Circuits, Systems, and Signal Processing

, Volume 37, Issue 4, pp 1562–1574 | Cite as

A Performance Guarantee for Orthogonal Matching Pursuit Using Mutual Coherence

  • Mohammad EmadiEmail author
  • Ehsan MiandjiEmail author
  • Jonas Unger


In this paper, we present a new performance guarantee for the orthogonal matching pursuit (OMP) algorithm. We use mutual coherence as a metric for determining the suitability of an arbitrary overcomplete dictionary for exact recovery. Specifically, a lower bound for the probability of correctly identifying the support of a sparse signal with additive white Gaussian noise and an upper bound for the mean square error is derived. Compared to the previous work, the new bound takes into account the signal parameters such as dynamic range, noise variance, and sparsity. Numerical simulations show significant improvements over previous work and a much closer correlation to empirical results of OMP.


Compressed sensing Sparse representation Orthogonal matching pursuit Sparse recovery 


  1. 1.
    Z. Ben-Haim, Y. Eldar, M. Elad, Coherence-based performance guarantees for estimating a sparse vector under random noise. IEEE Trans. Signal Process. 58(10), 5030–5043 (2010). doi: 10.1109/TSP.2010.2052460 MathSciNetCrossRefGoogle Scholar
  2. 2.
    G. Bennett, Probability inequalities for the sum of independent random variables. J. Am. Stat. Assoc. 57(297), 33–45 (1962)CrossRefzbMATHGoogle Scholar
  3. 3.
    P. Bofill, M. Zibulevsky, Underdetermined blind source separation using sparse representations. Signal Process. 81(11), 2353–2362 (2001). doi: 10.1016/S0165-1684(01)00120-7 CrossRefzbMATHGoogle Scholar
  4. 4.
    E.J. Candès, J.K. Romberg, T. Tao, Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math. 59(8), 1207–1223 (2006). doi: 10.1002/cpa.20124 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    A. Castrodad, Z. Xing, J.B. Greer, E. Bosch, L. Carin, G. Sapiro, Learning discriminative sparse representations for modeling, source separation, and mapping of hyperspectral imagery. IEEE Trans. Geosci. Remote Sens. 49(11), 4263–4281 (2011). doi: 10.1109/TGRS.2011.2163822 CrossRefGoogle Scholar
  6. 6.
    S.S. Chen, D.L. Donoho, M.A. Saunders, Atomic decomposition by basis pursuit. SIAM J. Sci. Comput. 20, 33–61 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    D. Donoho, Compressed sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006). doi: 10.1109/TIT.2006.871582 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    D. Donoho, X. Huo, Uncertainty principles and ideal atomic decomposition. IEEE Trans. Inf. Theory 47(7), 2845–2862 (2001). doi: 10.1109/18.959265 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    D. Donoho, M. Elad, V. Temlyakov, Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Trans. Inf. Theory 52(1), 6–18 (2006). doi: 10.1109/TIT.2005.860430 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    D.L. Donoho, M. Elad, Optimally sparse representation in general (nonorthogonal) dictionaries via l1 minimization. Proc. Natl. Acad. Sci. 100(5), 2197–2202 (2003). doi: 10.1073/pnas.0437847100.
  11. 11.
    M.F. Duarte, M.A. Davenport, D. Takbar, J.N. Laska, T. Sun, K.F. Kelly, R.G. Baraniuk, Single-pixel imaging via compressive sampling. IEEE Signal Process. Mag. 25(2), 83–91 (2008). doi: 10.1109/MSP.2007.914730 CrossRefGoogle Scholar
  12. 12.
    B. Efron, T. Hastie, I. Johnstone, R. Tibshirani, Least angle regression. Ann. Stat. 32(2), 407–499 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    M. Elad, Sparse and Redundant Representations (Springer, New York, 2010). doi: 10.1007/978-1-4419-7011-4 CrossRefzbMATHGoogle Scholar
  14. 14.
    M. Elad, M. Aharon, Image denoising via sparse and redundant representations over learned dictionaries. IEEE Trans. Image Process. 15(12), 3736–3745 (2006). doi: 10.1109/TIP.2006.881969 MathSciNetCrossRefGoogle Scholar
  15. 15.
    Y.C. Eldar, G. Kutyniok (eds.), Compressed Sensing Theory and Applications (Cambridge University Press, Cambridge, 2012)Google Scholar
  16. 16.
    M. Emadi, K. Sadeghi, DOA estimation of multi-reflected known signals in compact arrays. EEE Trans. Aerosp. Electron. Syst. 49(3), 1920–1931 (2013). doi: 10.1109/TAES.2013.6558028 CrossRefGoogle Scholar
  17. 17.
    M.A.T. Figueiredo, R.D. Nowak, S.J. Wright, Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems. IEEE J. Sel. Top. Signal Process. 1(4), 586–597 (2007). doi: 10.1109/JSTSP.2007.910281 CrossRefGoogle Scholar
  18. 18.
    G. Golub, C. Van Loan, Matrix Computations. Johns Hopkins Studies in the Mathematical Sciences (Johns Hopkins University Press, Baltimore, Ann Arbor, MI, 1996)Google Scholar
  19. 19.
    S.H. Hsieh, C.S. Lu, S.C. Pei, Fast omp: reformulating omp via iteratively refining l2-norm solutions, in 2012 IEEE Statistical Signal Processing Workshop (SSP), 2012, pp. 189–192. doi: 10.1109/SSP.2012.6319656
  20. 20.
    Z.M. Liu, Z.T. Huang, Y.Y. Zhou, Direction-of-arrival estimation of wideband signals via covariance matrix sparse representation. IEEE Trans. Signal Process. 59(9), 4256–4270 (2011). doi: 10.1109/TSP.2011.2159214 MathSciNetCrossRefGoogle Scholar
  21. 21.
    D. Malioutov, M. Cetin, A. Willsky, A sparse signal reconstruction perspective for source localization with sensor arrays. IEEE Trans. Signal Process. 53(8), 3010–3022 (2005). doi: 10.1109/TSP.2005.850882 MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    S. Mallat, Z. Zhang, Matching pursuits with time-frequency dictionaries. IEEE Trans. Signal Process. 41(12), 3397–3415 (1993). doi: 10.1109/78.258082 CrossRefzbMATHGoogle Scholar
  23. 23.
    F. Marvasti, A. Amini, F. Haddadi, M. Soltanolkotabi, B.H. Khalaj, A. Aldroubi, S. Sanei, J. Chambers, A unified approach to sparse signal processing. EURASIP J. Adv. Signal Process. 2012, 44 (2012)CrossRefGoogle Scholar
  24. 24.
    E. Miandji, J. Kronander, J. Unger, Compressive image reconstruction in reduced union of subspaces. Comput. Graph. Forum 34(2), 33–44 (2015). doi: 10.1111/cgf.12539 CrossRefGoogle Scholar
  25. 25.
    D. Needell, J. Tropp, CoSaMP: iterative signal recovery from incomplete and inaccurate samples. Appl. Comput. Harmon. Anal. 26(3), 301–321 (2009). doi: 10.1016/j.acha.2008.07.002 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    D. Needell, R. Vershynin, Signal recovery from incomplete and inaccurate measurements via regularized orthogonal matching pursuit. IEEE J. Sel. Top. Signal Process. 4(2), 310–316 (2010). doi: 10.1109/JSTSP.2010.2042412 CrossRefGoogle Scholar
  27. 27.
    Y. Pati, R. Rezaiifar, P.S. Krishnaprasad, Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition, in Conference Record of the Twenty-Seventh Asilomar Conference on Signals, Systems and Computers, 1993, pp. 40–44. doi: 10.1109/ACSSC.1993.342465
  28. 28.
    G. Pope, Compressive sensing: a summary of reconstruction algorithms, Master’s thesis, ETH Zürich, 2009Google Scholar
  29. 29.
    R. Tibshirani, Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B 58, 267–288 (1994)MathSciNetzbMATHGoogle Scholar
  30. 30.
    J. Tropp, Greed is good: algorithmic results for sparse approximation. IEEE Trans. Inf. Theory 50(10), 2231–2242 (2004). doi: 10.1109/TIT.2004.834793 MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    J. Tropp, Just relax: convex programming methods for identifying sparse signals in noise. IEEE Trans. Inf. Theory 52(3), 1030–1051 (2006). doi: 10.1109/TIT.2005.864420 MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    E. van den Berg, M.P. Friedlander, Probing the pareto frontier for basis pursuit solutions. SIAM J. Sci. Comput. 31(2), 890–912 (2009). doi: 10.1137/080714488 MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    S. Wright, R. Nowak, M. Figueiredo, Sparse reconstruction by separable approximation. IEEE Trans. Signal Process. 57(7), 2479–2493 (2009). doi: 10.1109/TSP.2009.2016892 MathSciNetCrossRefGoogle Scholar
  34. 34.
    J. Yang, J. Wright, T.S. Huang, Y. Ma, Image super-resolution via sparse representation. IEEE Trans. Image Process. 19(11), 2861–2873 (2010). doi: 10.1109/TIP.2010.2050625 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Qualcomm Technologies Inc.San JoseUSA
  2. 2.Department of Science and TechnologyLinköping UniversityLinköpingSweden

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