Orthogonal Least Absolute Value for Sparse Spike Deconvolution

Abstract

Several phenomena encountered in nature are characterized by very localized events occurring randomly at given times. Random pulses are an appropriate modelling tool for such events. Usually, the impulses are hidden in the noise due to unwanted convolution. In some cases, the problem is more complex because of the short time lag between the pulses. Considering these problems, the resulting signal is unclear and can lead to an erroneous analysis. Hence the need for deconvolution to restore the pulsed signal in order to obtain a more accurate diagnosis. The main objective of this study is to propose a new algorithm called orthogonal least absolute value. The particularity of this algorithm lies in its selection criterion. The algorithm iteratively selects the atom minimizing the absolute value of the approximation error. This allows the proposed algorithm to outperform classical greedy algorithms when the peaks are very close to each other. Numerical and experimental simulations are performed to study the proposed algorithm and compare its behavior to other greedy algorithms in deconvolution framework. Simulations results prove the performance of the proposed algorithm, especially when the impulses are very close to each other.

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Data Availability Statement

The generated data and the codes of the proposed work are available from the corresponding author on reasonable request.

References

  1. 1.

    P. Bentley, G. Nordehn, M. Coimbra, S. Mannor. The PASCAL classifying heart sounds challenge 2011 (CHSC2011) results. http://www.peterjbentley.com/heartchallenge/index.html

  2. 2.

    J. Bobin, J.L. Starck, Y. Moudden, M.J. Fadili. Blind source separation: the sparsity revolution. In Advances in Imaging and Electron Physics (Elsevier, 2008), pp. 221–302

  3. 3.

    S. Chen, S.A. Billings, W. Luo, Orthogonal least squares methods and their application to non-linear system identification. Int. J. Control 50(5), 1873 (1989)

    Article  Google Scholar 

  4. 4.

    S.S. Chen, D.L. Donoho, M.A. Saunders, Atomic decomposition by basis pursuit. SIAM Rev. 43(1), 129 (2001)

    MathSciNet  Article  Google Scholar 

  5. 5.

    A. Choklati, K. Sabri, Cyclic analysis of extra heart sounds: Gauss kernel based model. Int. J. Image Graph. Signal Process. 10(5), 1 (2018)

    Article  Google Scholar 

  6. 6.

    C. Dossal, S. Mallat. Sparse spike deconvolution with minimum scale. In Signal Processing with Adaptive Sparse Structured Representations (SPARS workshop) (Citeseer, 2005), pp. 1–4

  7. 7.

    M. Ehler, Shrinkage rules for variational minimization problems and applications to analytical ultracentrifugation. J. Inverse Ill-Posed Probl. 19(4–5), 593 (2011)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    R. Fernandes, H. Lopes, M. Gattass, Lobbes: an algorithm for sparse-spike deconvolution. IEEE Geosci. Remote Sens. Lett. 14(12), 2240 (2017)

    Article  Google Scholar 

  9. 9.

    R. Gribonval, M. Zibulevsky, in Handbook of Blind Source Separation (Elsevier, 2010), pp. 367–420

  10. 10.

    A. Had, K. Sabri, A two-stage blind deconvolution strategy for bearing fault vibration signals. Mech. Syst. Signal Process. 134(1), 106307 (2019)

    Article  Google Scholar 

  11. 11.

    A. Had, K. Sabri, M. Aoutoul, Detection of heart valves closure instants in phonocardiogram signals. Wirel. Pers. Commun. 112(3), 1569 (2020)

    Article  Google Scholar 

  12. 12.

    J. Idier, Bayesian Approach to Inverse Problems (Wiley, London, 2013)

    Google Scholar 

  13. 13.

    N.B. Karahanoglu, H. Erdogan, Compressed sensing signal recovery via forward–backward pursuit. Digit. Signal Proc. 23(5), 1539 (2013)

    MathSciNet  Article  Google Scholar 

  14. 14.

    A. Kazemipour, J. Liu, K. Solarana, D.A. Nagode, P.O. Kanold, M. Wu, B. Babadi, Fast and stable signal deconvolution via compressible state-space models. IEEE Trans. Biomed. Eng. 65(1), 74 (2018)

    Article  Google Scholar 

  15. 15.

    S. Kwon, J. Wang, B. Shim, Multipath matching pursuit. IEEE Trans. Inf. Theory 60(5), 2986 (2014)

    MathSciNet  Article  Google Scholar 

  16. 16.

    C. Liu, D. Wang, T. Wang, F. Feng, Y. Wang, Multichannel sparse deconvolution of seismic data with Shearlet–Cauchy constrained inversion. J. Geophys. Eng. 14(5), 1275 (2017)

    Article  Google Scholar 

  17. 17.

    S. Mallat, Z. Zhang, Matching pursuits with time-frequency dictionaries. IEEE Trans. Signal Process. 41(12), 3397 (1993)

    Article  Google Scholar 

  18. 18.

    A.R. Maud, M.R. Bell. A mismatched greedy pursuit algorithm for sparse spike deconvolution. in 49th Asilomar Conference on Signals, Systems and Computers (IEEE, 2015), pp. 1477–1481

  19. 19.

    D. Needell, J. Tropp, CoSaMP: Iterative signal recovery from incomplete and inaccurate samples. Appl. Comput. Harmonic Anal. 26(3), 301 (2009)

    MathSciNet  Article  Google Scholar 

  20. 20.

    S.L. Pan, K. Yan, H.Q. Lan, Z.Y. Qin, A Bregman adaptive sparse-spike deconvolution method in the frequency domain. Appl. Geophys. 16(4), 463 (2019)

    Article  Google Scholar 

  21. 21.

    Y. Pati, R. Rezaiifar, P. Krishnaprasad. Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition. in Proceedings of 27th Asilomar Conference on Signals, Systems and Computers (IEEE Comput. Soc. Press, 1993), pp. 40–44

  22. 22.

    B. Rao, K. Kreutz-Delgado, An affine scaling methodology for best basis selection. IEEE Trans. Signal Process. 47(1), 187 (1999)

    MathSciNet  Article  Google Scholar 

  23. 23.

    K. Sabri, New cyclic sparsity measures for deconvolution based on convex relaxation. Circuits Syst. Signal Process. 34(9), 2911 (2015)

    MathSciNet  Article  Google Scholar 

  24. 24.

    R. Tibshirani. Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B (Methodol.) 58(1), 267 (1996)

  25. 25.

    J. Tropp, Greed is good: algorithmic results for sparse approximation. IEEE Trans. Inf. Theory 50(10), 2231 (2004)

    MathSciNet  Article  Google Scholar 

  26. 26.

    F. Wang, J. Zhang, G. Sun, T. Geng, Iterative forward-backward pursuit algorithm for compressed sensing. J. Electr. Comput. Eng. 2016(1), 1 (2016)

    Google Scholar 

  27. 27.

    L. Wang, Q. Zhao, J. Gao, Z. Xu, M. Fehler, X. Jiang, Seismic sparse-spike deconvolution via toeplitz-sparse matrix factorization. Geophysics 81(3), 169 (2016)

    Article  Google Scholar 

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Correspondence to A. Had.

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Had, A., Sabri, K. Orthogonal Least Absolute Value for Sparse Spike Deconvolution. Circuits Syst Signal Process (2021). https://doi.org/10.1007/s00034-021-01667-z

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Keywords

  • Sparsity
  • Random impulses
  • Deconvolution
  • Greedy algorithm
  • Phonocardiogram signal