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Digital Signal Processing with Matlab Examples, Volume 2 pp 115–241Cite as

Wavelets

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Part of the book series: Signals and Communication Technology ((SCT))

Abstract

Wavelets have attracted a lot of attention from people involved in time-frequency analysis of signals. The literature on wavelets, books, papers, is quite extensive. Many practical applications of wavelets have been found. Signals can be decomposed into wavelets, which capture frequency and time punctual characteristics of nonstationary signals, which is an important advantage compared with the Fourier transform. The first section of this chapter presents the Haar wavelet, being an important archetype of wavelet that also fits well with the introductory purpose of this section. Once a mathematical approach, in terms of functional decomposition, has been introduced, the second section deals directly with the heart of the matter: the multiresolution analysis equation. From this equation a series of different types of wavelets can be deduced, as it will be described in the third and fourth sections. Then, the next sections are devoted to the continuous wavelet transform, the lifting method, wavelet packets, and multiwavelets. There are two sections for experiments and applications, including signal denoising and compression. The final section introduces the MATLAB wavelet toolbox.

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References

  1. P.S. Addison, J. Walker, R.C. Guido, Time-frequency analysis of biosignals. IEEE Eng. Med. Biol. Mgz. pp. 14–29 (2009)

    Google Scholar 

  2. S.A. Adewuai, B.O. Al-Bedoor, Wavelet analysis of vibration signals of an overhang rotor with a propagating transverse crack. J. Sound Vibr. 246(5), 777–793 (2001)

    Google Scholar 

  3. L. Aguiar-Conraria, M.J. Soares, The continuous wavelet transform: Moving beyond uni-and bivariate analysis. J. Econ. Surv. 28(2), 344–375 (2014)

    Google Scholar 

  4. A.N. Akansu, W.A. Serdijn, I.W. Selesnick, Emerging applications of wavelets: A review. Phys. Commun. 3, 1–18 (2010)

    Google Scholar 

  5. M. Akay, Wavelet applications in medicine. IEEE Spectrum 34(5), 50–56 (1997)

    Google Scholar 

  6. A. Aldroubi, M. Unser, Wavelets in Medicine and Biology (CRC Press, 1996)

    Google Scholar 

  7. O. Alkin, H. Caglar, Design of efficient M-band coders with linear-phase and perfect-reconstruction properties. IEEE Trans. Sign. Process. 43(7), 1579–1590 (1995)

    Google Scholar 

  8. B.K. Alpert, A class of bases in L2 for the sparse representation of integral operators. SIAM J. Math. Anal. 24(1), 246–262 (1993)

    Google Scholar 

  9. A. Alvandi, J. Bastien, E. Gregoire, M. Jolin, Bridge integrity assessment by continuous wavelet transforms. Intl. J. Struct. Stab. Dyn. 9(11) (2009)

    Google Scholar 

  10. V. Aniket, Biosignal Processing Challenges in Emotion Recognition for Adaptive Learning. PhD thesis (Univ. Central Florida, 2010)

    Google Scholar 

  11. D. Balenau, Wavelet Transforms and Their Recent Applications in Biology and Geoscience (InTech., 2012)

    Google Scholar 

  12. R.P. Boyer, Generalized Bernstein polynomials and symmetric functions. Adv. Appl. Math. 28, 17–39 (2002)

    Google Scholar 

  13. J. Bradley, C. Brislawn, T. Hopper, The FBI wavelet/scalar quantization standard for gray-scale fingerprint image compression, in SPIE v.1961: Visual Image Processing (1993), pp. 293–304

    Google Scholar 

  14. C.S. Burrus, R.A. Gopinath, H. Guo, Wavelets and Wavelet Transforms (Prentice-Hall, 1998)

    Google Scholar 

  15. P. Burt, E. Adelson, The Laplacian pyramid as a compact image code. IEEE Trans. Commun. 31, 482–540 (1983)

    Google Scholar 

  16. H. Caglar, A.N. Akansu, A generalized parametric PR-QMF design technique based on Bernstein polynomial approximation. IEEE Trans. Sign. Process. 41(7), 2314–2321 (1993)

    Google Scholar 

  17. T.T. Cai, D. Zhang, D. Ben-Amotz, Enhanced chemical classification of Raman images using multiresolution wavelet transformation. Appl. Spectrosc. 55(9), 1124–1130 (2001)

    Google Scholar 

  18. R. Capobianco, Emergent Applications of Fractals and Wavelets in Biology and Biomedicine (Elsevier, 2009)

    Google Scholar 

  19. A. Chamoli, V.S. Rani, K. Srivastava, D. Srinagesh, V.P. Dimri, Wavelet analysis of the seismograms for tsunami warning. Nonlinear Process. Geophys. 17, 569–574 (2010)

    Google Scholar 

  20. A. Cohen, I. Daubechies, J.C. Feauveau, Biorthogonal bases of compactly supported wavelets. Commun. Pure Appl. Math. 45, 485–560 (1992)

    Google Scholar 

  21. R.R. Coifman, M.V. Wickerhauser, Entropy-based algorithms for best basis selection. IEEE Trans. Inform. Theory 38(2), 713–718 (1992)

    Google Scholar 

  22. T. Cooklev, A. Nishihara, M. Sablatash, Regular orthonormal and biorthogonal wavelet filters. Sign. Process. 57, 121–137 (1997)

    Google Scholar 

  23. A. Cour-Harbo, A. Jensen. Wavelets and the lifting scheme, in Encyclopedia of Complexity and Systems Science, ed. by R.A. Meyers (Springer, 2009), pp. 10007–10031

    Google Scholar 

  24. I. Daubechies, The wavelet transform, time-frequency localization and signal analysis. IEEE Trans. Inform. Theory 36(5), 961–1005 (1990)

    Google Scholar 

  25. I. Daubechies, Ten Lectures on Wavelets (SIAM, Philadelphia, 1992)

    Google Scholar 

  26. I. Daubechies, Where do wavelets come from?- A personal point of view. Proc. IEEE 84(4), 510–513 (1996)

    Google Scholar 

  27. I. Daubechies, W. Sweldens, Factoring wavelet and subband transforms into lifting steps. Technical report, TechnicalBell Laboratories, Lucent Technologies (1996)

    Google Scholar 

  28. M.O. Domingues, O. Jr. Mendes, A. Mendes da Costa, On wavelet techniques in atmospheric sciences. Adv. Space Res. 35, 831–842 (2005)

    Google Scholar 

  29. D.L. Donoho, Denoising by soft-thresholding. IEEE Trans. Inform. Theory 41(3), 613–627 (1995)

    Google Scholar 

  30. F. Ebrahimi, M. Mikaeili, E. Estrada, H. Nazeran, Automatic sleep stage classification based on EEG signals by using neural networks and wavelet packet coefficients. Proc. IEEE Int. Conf. EMBS 1151–1154 (2008)

    Google Scholar 

  31. F. Ehrentreich, Wavelet transform applications in analytical chemistry. Anal. Bioanal. Chem. 372(1), 115–121 (2002)

    Google Scholar 

  32. M. Elsayed, An overview of wavelet analysis and its application to ocean wind waves. J. Coast. Res. 26(3), 535–540 (2010)

    Google Scholar 

  33. Foufola-Georgiou, E., P. Kumar, Wavelets in Geophysics (Academic Press, 1994)

    Google Scholar 

  34. K. Gurley, A. Kareem, Applications of wavelet transforms in earthquakes, wind and ocean engineering. Eng. Struct. 21, 149–167 (1999)

    Google Scholar 

  35. A. Jensen, A. la Cour-Harbo, Ripples in Mathematics (Springer, 2001)

    Google Scholar 

  36. Z. Jiang, X. Guo, A note on the extension of a family of biorthogonal Coifman wavelet systems. The ANZIAM J. 46, 111–120 (2004)

    Google Scholar 

  37. M. Kobayashi, Wavelets and their applications: Case studies. Technical report, IBM Tokyo Research Lab (1998)

    Google Scholar 

  38. P. Kumar, Wavelet analysis for geophysical applications. Rev. Geophys. 35(4), 385–412 (1997)

    Google Scholar 

  39. F. Kurth, M. Clausen, Filter bank tree and M-band wavelet packet algorithms in audio signal processing. IEEE Trans. Sign. Process. 47(2), 549–554 (1999)

    Google Scholar 

  40. M.S. Lewicki, Efficient coding of natural sounds. Nat. Neurosci. 5(4), 356–363 (2002)

    Google Scholar 

  41. T. Lin, S. Xu, Q. Shi, P. Hao, An algebraic construction of orthonormal M-band wavelets with perfect reconstruction. Appl. Math. Comput. 172, 717–730 (2006)

    Google Scholar 

  42. P. Lio, Wavelets in bioinformatics and computational biology: State of art and perspectives. Bioinform. Rev. 19(1), 2–9 (2003)

    Google Scholar 

  43. Z. Liu, N. Zheng, Parametrization construction of biorthogonal wavelet filter banks for image coding. Sign. Image Video Process. 1, 63–76 (2007)

    Google Scholar 

  44. S. Mallat, A Wavelet Tour of Signal Processing: The Sparse Way (Academic Press, 2008)

    Google Scholar 

  45. S.G. Mallat, A theory for multiresolution signal decomposition: The wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell. 11(7), 674–693 (1989)

    Google Scholar 

  46. M. Maslen, P. Abbott, Automation of the lifting factorization of wavelet transforms. Comput. Phys. Commun. 127, 309–326 (2000)

    Google Scholar 

  47. Y. Meyer. Principe D’incertitude, Bases Hilbertiennes Et Algebres D’operateurs (1985). Seminaire Bourbaki, n. 662. http://archive.numdam.org/article/SB_1985-1986_28_209_0.pdf

  48. M. Misiti, Y. Misiti, G. Oppenheim, J.M. Poggi, Wavelets and Their Applications (ISTE, London, 20070

    Google Scholar 

  49. J. Morlet, A. Grossman, Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM J. Math. Anal. 15, 723–736 (1984)

    Google Scholar 

  50. H. Nagendra, S. Mukherjee, V. Kumar, Application of wavelet techniques in ECG signal processing: An overview. Int. J. Eng. Sci. Technol. 3(10), 7432–7443 (2011)

    Google Scholar 

  51. A. Nait-Ali, Advanced Biosignal Processing (Springer, 2009)

    Google Scholar 

  52. K. Najarian, R. Splinter, Biomedical Signal and Image Processing (CRC Press, 2012)

    Google Scholar 

  53. B.D. Patil, P.G. Patwardhan, V.M. Gadre, On the design of FIR wavelet filter banks using factorization of a halfband polynomial. IEEE Sign. Process. Lett. 15, 485–488 (2008)

    Google Scholar 

  54. Z.K. Peng, F.L. Chu, Application of the wavelet transform in machine condition monitoring and fault diagnostics: A review with bibliography. Mech. Syst. Sign. Process. 18, 199–221 (2004)

    Google Scholar 

  55. G. Qi, Wavelet-based AE characterization of composite materials. NDT E Int. 33(3), 133–144 (2000)

    Google Scholar 

  56. M. Raphan, E.P. Simoncelli, Optimal denoising in redundant representations. IEEE Trans. Image Process. 17(8), 1342–1352 (2008)

    Google Scholar 

  57. B. Rivard, J. Feng, A. Gallie, A. Sanchez-Azofeifa, Continuous wavelets for the improved use of spectral libraries and hyperspectral data. Remote Sens. Environ. 112, 2850–2862 (2008)

    Google Scholar 

  58. J.L. Semmlow, Biosignal and Biomedical Image Processing (CRC Press, 2008)

    Google Scholar 

  59. X.G. Shao, A.K. Leung, F.T. Chau, Wavelet: A new trend in chemistry. Acc. Chem. Res 36(4), 276–283 (2003)

    Google Scholar 

  60. M.C. Shou, L.P. Leu, Energy of power spectral density function and wavelet analysis of absolute pressure fluctuation measurements in fluidized beds. Chem. Eng. Res. Des. 83(5), 478–491 (2005)

    Google Scholar 

  61. F.J. Simons, B.D.E. Dando, R.M. Allen, Automatic detection and rapid determination of earthquake magnitude by wavelet multiscale analysis of the primary arrival. EarthPlanet. Sci. Lett. 250, 214–223 (2006)

    Google Scholar 

  62. F.J. Simons, I. Loris, E. Brevdo, I. Daubechies, Wavelets and wavelet-like transforms on the sphere and their application to geophysical data inversion. Proc. SPIE 8138, 1–15 (2011)

    Google Scholar 

  63. S. Sinha, P.S. Routh, P.D. Anno, J.P. Castagna, Spectral decomposition of seismic data with continuous-wavelet transform. Geophysics 70, 19–25 (2005)

    Google Scholar 

  64. A.N. Skodras, C.A. Christopoulos, T. Ebrahimi, JPEG2000: The upcoming still image compression standard. Pattern Recogn. Lett. 22(12), 1337–1345 (2001)

    Google Scholar 

  65. V. Strela, P.N. Heller, G. Strang, P. Topiwala, C. Heil, The application of multiwavelet filterbanks to image processing. IEEE T. Image Proc. 8(4), 548–563 (1999)

    Google Scholar 

  66. Z. Sun, C.C. Chang, Structural damage assessment based on wavelet packet transform. J. Struct. Eng. 128(10), 1354–1361 (2002)

    Google Scholar 

  67. W. Sweldens, The lifting scheme: A construction of second generation wavelets. SIAM. J. Math. Anal. 29(2), 511–546 (1997)

    Google Scholar 

  68. D.B.H. Tay, Rationalizing the coefficients of popular biorthogonal wavelet filters. IEEE Trans. Circ. Syst. Video Technol. 10(6), 998–1005 (2000)

    Google Scholar 

  69. D.B.H. Tay, M. Palaniswami, A novel approach to the design of the class of triplet halfband filterbanks. IEEE Trans. Circ. Syst.-II: Express Briefs 51(7), 378–383 (2004)

    Google Scholar 

  70. J. Tian, R.O. Wells Jr, Vanishing moments and biorthogonal wavelet systems, in Mathematics in Signal Processing IV, ed. by Mc.Whirter (Oxford University Press, 1997)

    Google Scholar 

  71. A.C. To, J.R. Moore, S.D. Glaser, Wavelet denoising techniques with applications to experimental geophysical data. Sign. Process. 89, 144–160 (2009)

    Google Scholar 

  72. F. Truchelet, O. Laligant, Wavelets in industrial applications: A review, in Proceedings SPIE, vol. 5607 (2004), pp. 1–14

    Google Scholar 

  73. F. Truchelet, O. Laligant, Review of industrial applications of wavelet and multiresolution-based signal and image processing. J. Electron. Imaging 17(3) (2008)

    Google Scholar 

  74. M. Unser, A. Aldroubi, A review of wavelets in biomedical applications. Proc. IEEE 84(4), 626–638 (1996)

    Google Scholar 

  75. M. Unser, T. Blu, Mathematical properties of the JPEG2000 wavelet filters. IEEE Trans. Image Process. 12(9), 1080–1090 (2003)

    Google Scholar 

  76. G. Uytterhoeven, D. Roose, A. Bultheel, Wavelet transforms using the lifting scheme. Technical report, Katholieke Universiteit Leuven, 1997. ITA-Wavelets-WP.1.1

    Google Scholar 

  77. M. Vetterli, J. Kovacevic. Wavelets and Subband Coding (Prentice Hall, 1995)

    Google Scholar 

  78. D. Wei. Coiflet-type Wavelets: Theory, Design, and Applications. PhD thesis, University of Texas at Austin (1998)

    Google Scholar 

  79. X. Yang, Y. Shi, B. Yang, General framework of the construction of biorthogonal wavelets based on Bernstein bases: Theory analysis and application in image compression. IET Comput. Vision 5(1), 50–67 (2011)

    Google Scholar 

  80. R. Yu, A. Baradarani, Design of halfband filters for ortogonal wavelets via sum of squares decomposition. IEEE Sign. Process. 15, 437–440 (2008)

    Google Scholar 

  81. X. Zhang, Design of FIR halfband filters for orthonormal wavelets using Remez exchange algorithm. IEEE Sign. Proces. Lett. 16(9), 814–817 (2009)

    Google Scholar 

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  1. Systems Engineering and Automatic Control, Universidad Complutense de Madrid, Madrid, Spain

    Jose Maria Giron-Sierra

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Giron-Sierra, J.M. (2017). Wavelets. In: Digital Signal Processing with Matlab Examples, Volume 2. Signals and Communication Technology. Springer, Singapore. https://doi.org/10.1007/978-981-10-2537-2_2

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  • DOI: https://doi.org/10.1007/978-981-10-2537-2_2

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