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Constructing Wavelets

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From Fourier Analysis to Wavelets

Part of the book series: IMPA Monographs ((IMPA,volume 3))

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Abstract

The proof of the existence of a wavelet associated with a multiresolution representation described in Chap. 6 has a constructive flavor. In this chapter we will go over details of this proof with the purpose of obtaining a recipe to construct wavelets.

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References

  1. Andreas Antoniou. Digital Filters. McGraw-Hill, second edition, 1993.

    Google Scholar 

  2. Emmanual Bacry, Stephane Mallat, and George Papanicolaou. A wavelet based space–time adaptive numerical method for partial differential equations. Mathematical Modelling and Numerical Analysis, 26:793–834, 1992.

    MATH  MathSciNet  Google Scholar 

  3. G. Beylkin, R. Coifman, and V. Rokhlin. Fast wavelet transforms and numerical algorithms. Comm. in Pure and Applied Math., 44:141–183, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  4. G. Beylkin, R. Coifman, and V. Rokhlin. Fast wavelet transforms and numerical algorithms I. Communications on Pure and Applied Mathematics, XLIV:141–183, 1991. transformation en ondelettes.

    Google Scholar 

  5. L. Blum. Lectures on a Theory of Computation and Complexity over the Reals. 18 Coloquio Brasileiro de Matematica – IMPA, 1991.

    Google Scholar 

  6. Mikael Bourges-Sévenier. Réalisation d’une bibliothèque c de fonctions ondelettes. Technical report, IRISA – INRIA, 1994.

    Google Scholar 

  7. W. L. Briggs and V. E. Henson. DFT: an owner’s manual for the discrete Fourier transform. SIAM books, Philadelphia, 1995.

    Book  MATH  Google Scholar 

  8. P. J. Burt and E. H. Adelson. The Laplacian pyramid as a compact image code. IEEE Trans. Commun., 31(4):532–540, April 1983.

    Article  Google Scholar 

  9. P. L. Butzer and R. L. Stens. Sampling theory for not necessarily band-limited functions: a historical overview. SIAM Reviews, 34(1):40–53, march 1992.

    Google Scholar 

  10. Claudio Canuto and Anita Tabacco. Multilevel decompositions of functional spaces. The Journal of Fourier Analysis and Applications, 3(6), 1997.

    Google Scholar 

  11. Shaobing Chen and David L. Donoho. Basis pursuit. Technical report, Stanford University, 1994.

    Google Scholar 

  12. Shaobing Chen and David L. Donoho. Atomic decomposition by basis pursuit. Technical report, Stanford University, 1995.

    Google Scholar 

  13. C. K. Chui. An introduction to wavelets. Academic Press, 1992.

    Google Scholar 

  14. A. Cohen, I. Daubechies, and J. C. Feauveau. Biorthogonal bases of compactly supported wavelets. Comm. Pure Applied Math., 1992.

    Google Scholar 

  15. A. Cohen, I. Daubechies, and P. Vial. Wavelets on the interval and fast wavelet transforms. Applied and Computational Harmonic Analysis, 1(1):54–81, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  16. Ronald R. Coifman and Mladen Victor Wickerhauser. Entropy-based algorithms for best basis selection. IEEE Transactions on Information Theory, 38(2):713–718, 1992.

    Google Scholar 

  17. Wolfgang Dahmen and Reinhold Schneider. Wavelets on manifolds i: Construction and domain decomposition. Technical report, Institut fur Geometrie und Praktische Mathematik, 1998.

    Google Scholar 

  18. I. Daubechies. Orthonormal bases of compactly supported wavelets. Comm. Pure Applied Math., XLI(41):909–996, November 1988.

    Google Scholar 

  19. I. Daubechies, A. Grossmann, and Y. Meyer. Painless non-orthogonal expansions. J. Math. Phys., (27):1271–1283, 1986.

    Google Scholar 

  20. Ingrid Daubechies. Ten Lectures on Wavelets. SIAM Books, Philadelphia, PA, 1992.

    Book  MATH  Google Scholar 

  21. P. Duhamel and M. Vetterli. Fast fourier transform: A tutorial review and a state of the art. Signal Proc., 19(4):259–299, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  22. Alain Fournier. Wavelets and their applications in computer graphics. SIGGRAPH Course Notes, http://www.cs.ubc.ca/nest/imager/contributions/bobl/wvlt/download/notes.ps.Z.saveme, 1994.

  23. D. Gabor. Theory of communication. J. of the IEEE, (93):429–457, 1946.

    Google Scholar 

  24. J. Gomes and Luiz Velho. Image Processing for Computer Graphics. Springer-Verlag, 1997.

    Google Scholar 

  25. Jonas Gomes, Bruno Costa, Lucia Darsa, and Luiz Velho. Graphical objects. The Visual Computer, 12:269–282, 1996.

    Article  Google Scholar 

  26. Jonas Gomes and Luiz Velho. Abstraction paradigms for computer graphics. The Visual Computer, 11:227–239, 1995.

    Article  Google Scholar 

  27. Donald Greenspan. Computer Oriented Mathematical Physics. International Series in Nonlinear Mathematics, methods and Applications. Pergamon Press, New York., 1981.

    Google Scholar 

  28. E. Hernandez and G. Weiss. A First Course on Wavelets. CRC Press, Boca Raton, 1996.

    Book  MATH  Google Scholar 

  29. Nicholas J. Higham. Accuracy and Stability of Numerical Algorithms. SIAM Books, Philadelphia, 1996.

    MATH  Google Scholar 

  30. B. Jawerth and Wim Sweldens. An overview of wavelet based multiresolution analyses. SIAM Rev., 36(3):377–412, 1994.

    Google Scholar 

  31. G. Kaiser. A Friendly Guide to Wavelets. Birkhauser, Boston, 1994.

    MATH  Google Scholar 

  32. Bhagwandas P. Lathi. Signals, Systems, and Control. Harper & Row, 1974.

    Google Scholar 

  33. Jae S. Lim. Two Dimensional Signal and Image Processing. Prentice-Hall, New York, 1990.

    Google Scholar 

  34. Van Loan. Computational Frameworks for the Fast Fourier Transform. SIAM books, Philadelphia, 1996.

    Google Scholar 

  35. S. Mallat. Multifrequency channel decomposition of images and wavelet models. IEEE Transaction on ASSP, 37:2091–2110, 1989.

    Article  Google Scholar 

  36. S. Mallat. Multiresolution approximation and wavelets. Trans. Amer. Math. Soc., 315:69–88, 1989.

    MATH  MathSciNet  Google Scholar 

  37. S. Mallat. A Wavelet Tour of Signal Processing. Academic Press, 1998.

    Google Scholar 

  38. S. Mallat and A. Zhang. Matching pursuit with time-frequency dictionaries. Technical report, Courant Institute of Mathematical Sciences, 1993.

    Google Scholar 

  39. S. Mallat and S. Zhong. Characterization of signals from multiscale edges. IEEE Transactions on Pattern Analysis and Machine Intelligence, 14(7):710–732, 1992.

    Article  Google Scholar 

  40. S. G. Mallat and W. L. Hwang. Singularity detection and processing with wavelets. IEEE Transactions on Information Theory, 38(2):617–643, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  41. H. S. Malvar. Modulated QMF filter banks with perfect reconstruction. Electronics Letters, 26:906–907, June 1990.

    Article  Google Scholar 

  42. A. I. Markushevich. Theory of Functions of a Complex Variable. Chelsea Publishing Co., 1977.

    Google Scholar 

  43. Y. Meyer. Construction de bases orthonormées d’ondelettes. Colloq. Math., 60/61:141–149, 1990. bib.

    Google Scholar 

  44. Alan V. Oppenhein and Alan S. Willsky. Signal and Systems. Prentice-Hall inc, 1983.

    Google Scholar 

  45. William H. Press, Saul A. Teukolsky, and William T. Vetterling. Numerical Recipes: The Art of Scientific Computing, chapter 13, pages 591–606. Cambridge Univ Press, 1996.

    Google Scholar 

  46. A. A. G. Requicha. Representations for rigid solids: Theory methods, and systems. ACM Computing Surveys, 12:437–464, December 1980.

    Article  Google Scholar 

  47. Peter Schroder. Wavelets in computer graphics. In Proceedings of the IEEE, volume 84, pages 615–625, 1996.

    Google Scholar 

  48. Peter Schroder and Wim Sweldens. Wavelets in computer graphics. SIGGRAPH Course Notes, http://www.multires.caltech.edu/teaching/courses/waveletcourse/, 1995.

  49. Eero P. Simoncelli and William T. Freeman. The steerable pyramid: A flexible architecture for multi-scale derivative computation. In International Conference on Image Processing, volume 3, pages 444–447, 23–26 Oct. 1995, Washington, DC, USA, October 1995.

    Google Scholar 

  50. J. Stöckler. Multivariate wavelets. In C. K. Chui, editor, Wavelets: A Tutorial in Theory and Applications, pages 325–356. Academic Press, 1992.

    Google Scholar 

  51. Eric J. Stollnitz, Tony D. DeRose, and David H. Salesin. Wavelets for computer graphics: a primer, part 1. IEEE Computer Graphics and Applications, 15(3):76–84, May 1995.

    Article  Google Scholar 

  52. Eric J. Stollnitz, Tony D. DeRose, and David H. Salesin. Wavelets for computer graphics: a primer, part 2. IEEE Computer Graphics and Applications, 15(4):75–85, July 1995.

    Article  Google Scholar 

  53. Eric J. Stollnitz, Tony D. DeRose, and David H. Salesin. Wavelets for Computer Graphics: Theory and Applications. Morgann Kaufmann, San Francisco, CA, 1996.

    Google Scholar 

  54. G. Strang and J. Shen. The zeros of the Daubechies polynomials. Proc. Amer. Math. Soc.), 1996.

    Google Scholar 

  55. Gilbert Strang and Truong Nguyen. Wavelets and Filter Banks. Wellesley-Cambridge Press, Wellesley, MA, 1996.

    MATH  Google Scholar 

  56. Gilbert Strang and Vasily Strela. Orthogonal multiwavelets with vanishing moments. Optical Engineering, 33(7):2104–2107, 1994.

    Article  Google Scholar 

  57. W. Sweldens. The lifting scheme: A construction of second generation wavelets. Technical Report 1995:6, Department of Mathematics, University of South Carolina, 1995.

    Google Scholar 

  58. Carl Taswell. Algorithms for the generation of Daubechies orthogonal least asymmetric wavelets and the computation of their Holder regularity. Technical report, Scientific Computing and Computational Mathematics, Stanford University, Stanford, CA, August 1995.

    Google Scholar 

  59. Carl Taswell. Computational Algorithms for Daubechies Least-Asymmetric, Symmetric, and Most-Symmetric Wavelets, pages 1834–1838. Miller Freeman, September 1997.

    Google Scholar 

  60. Carl Taswell. The systematized collection of wavelet filters computable by spectral factorization of the Daubechies polynomial. Technical report, Computational Toolsmiths, www.toolsmiths.com, December 1997.

  61. P. P. Vaidyanathan. Multirate Systems and Filter Banks. Prentice Hall PTR, Englewood Cliffs, New Jersey, 1993.

    MATH  Google Scholar 

  62. M. Vetterli and C. Herley. Wavelets and filter banks: theory and design. IEEE Trans. Acoust. Speech Signal Process., 40(9), 1992.

    Google Scholar 

  63. Martin Vetterli and Jelena Kovacevic. Wavelets and Subband Coding. Prentice Hall PTR, Englewood Cliffs, New Jersey, 1995.

    MATH  Google Scholar 

  64. J. Weaver. Theory of Discrete and Continuous Fourier Transform. John Wiley & Sons, New York, 1989.

    Google Scholar 

  65. Mladen Victor Wickerhauser. Adapted Wavelet Analysis from Theory to Software. A. K. Peters, Wellesley, MA, 1994.

    Google Scholar 

  66. Ahmed Zayed. Advances in Shannon’s Sampling Theory. CRC Press, Boca Raton, 1993.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Offices Shopping Leblon, BOZANO Investimentos, Rio de Janeiro, Brazil

    Jonas Gomes

  2. Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil

    Luiz Velho

Authors
  1. Jonas Gomes
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  2. Luiz Velho
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Gomes, J., Velho, L. (2015). Constructing Wavelets. In: From Fourier Analysis to Wavelets. IMPA Monographs, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-319-22075-8_9

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