Random Measurements

  • Rateb, Ahmad M. (Telecommunications Research Group, Faculty of Electrical Engineering, Universiti Teknologi Malaysia) ;
  • Syed-Yusof, Sharifah Kamilah (Telecommunications Research Group, Faculty of Electrical Engineering, Universiti Teknologi Malaysia)
  • Received : 2012.05.22
  • Accepted : 2012.10.26
  • Published : 2013.04.01
Download PDF
⟨ Previous Next ⟩

Abstract

Most of the literature on compressed sensing has not paid enough attention to scenarios in which the number of acquired measurements is insufficient to satisfy minimal exact reconstruction requirements. In practice, encountering such scenarios is highly likely, either intentionally or unintentionally, that is, due to high sensing cost or to the lack of knowledge of signal properties. We analyze signal reconstruction performance in this setting. The main result is an expression of the reconstruction error as a function of the number of acquired measurements.

Keywords

References

  1. D.L. Donoho, "Compressed Sensing," IEEE Trans. Inf. Theory, vol. 52, no. 4, 2006, pp. 1289-1306. https://doi.org/10.1109/TIT.2006.871582
  2. M. Lustig, D. Donoho, and J.M. Pauly, "Sparse MRI: The Application of Compressed Sensing for Rapid MR Imaging," Magn. Resonance Medicine, vol. 58, no. 6, 2007, pp. 1182-1195. https://doi.org/10.1002/mrm.21391
  3. J. Bobin, J.L. Starck, and R. Ottensamer, "Compressed Sensing in Astronomy," IEEE J. Sel. Topics Signal Process., vol. 2, no. 5, 2008, pp. 718-726. https://doi.org/10.1109/JSTSP.2008.2005337
  4. M.A. Herman and T. Strohmer, "High-Resolution Radar via Compressed Sensing," IEEE Trans. Signal Process., vol. 57, no. 6, 2009, pp. 2275-2284. https://doi.org/10.1109/TSP.2009.2014277
  5. C.R. Berger et al., "Sparse Channel Estimation for Multicarrier Underwater Acoustic Communication: From Subspace Methods to Compressed Sensing," IEEE Trans. Signal Process., vol. 58, no. 3, 2010, pp. 1708-1721. https://doi.org/10.1109/TSP.2009.2038424
  6. J. Ma, "Single-Pixel Remote Sensing," IEEE Geosci. Remote Sensing Lett., vol. 6, no. 2, 2009, pp. 199-203. https://doi.org/10.1109/LGRS.2008.2010959
  7. E. Candès, J. Romberg, and T. Tao, "Stable Signal Recovery from Incomplete and Inaccurate Measurements," Commun. Pure Appl. Mathematics, vol. 59, no. 8, 2006, pp. 1207-1223. https://doi.org/10.1002/cpa.20124
  8. S.S. Chen, D.L. Donoho, and M.A. Saunders, "Atomic Decomposition by Basis Pursuit," SIAM J. Sci. Computing, vol. 20, no. 1, 1999, pp. 33-61.
  9. E. Candès and T. Tao, "Decoding by Linear Programming," IEEE Trans. Inf. Theory, vol. 51, no. 12, 2005, pp. 4203-4215. https://doi.org/10.1109/TIT.2005.858979
  10. E. Candès, "The Restricted Isometry Property and Its Implications for Compressed Sensing," Comptes Rendus Mathematique, vol. 346, no. 9-10, 2008, pp. 589-592. https://doi.org/10.1016/j.crma.2008.03.014
  11. J. Tropp and A.C. Gilbert, "Signal Recovery from Random Measurements via Orthogonal Matching Pursuit," IEEE Trans. Inf. Theory, vol. 53, no. 12, 2007, pp. 4655-4666. https://doi.org/10.1109/TIT.2007.909108
  12. S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge: Cambridge University Press, 2004.
  13. E. Candès and T. Tao, "Near Optimal Signal Recovery from Random Projections: Universal Encoding Strategies?" IEEE Trans. Inf. Theory, vol. 52, no. 12, 2006, pp. 5406-5425. https://doi.org/10.1109/TIT.2006.885507
  14. R. Baraniuk et al., "A Simple Proof of the Restricted Isometry Property for Random Matrices," Constructive Approximation, vol. 28, no. 3, 2008, pp. 253-263. https://doi.org/10.1007/s00365-007-9003-x
  15. J. Tropp et al., "Beyond Nyquist: Effcient Sampling of Sparse, Bandlimited Signals," IEEE Trans. Inf. Theory, vol. 56, no. 1, 2010, pp. 520-544. https://doi.org/10.1109/TIT.2009.2034811
  16. R. Hecht-Nielsen, "Context Vectors: General Purpose Approximate Meaning Representations Self-Organized from Raw Data," Computational Intelligence: Imitating Life, J. Zurada, R. Marks, and C. Robinson, Eds., IEEE Press, 1994, pp. 43-56.
  17. D. Achlioptas, "Database-Friendly Random Projections," Proc. Symp. Principles Database Syst., Santa Barbara, CA, USA, May 2001, pp. 274-281.
  18. J.G. Proakis and D.K. Manolakis, Digital Signal Processing, 4th ed., Upper Saddle River, NJ: Prentice Hall, 2006.
  19. S.M. Ross, Introduction to Probability Theory, New York: Academic Press, 1981.
  20. E. Candes and J. Romberg, $\ell$1-Magic: reconstruction of sparse signals. Available: http://users.ece.gatech.edu/-justin/l1magic/

Cited by

  1. 2D DOA Estimation Using Sparse Representation of Higher-Order Power of Covariance Matrix vol.998, pp.None, 2013, https://doi.org/10.4028/www.scientific.net/amr.998-999.779
  2. Texture classification with cross-covariance matrices in compressive measurement domain vol.10, pp.8, 2013, https://doi.org/10.1007/s11760-016-0902-9
  3. Compressive sensing-based two-dimensional scattering-center extraction for incomplete RCS data vol.42, pp.6, 2013, https://doi.org/10.4218/etrij.2019-0017