Skip to main content

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

Circuits, Systems, and Signal Processing volume 38, pages 1206–1263 (2019)Cite this 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.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price includes VAT (USA)
Tax calculation will be finalised during checkout.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Notes

  1. Consider a signal x(t) of a duration T and its samples \(x(n\Delta t)\) satisfying the sampling theorem. The periodic extension of this signal can be written in a Fourier series (FS) form

    $$\begin{aligned} x(t) = \frac{1}{N}\sum \limits _{k = 0}^{N-1} {{X(k)}{\hbox {e}^{j2\pi k \frac{t}{T}}}}, \end{aligned}$$

    where the FS coefficients X(k) are equal to the DFT coefficients if we use the notation x(n) for \(x(n\Delta t)\) and \(\Delta t=T/N\) as the sampling interval. When the sampling theorem is satisfied then

    $$\begin{aligned} X(k)=\frac{N}{T}\int \limits _0^Tx(t)\hbox {e}^{-j2 \pi kt/T}\hbox {d}t=\sum _{n=0}^{N-1}x(n)\hbox {e}^{-j2\pi kn/N}, \quad k=0,1,2,\dots ,N-1. \end{aligned}$$

    This is the DFT of a signal x(n).

References

  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)

    MATH  Google Scholar 

  2. N. Ahmed, T. Natarajan, K.R. Rao, Discrete cosine transform. IEEE Trans. Comput. C–23(1), 90–93 (1974)

    MathSciNet  MATH  Google Scholar 

  3. M.G. Amin, Compressive Sensing for Urban Radar (CRC Press, Boca Raton, 2014)

    Google Scholar 

  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. 1–8

  5. E. Arias-Castro, Y. Eldar, Noise folding in compressed sensing. IEEE Signal Process. Lett. 18(8), 478–481 (2011)

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  8. R. Baraniuk, Compressive sensing. IEEE Signal Process. Mag. 24(4), 118–121 (2007)

    Google Scholar 

  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. D. Baron, S. Sarvotham, R.G. Baraniuk, Bayesian compressive sensing via belief propagation. IEEE Trans. Signal Process. 58(1), 269–280 (2010)

    MathSciNet  MATH  Google Scholar 

  11. J. Bazerque, G. Giannakis, Distributed spectrum sensing for cognitive radio networks by exploiting sparsity. IEEE Trans. Signal Process. 58(3), 1847–1862 (2010)

    MathSciNet  MATH  Google Scholar 

  12. A. Beck, M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–197 (2009)

    MathSciNet  MATH  Google Scholar 

  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. 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)

    MathSciNet  Google Scholar 

  15. J.D. Blanchard, Cartis, J. Tanner, Compressed sensing: how sharp is the restricted isometry property? SIAM Rev. 53(1), 105–125 (2011)

    MathSciNet  MATH  Google Scholar 

  16. T. Blumensath, M.E. Davies, Gradient pursuits. IEEE Trans. Signal Process. 56(6), 2370–2382 (2008)

    MathSciNet  MATH  Google Scholar 

  17. T. Blumensath, M.E. Davies, Iterative thresholding for sparse approximations. J. Fourier Anal. Appl. 14(5–6), 629–654 (2008)

    MathSciNet  MATH  Google Scholar 

  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. S. Boyd, L. Vandenberghe, Convex Optimization (Cambridge University Press, Cambridge, 2004)

    MATH  Google Scholar 

  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. 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. 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. L. Breiman, Better subset regression using the nonnegative garrote. Technometrics 37(4), 373–384 (1995)

    MathSciNet  MATH  Google Scholar 

  24. E.J. Candès, The restricted isometry property and its implications for compressed sensing. C. R. Math. 346(9–10), 589–592 (2008)

    MathSciNet  MATH  Google Scholar 

  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. 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)

    MathSciNet  MATH  Google Scholar 

  27. E.J. Candès, M. Wakin, An introduction to compressive sampling. IEEE Signal Process. Mag. 25(2), 21–30 (2008)

    Google Scholar 

  28. R. Chartrand, V. Staneva, Restricted isometry properties and nonconvex compressive sensing. Inverse Probl. 24(3), 035020-1-14 (2008)

    MathSciNet  MATH  Google Scholar 

  29. S.S. Chen, D.L. Donoho, M.A. Saunders, Atomic decomposition by basis pursuit. SIAM Rev. 43(1), 129–159 (2001)

    MathSciNet  MATH  Google Scholar 

  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. S. Costanzo, A. Rocha, M.D. Migliore, Compressed sensing: applications in radar and communications. Sci. World J. 2016, 2. Article ID 5407415 (2016)

  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)

    MathSciNet  MATH  Google Scholar 

  33. G. Davis, S. Mallat, M. Avellaneda, Adaptive greedy approximations. Constr. Approx. 13(1), 57–98 (1997)

    MathSciNet  MATH  Google Scholar 

  34. D.L. Donoho, Compressed sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  36. M. Elad, Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing (Springer, Berlin, 2010)

    MATH  Google Scholar 

  37. Y.C. Eldar, G. Kutyniok, Compressed Sensing: Theory and Applications (Cambridge University Press, Cambridge, 2012)

    Google Scholar 

  38. J. Ender, On compressive sensing applied to radar. Signal Process. 90(5), 1402–1414 (2010)

    MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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. P. Flandrin, P. Borgnat, Time-frequency energy distributions meet compressed sensing. IEEE Trans. Signal Process. 58(6), 2974–2982 (2010)

    MathSciNet  MATH  Google Scholar 

  42. M. Fornsaier, H. Rauhut, Iterative thresholding algorithms. Appl. Comput. Harmon. Anal. 25(2), 187–208 (2008)

    MathSciNet  MATH  Google Scholar 

  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. G. Hua, Y. Hiang, G. Bi, When compressive sensing meets data hiding. IEEE Signal Process. Lett. 23(4), 473–477 (2016)

    Google Scholar 

  45. S. Ji, Y. Xue, L. Carin, Bayesian compressive sensing. IEEE Trans. Signal Process. 56(6), 2346–2356 (2008)

    MathSciNet  MATH  Google Scholar 

  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. 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. 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. 1158–1162

  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. S. Liu, Y.D. Zhang, T. Shan, Detection of weak astronomical signals with frequency-hopping interference suppression. Digit. Signal Process. 72, 1–8 (2018)

    MathSciNet  Google Scholar 

  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. 98570N

  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)

    MathSciNet  Google Scholar 

  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)

  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. 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)

    MathSciNet  MATH  Google Scholar 

  56. S. Luo, P. Johnston, A review of electrocardiogram filtering. J. Electrocardiol. 43(6), 486–496 (2010)

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  58. S. Mallat, Z. Zhang, Matching pursuits with time-frequency dictionaries. IEEE Trans. Signal Process. 41(12), 3397–3415 (1993)

    MATH  Google Scholar 

  59. J.B. Martens, The Hermite transform—theory. IEEE Trans. Acoust. Speech Signal Process. 38(9), 1595–1606 (1990)

    MATH  Google Scholar 

  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. 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. 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. 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

    MATH  Google Scholar 

  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)

  65. D. Needell, J.A. Tropp, CoSaMP: iterative signal recovery from incomplete and inaccurate samples. Commun. ACM 53(12), 93–100 (2010)

    MATH  Google Scholar 

  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. 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)

    MathSciNet  MATH  Google Scholar 

  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. 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)

  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. C. Ozdemir, Inverse Synthetic Aperture Radar Imaging with MATLAB Algorithms (Wiley, Hoboken, 2012)

    Google Scholar 

  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. 562–566

  73. V.M. Patel, R. Chellappa, Sparse Representations and Compressive Sensing for Imaging and Vision (Springer, Berlin, 2013)

    MATH  Google Scholar 

  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. 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. 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. R. Rubinstein, A.M. Bruckstein, M. Elad, Dictionaries for sparse representation modeling. Proc. IEEE 98(6), 1045–1057 (2010)

    Google Scholar 

  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. 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)

    MathSciNet  MATH  Google Scholar 

  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. 581–584

  81. E. Sejdic, Time-frequency compressive sensing, in Frequency Signal Analysis and Processing, ed. B. Boashash (Academic Press, 2015), pp. 424–429

  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. 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. 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. L. Stanković, A measure of some time-frequency distributions concentration. Signal Process. 81, 621–631 (2001)

    MATH  Google Scholar 

  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. 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. 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. 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. 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. L. Stanković, M. Daković, T. Thayaparan, Time-Frequency Signal Analysis with Applications (Artech House, Boston, 2013)

    MATH  Google Scholar 

  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. L. Stanković, M. Daković, S. Vujovic, Reconstruction of sparse signals in impulsive disturbance environments. Circuits Syst. Signal Process. 36, 1–28 (2016)

    MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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. 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. 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. 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. 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. 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. 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)

  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. 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. S. Stanković, I. Orović, L. Stanković, Polynomial Fourier domain as a domain of signal sparsity. Signal Process. 130, 243–253 (2017)

    Google Scholar 

  105. S. Stanković, L. Stanković, I. Orović, Compressive sensing approach in the Hermite transform domain. Math. Probl. Eng., p 9. Article ID 286590 (2015)

  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. R. Tibshirani, Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B (Methodol.) 58(1), 267–88 (1996)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  109. M. Tipping, Sparse Bayesian learning and the relevance vector machine. J. Mach. Learn. Res. 1, 211–244 (2001)

    MathSciNet  MATH  Google Scholar 

  110. J.A. Tropp, Greed is good: algorithmic results for sparse approximation. IEEE Trans. Inf. Theory 50(10), 2231–2242 (2004)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

  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. 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. 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. T. Zhang, Sparse recovery with orthogonal matching pursuit under RIP. IEEE Trans. Inf. Theory 57(9), 6215–6221 (2011)

    MathSciNet  MATH  Google Scholar 

  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. 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. 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. 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. 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)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Electrical Engineering, University of Montenegro, Podgorica, Montenegro

    Ljubiša Stanković, Srdjan Stanković, Miloš Daković & Irena Orović

  2. Electrical and Computer Engineering, University of Pittsburgh, Pittsburgh, PA, USA

    Ervin Sejdić

Authors
  1. Ljubiša Stanković
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ervin Sejdić
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Srdjan Stanković
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Miloš Daković
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Irena Orović
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ljubiša Stanković.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Stanković, L., Sejdić, E., Stanković, S. et al. A Tutorial on Sparse Signal Reconstruction and Its Applications in Signal Processing. Circuits Syst Signal Process 38, 1206–1263 (2019). https://doi.org/10.1007/s00034-018-0909-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-018-0909-2

Keywords

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