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Circuits, Systems, and Signal Processing

, Volume 38, Issue 3, pp 1206–1263 | Cite as

A Tutorial on Sparse Signal Reconstruction and Its Applications in Signal Processing

  • Ljubiša StankovićEmail author
  • Ervin Sejdić
  • Srdjan Stanković
  • Miloš Daković
  • Irena Orović
Article

Abstract

Sparse signals are characterized by a few nonzero coefficients in one of their transformation domains. This was the main premise in designing signal compression algorithms. Compressive sensing as a new approach employs the sparsity property as a precondition for signal recovery. Sparse signals can be fully reconstructed from a reduced set of available measurements. The description and basic definitions of sparse signals, along with the conditions for their reconstruction, are discussed in the first part of this paper. The numerous algorithms developed for the sparse signals reconstruction are divided into three classes. The first one is based on the principle of matching components. Analysis of noise and nonsparsity influence on reconstruction performance is provided. The second class of reconstruction algorithms is based on the constrained convex form of problem formulation where linear programming and regression methods can be used to find a solution. The third class of recovery algorithms is based on the Bayesian approach. Applications of the considered approaches are demonstrated through various illustrative and signal processing examples, using common transformation and observation matrices. With pseudocodes of the presented algorithms and compressive sensing principles illustrated on simple signal processing examples, this tutorial provides an inductive way through this complex field to researchers and practitioners starting from the basics of sparse signal processing up to the most recent and up-to-date methods and signal processing applications.

Keywords

Sparse signals Compressive sensing Signal sampling Signal representation Signal reconstruction Discrete Fourier transform 

References

  1. 1.
    M. Aharon, M. Elad, A. Bruckstein, K-SVD: an algorithm for designing overcomplete dictionaries for sparse representation. IEEE Trans. Signal Process. 54(11), 4311–4322 (2006)zbMATHGoogle Scholar
  2. 2.
    N. Ahmed, T. Natarajan, K.R. Rao, Discrete cosine transform. IEEE Trans. Comput. C–23(1), 90–93 (1974)MathSciNetzbMATHGoogle Scholar
  3. 3.
    M.G. Amin, Compressive Sensing for Urban Radar (CRC Press, Boca Raton, 2014)Google Scholar
  4. 4.
    D. Angelosante, G.B. Giannakis, E. Grossi, Compressed sensing of time-varying signals, in Proceedings of the 16th international conference on digital signal processing (DSP ’09) (Santorini-Hellas, Greece, 2009), pp. 1–8Google Scholar
  5. 5.
    E. Arias-Castro, Y. Eldar, Noise folding in compressed sensing. IEEE Signal Process. Lett. 18(8), 478–481 (2011)Google Scholar
  6. 6.
    S.D. Babacan, R. Molina, A.K. Katsaggelos, Bayesian Compressive Sensing Using Laplace Priors. IEEE Transactions on Image Processing 19(1), 53–63 (2010)MathSciNetzbMATHGoogle Scholar
  7. 7.
    A.S. Bandeira, E. Dobriban, D.G. Mixon, W.F. Sawin, Certifying the restricted isometry property is hard. IEEE Trans. Inf. Theory 59(6), 3448–3450 (2013)MathSciNetzbMATHGoogle Scholar
  8. 8.
    R. Baraniuk, Compressive sensing. IEEE Signal Process. Mag. 24(4), 118–121 (2007)Google Scholar
  9. 9.
    R.G. Baraniuk, T. Goldstein, A.C. Sankaranarayanan, C. Studer, A. Veeraraghavan, M.B. Wakin, Compressive video sensing: algorithms, architectures, and applications. IEEE Signal Process. Mag. 34(1), 52–66 (2017)Google Scholar
  10. 10.
    D. Baron, S. Sarvotham, R.G. Baraniuk, Bayesian compressive sensing via belief propagation. IEEE Trans. Signal Process. 58(1), 269–280 (2010)MathSciNetzbMATHGoogle Scholar
  11. 11.
    J. Bazerque, G. Giannakis, Distributed spectrum sensing for cognitive radio networks by exploiting sparsity. IEEE Trans. Signal Process. 58(3), 1847–1862 (2010)MathSciNetzbMATHGoogle Scholar
  12. 12.
    A. Beck, M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–197 (2009)MathSciNetzbMATHGoogle Scholar
  13. 13.
    C.R. Berger, Z. Wang, J. Huang, S. Zhou, Application of compressive sensing to sparse channel estimation. IEEE Commun. Mag. 48(11), 164–174 (2010)Google Scholar
  14. 14.
    J.M. Bioucas-Dias, M.A.T. Figueiredo, A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration. IEEE Trans. Image Process. 16(12), 2992–3004 (2007)MathSciNetGoogle Scholar
  15. 15.
    J.D. Blanchard, Cartis, J. Tanner, Compressed sensing: how sharp is the restricted isometry property? SIAM Rev. 53(1), 105–125 (2011)MathSciNetzbMATHGoogle Scholar
  16. 16.
    T. Blumensath, M.E. Davies, Gradient pursuits. IEEE Trans. Signal Process. 56(6), 2370–2382 (2008)MathSciNetzbMATHGoogle Scholar
  17. 17.
    T. Blumensath, M.E. Davies, Iterative thresholding for sparse approximations. J. Fourier Anal. Appl. 14(5–6), 629–654 (2008)MathSciNetzbMATHGoogle Scholar
  18. 18.
    J. Bobin, J.L. Starck, R. Ottensamer, Compressed sensing in astronomy. IEEE J. Sel. Top. Signal Process. 2(5), 718–726 (2008)Google Scholar
  19. 19.
    S. Boyd, L. Vandenberghe, Convex Optimization (Cambridge University Press, Cambridge, 2004)zbMATHGoogle Scholar
  20. 20.
    M. Brajovic, I. Orović, M. Daković, S. Stanković, On the parameterization of Hermite transform with application to the compression of QRS complexes. Signal Process. 131, 113–119 (2017)Google Scholar
  21. 21.
    M. Brajovic, I. Orović, M. Daković, S. Stanković, Gradient-based signal reconstruction algorithm in the Hermite transform domain. Electron. Lett. 52(1), 41–43 (2016)Google Scholar
  22. 22.
    M. Brajovic, I. Stanković, M. Daković, C. Ioana, L. Stanković, Error in the reconstruction of nonsparse images. Math. Probl. Eng. 2018, 10. Article ID 4314527 (2018).  https://doi.org/10.1155/2018/4314527
  23. 23.
    L. Breiman, Better subset regression using the nonnegative garrote. Technometrics 37(4), 373–384 (1995)MathSciNetzbMATHGoogle Scholar
  24. 24.
    E.J. Candès, The restricted isometry property and its implications for compressed sensing. C. R. Math. 346(9–10), 589–592 (2008)MathSciNetzbMATHGoogle Scholar
  25. 25.
    E.J. Candès, J. Romberg, \(\ell_1\)-magic: recovery of sparse signals via convex programming. Caltech, http://users.ece.gatech.edu/justin/l1magic/downloads/l1magic.pdf. Oct 2005
  26. 26.
    E.J. Candès, J. Romberg, T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52(2), 489–509 (2006)MathSciNetzbMATHGoogle Scholar
  27. 27.
    E.J. Candès, M. Wakin, An introduction to compressive sampling. IEEE Signal Process. Mag. 25(2), 21–30 (2008)Google Scholar
  28. 28.
    R. Chartrand, V. Staneva, Restricted isometry properties and nonconvex compressive sensing. Inverse Probl. 24(3), 035020-1-14 (2008)MathSciNetzbMATHGoogle Scholar
  29. 29.
    S.S. Chen, D.L. Donoho, M.A. Saunders, Atomic decomposition by basis pursuit. SIAM Rev. 43(1), 129–159 (2001)MathSciNetzbMATHGoogle Scholar
  30. 30.
    D. Craven, B. McGinley, L. Kilmartin, M. Glavin, E. Jones, Compressed sensing for bioelectric signals: a review. IEEE J. Biomed. Health Inf. 19(2), 529–540 (2015)Google Scholar
  31. 31.
    S. Costanzo, A. Rocha, M.D. Migliore, Compressed sensing: applications in radar and communications. Sci. World J. 2016, 2. Article ID 5407415 (2016)Google Scholar
  32. 32.
    I. Daubechies, M. Defrise, C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math. 57(11), 1413–1457 (2004)MathSciNetzbMATHGoogle Scholar
  33. 33.
    G. Davis, S. Mallat, M. Avellaneda, Adaptive greedy approximations. Constr. Approx. 13(1), 57–98 (1997)MathSciNetzbMATHGoogle Scholar
  34. 34.
    D.L. Donoho, Compressed sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006)MathSciNetzbMATHGoogle Scholar
  35. 35.
    D.L. 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)MathSciNetzbMATHGoogle Scholar
  36. 36.
    M. Elad, Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing (Springer, Berlin, 2010)zbMATHGoogle Scholar
  37. 37.
    Y.C. Eldar, G. Kutyniok, Compressed Sensing: Theory and Applications (Cambridge University Press, Cambridge, 2012)Google Scholar
  38. 38.
    J. Ender, On compressive sensing applied to radar. Signal Process. 90(5), 1402–1414 (2010)zbMATHGoogle Scholar
  39. 39.
    N. Eslahi, A. Aghagolzadeh, Compressive sensing image restoration using adaptive curvelet thresholding and nonlocal sparse regularization. IEEE Trans. Image Process. 25(7), 3126–3140 (2016)MathSciNetGoogle Scholar
  40. 40.
    M.A. 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)Google Scholar
  41. 41.
    P. Flandrin, P. Borgnat, Time-frequency energy distributions meet compressed sensing. IEEE Trans. Signal Process. 58(6), 2974–2982 (2010)MathSciNetzbMATHGoogle Scholar
  42. 42.
    M. Fornsaier, H. Rauhut, Iterative thresholding algorithms. Appl. Comput. Harmon. Anal. 25(2), 187–208 (2008)MathSciNetzbMATHGoogle Scholar
  43. 43.
    M.A. Hadi, S. Alshebeili, K. Jamil, F.E. Abd El-Samie, Compressive sensing applied to radar systems: an overview. Signal Image Video Process. 9, 25–39 (2015)Google Scholar
  44. 44.
    G. Hua, Y. Hiang, G. Bi, When compressive sensing meets data hiding. IEEE Signal Process. Lett. 23(4), 473–477 (2016)Google Scholar
  45. 45.
    S. Ji, Y. Xue, L. Carin, Bayesian compressive sensing. IEEE Trans. Signal Process. 56(6), 2346–2356 (2008)MathSciNetzbMATHGoogle Scholar
  46. 46.
    P. Lander, E.J. Berbari, Principles and signal processing techniques of the high-resolution electrocardiogram. Prog. Cardiovasc. Dis. 35(3), 169–188 (1992)Google Scholar
  47. 47.
    C. Li, G. Zhao, W. Zhang, Q. Qiu, H. Sun, ISAR imaging by two-dimensional convex optimization-based compressive sensing. IEEE Sens. J. 16(19), 7088–7093 (2016)Google Scholar
  48. 48.
    X. Li, G. Bi, Time-frequency representation reconstruction based on the compressive sensing, in 9th IEEE Conference on Industrial Electronics and Applications (Hangzhou, 2014), pp. 1158–1162Google Scholar
  49. 49.
    X. Liao, K. Li, J. Yin, Separable data hiding in encrypted image based on compressive sensing and discrete Fourier transform. Multimed. Tools Appl. 76, 1–15 (2016)Google Scholar
  50. 50.
    S. Liu, Y.D. Zhang, T. Shan, Detection of weak astronomical signals with frequency-hopping interference suppression. Digit. Signal Process. 72, 1–8 (2018)MathSciNetGoogle Scholar
  51. 51.
    S. Liu, Y.D. Zhang, T. Shan, S. Qin, M.G. Amin, Structure-aware Bayesian compressive sensing for frequency-hopping spectrum estimation, in Proceedings of SPIE 9857, Compressive Sensing V: From Diverse Modalities to Big Data Analytics (2016), p. 98570NGoogle Scholar
  52. 52.
    S. Liu, Y.D. Zhang, T. Shan, R. Tao, Structure-aware Bayesian compressive sensing for frequency-hopping spectrum estimation with missing observations. IEEE Trans. Signal Process. 66(8), 2153–2166 (2018)MathSciNetGoogle Scholar
  53. 53.
    S. Liu, J.B. Jia, Y.J. Yang, Image reconstruction algorithm for electrical impedance tomography based on block sparse Bayesian learning, in Proceedings of IEEE International Conference on Imaging Systems and Techniques (IST) (Beijing, China, Oct. 18–20, 2017)Google Scholar
  54. 54.
    Y. Liu, M. De Vos, S. Van Huffel, Compressed sensing of multichannel EEG signals: the simultaneous cosparsity and low-rank optimization. IEEE Trans. Biomed. Eng. 62(8), 2055–2061 (2015)Google Scholar
  55. 55.
    W. Lu, N. Vaswani, Regularized modified BPDN for noisy sparse reconstruction with partial erroneous support and signal value knowledge. IEEE Trans. Signal Process. 60(1), 182–196 (2012)MathSciNetzbMATHGoogle Scholar
  56. 56.
    S. Luo, P. Johnston, A review of electrocardiogram filtering. J. Electrocardiol. 43(6), 486–496 (2010)Google Scholar
  57. 57.
    X. Lv, G. Bi, C. Wan, The group lasso for stable recovery of block-sparse signal representations. IEEE Trans. Signal Process. 59(4), 1371–1382 (2011)MathSciNetzbMATHGoogle Scholar
  58. 58.
    S. Mallat, Z. Zhang, Matching pursuits with time-frequency dictionaries. IEEE Trans. Signal Process. 41(12), 3397–3415 (1993)zbMATHGoogle Scholar
  59. 59.
    J.B. Martens, The Hermite transform—theory. IEEE Trans. Acoust. Speech Signal Process. 38(9), 1595–1606 (1990)zbMATHGoogle Scholar
  60. 60.
    S.A. Martucci, Symmetric convolution and the discrete sine and cosine transforms. IEEE Trans. Signal Process. 42(5), 1038–1051 (1994)Google Scholar
  61. 61.
    J. Music, T. Marasovic, V. Papic, I. Orović, S. Stanković, Performance of compressive sensing image reconstruction for search and rescue. IEEE Geosci. Remote Sens. Lett. 13(11), 1739–1743 (2016)Google Scholar
  62. 62.
    J. Music, I. Orović, T. Marasovic, V. Papic, S. Stanković, Gradient compressive sensing for image data reduction in UAV based search and rescue in the wild. Math. Probl. Eng. 2016, 6827414 (2016)Google Scholar
  63. 63.
    D. Needell, J.A. Tropp, CoSaMP: iterative signal recovery from noisy samples. Appl. Comput. Harmon. Anal. (2008).  https://doi.org/10.1016/j.acha.2008.07.002 zbMATHGoogle Scholar
  64. 64.
    D. Needell, J.A. Tropp, CoSaMP: iterative signal recovery from incomplete and inaccurate samples. ACM Technical Report, 2008-01 (California Institute of Technology, Pasadena, 2008)Google Scholar
  65. 65.
    D. Needell, J.A. Tropp, CoSaMP: iterative signal recovery from incomplete and inaccurate samples. Commun. ACM 53(12), 93–100 (2010)zbMATHGoogle Scholar
  66. 66.
    B. Ophir, M. Lustig, M. Elad, Multi-scale dictionary learning using wavelets. IEEE J. Sel. Top. Signal Process. 5(5), 1014–1024 (2011)Google Scholar
  67. 67.
    I. Orović, V. Papic, C. Ioana, X. Li, S. Stanković, Compressive sensing in signal processing: algorithms and transform domain formulations. Math. Probl. Eng. 2016, 1 (2016)MathSciNetzbMATHGoogle Scholar
  68. 68.
    I. Orović, S. Stanković, Improved higher order robust distributions based on compressive sensing reconstruction. IET Signal Process. 8(7), 738–748 (2014)Google Scholar
  69. 69.
    I. Orović, S. Stanković, T. Chau, C.M. Steele, E. Sejdic, Time-frequency analysis and Hermite projection method applied to swallowing accelerometry signals. EURASIP J. Adv. Signal Process. 2010, p 7. Article ID 323125 (2010)Google Scholar
  70. 70.
    I. Orović, S. Stanković, T. Thayaparan, Time-frequency based instantaneous frequency estimation of sparse signals from an incomplete set of samples. IET Signal Process. Spec. Issue Compressive Sens. Robust Transforms 8(3), 239–245 (2014)Google Scholar
  71. 71.
    C. Ozdemir, Inverse Synthetic Aperture Radar Imaging with MATLAB Algorithms (Wiley, Hoboken, 2012)Google Scholar
  72. 72.
    M. Panic, J. Aelterman, V.S. Crnojevic, A. Pizurica, Compressed sensing in MRI with a Markov random field prior for spatial clustering of subband coefficients, in Proceedings of the EUSIPCO (2016), pp. 562–566Google Scholar
  73. 73.
    V.M. Patel, R. Chellappa, Sparse Representations and Compressive Sensing for Imaging and Vision (Springer, Berlin, 2013)zbMATHGoogle Scholar
  74. 74.
    G. Pope, Compressive sensing: a summary of reconstruction algorithms. Eidgenossische Technische Hochschule, Zurich, Switzerland (2008), http://e-collection.library.ethz.ch/eserv/eth:41464/eth-41464-01.pdf. Aug 2008
  75. 75.
    L.C. Potter, E. Ertin, J.T. Parker, M. Cetin, Sparsity and compressed sensing in radar imaging. Proc. IEEE 98(6), 1006–1020 (2010)Google Scholar
  76. 76.
    S. Qaisar, R.M. Bilal, W. Iqbal, M. Naureen, S. Lee, Compressive sensing: from theory to applications, a survey. J. Commun. Netw. 15(5), 443–456 (2013)Google Scholar
  77. 77.
    R. Rubinstein, A.M. Bruckstein, M. Elad, Dictionaries for sparse representation modeling. Proc. IEEE 98(6), 1045–1057 (2010)Google Scholar
  78. 78.
    R. Sameni, G.D. Clifford, A review of fetal ECG signal processing issues and promising directions. Open Pacing Electrophysiol. Therapy J. 3, 4–20 (2010)Google Scholar
  79. 79.
    A. Sandryhaila, S. Saba, M. Puschel, J. Kovacevic, Efficient compression of QRS complexes using Hermite expansion. IEEE Trans. Signal Process. 60(2), 947–955 (2012)MathSciNetzbMATHGoogle Scholar
  80. 80.
    A. Sandryhaila, J. Kovacevic, M. Puschel, Compression of QRS complexes using Hermite expansion, in IEEE International Conference on Acoustic, Speech and Signal Process, ICASSP (Prague, 2011), pp. 581–584Google Scholar
  81. 81.
    E. Sejdic, Time-frequency compressive sensing, in Frequency Signal Analysis and Processing, ed. B. Boashash (Academic Press, 2015), pp. 424–429Google Scholar
  82. 82.
    I. Stanković, C. Ioana, M. Daković, On the reconstruction of nonsparse time-frequency signals with sparsity constraint from a reduced set of samples. Signal Process. 142, 480–484 (2018)Google Scholar
  83. 83.
    I. Stanković, I. Orović, M. Daković, S. Stanković, Denoising of sparse images in impulsive disturbance environment. Multimed. Tools Appl. (2017).  https://doi.org/10.1007/s11042-017-4502-7 Google Scholar
  84. 84.
    L. Stanković, Digital Signal Processing with Applications: Adaptive Systems, Time-Frequency Analaysis, Sparse Signal Processing (CreateSpace Independent Publishing Platform, North Charlestone, 2015)Google Scholar
  85. 85.
    L. Stanković, A measure of some time-frequency distributions concentration. Signal Process. 81, 621–631 (2001)zbMATHGoogle Scholar
  86. 86.
    L. Stanković, On the ISAR image analysis and recovery with unavailable or heavily corrupted data. IEEE Trans. Aerosp. Electron. Syst. 51(3), 2093–2106 (2015)Google Scholar
  87. 87.
    L. Stanković, M. Brajovic, Analysis of the reconstruction of sparse signals in the DCT domain applied to audio signals. IEEE/ACM Trans. Audio Speech Lang. Process. 26(7), 1216–1231 (2018)Google Scholar
  88. 88.
    L. Stanković, M. Daković, On the uniqueness of the sparse signals reconstruction based on the missing samples variation analysis. Math. Probl. Eng. 2015, p 14 (2015). Article ID 629759.  https://doi.org/10.1155/2015/629759
  89. 89.
    L. Stanković, M. Daković, I. Stanković, S. Vujovic, On the errors in randomly sampled nonsparse signals reconstructed with a sparsity assumption. IEEE Geosci. Remote Sens. Lett. 14(12), 2453–2456 (2017)Google Scholar
  90. 90.
    L. Stanković, M. Daković, S. Stanković, I. Orović, Sparse Signal Processing—Introduction. Wiley Encyclopedia of Electrical and Electronics Engineering (Wiley, Hoboken, 2017)Google Scholar
  91. 91.
    L. Stanković, M. Daković, T. Thayaparan, Time-Frequency Signal Analysis with Applications (Artech House, Boston, 2013)zbMATHGoogle Scholar
  92. 92.
    L. Stanković, M. Daković, S. Vujovic, Adaptive variable step algorithm for missing samples recovery in sparse signals. IET Signal Process. 8(3), 246–256 (2014)Google Scholar
  93. 93.
    L. Stanković, M. Daković, S. Vujovic, Reconstruction of sparse signals in impulsive disturbance environments. Circuits Syst. Signal Process. 36, 1–28 (2016)zbMATHGoogle Scholar
  94. 94.
    L. Stanković, I. Orović, S. Stanković, M. Amin, Compressive sensing based separation of non-stationary and stationary signals overlapping in time-frequency. IEEE Trans. Signal Process. 61(18), 4562–4572 (2013)MathSciNetzbMATHGoogle Scholar
  95. 95.
    L. Stanković, I. Stanković, M. Daković, Nonsparsity influence on the ISAR recovery from reduced data. IEEE Trans. Aerosp. Electron. Syst. 52(6), 3065–3070 (2016)Google Scholar
  96. 96.
    L. Stanković, S. Stanković, M.G. Amin, Missing samples analysis in signals for applications to l-estimation and compressive sensing. Signal Process. 94, 401–408 (2014)Google Scholar
  97. 97.
    L. Stanković, S. Stanković, T. Thayaparan, M. Daković, I. Orović, Separation and reconstruction of the rigid body and micro-Doppler signal in ISAR part II—statistical analysis. IET Radar Sonar Navig. 9(9), 1155–1161 (2015)Google Scholar
  98. 98.
    L. Stanković, S. Stanković, T. Thayaparan, M. Daković, I. Orović, Separation and reconstruction of the rigid body and micro-Doppler signal in ISAR part I—theory. IET Radar Sonar Navig. 9(9), 1147–1154 (2015)Google Scholar
  99. 99.
    S. Stanković, I. Orović, An approach to 2D signals recovering in compressive sensing context. Circuits Syst. Signal Process. 36(4), 1700–1713 (2017)Google Scholar
  100. 100.
    S. Stanković, I. Orović, M. Amin, L-statistics based modification of reconstruction algorithms for compressive sensing in the presence of impulse noise. Signal Process. 93(11), 2927–2931 (2013)Google Scholar
  101. 101.
    S. Stanković, I. Orović, A. Krylov, Video frames reconstruction based on time-frequency analysis and Hermite projection method. EURASIP J. Adv. Signal Process. Spec. Issue Time Freq. Anal. Appl. Multimed. Signals, 11. Article ID 970105 (2010)Google Scholar
  102. 102.
    S. Stanković, I. Orović, E. Sejdic, Multimedia Signals and Systems: Basic and Advanced Algorithms for Signal Processing, 2nd edn. (Springer, Berlin, 2015)Google Scholar
  103. 103.
    S. Stanković, I. Orović, L. Stanković, An automated signal reconstruction method based on analysis of compressive sensed signals in noisy environment. Signal Process. 104, 43–50 (2014)Google Scholar
  104. 104.
    S. Stanković, I. Orović, L. Stanković, Polynomial Fourier domain as a domain of signal sparsity. Signal Process. 130, 243–253 (2017)Google Scholar
  105. 105.
    S. Stanković, L. Stanković, I. Orović, Compressive sensing approach in the Hermite transform domain. Math. Probl. Eng., p 9. Article ID 286590 (2015)Google Scholar
  106. 106.
    S. Stanković, L. Stanković, I. Orović, A relationship between the robust statistics theory and sparse compressive sensed signals reconstruction. IET Signal Process. 8(3), 223–229 (2014)Google Scholar
  107. 107.
    R. Tibshirani, Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B (Methodol.) 58(1), 267–88 (1996)MathSciNetzbMATHGoogle Scholar
  108. 108.
    R. Tibshirani, M. Saunders, S. Rosset, J. Zhu, K. Knight, Sparsity and smoothness via the fused lasso. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 67(1), 91–108 (2005)MathSciNetzbMATHGoogle Scholar
  109. 109.
    M. Tipping, Sparse Bayesian learning and the relevance vector machine. J. Mach. Learn. Res. 1, 211–244 (2001)MathSciNetzbMATHGoogle Scholar
  110. 110.
    J.A. Tropp, Greed is good: algorithmic results for sparse approximation. IEEE Trans. Inf. Theory 50(10), 2231–2242 (2004)MathSciNetzbMATHGoogle Scholar
  111. 111.
    J.A. Tropp, A.C. Gilbert, Signal recovery from random measurements via orthogonal matching pursuit. IEEE Trans. Inf. Theory 53(12), 4655–4666 (2007)MathSciNetzbMATHGoogle Scholar
  112. 112.
    D. Vukobratovic, A. Pizurica, Compressed sensing using sparse adaptive measurements, in Proceedings of the Symposium on Information Theory in the Benelux (SITB ’14) (Eindhoven, The Netherlands, 2014)Google Scholar
  113. 113.
    Y. Wang, J. Xiang, Q. Mo, S. He, Compressed sparse time-frequency feature representation via compressive sensing and its applications in fault diagnosis. Measurement 68, 70–81 (2015)Google Scholar
  114. 114.
    L. Wang, L. Zhao, G. Bi, C. Wan, Hierarchical sparse signal recovery by variational Bayesian inference. IEEE Signal Process. Lett. 21(1), 110–113 (2014)Google Scholar
  115. 115.
    L. Zhang, M. Xing, C.W. Qiu, J. Li, Z. Bao, Achieving higher resolution ISAR imaging with limited pulses via compressed sampling. IEEE Geosci. Remote Sens. Lett. 6(3), 567–571 (2009)Google Scholar
  116. 116.
    T. Zhang, Sparse recovery with orthogonal matching pursuit under RIP. IEEE Trans. Inf. Theory 57(9), 6215–6221 (2011)MathSciNetzbMATHGoogle Scholar
  117. 117.
    Z. Zhang, T.P. Jung, S. Makeig, B.D. Rao, Compressed sensing of EEG for wireless telemonitoring with low energy consumption and inexpensive hardware. IEEE Trans. Biomed. Eng. 60(1), 221–224 (2013)Google Scholar
  118. 118.
    Z. Zhang, B.D. Rao, Sparse signal recovery with temporally correlated source vectors using sparse Bayesian learning. IEEE J. Sel. Top. Signal Process. 5(5), 912–926 (2011)Google Scholar
  119. 119.
    Z. Zhang, B.D. Rao, Extension of SBL algorithms for the recovery of block sparse signals with intra-block correlation. IEEE Trans. Signal Process. 61(8), 2009–2015 (2013)Google Scholar
  120. 120.
    L. Zhu, E. Liu, J.H. McClellan, Sparse-promoting full-waveform inversion based on online orthonormal dictionary learning. Geophysiscs 82(2), 87–107 (2017)Google Scholar
  121. 121.
    Z. Zhu, K. Wahid, P. Babyn, D. Cooper, I. Pratt, Y. Carter, Improved compressed sensing-based algorithm for sparse-view CT image reconstruction. Comput. Math. Methods Med. 2013, 185750 (2013)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Ljubiša Stanković
    • 1
    Email author
  • Ervin Sejdić
    • 2
  • Srdjan Stanković
    • 1
  • Miloš Daković
    • 1
  • Irena Orović
    • 1
  1. 1.Faculty of Electrical EngineeringUniversity of MontenegroPodgoricaMontenegro
  2. 2.Electrical and Computer EngineeringUniversity of PittsburghPittsburghUSA

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