Skip to main content

Factoring wavelet transforms into lifting steps

  • Chapter
  • First Online:
Wavelets in the Geosciences

Part of the book series: Lecture Notes in Earth Sciences ((LNEARTH,volume 90))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Aldroubi and M. Unser. Families of multiresolution and wavelet spaces with optimal properties. Numer. Funct. Anal. Optim., 14:417–446, 1993.

    Google Scholar 

  2. H. Bass. Algebraic K-theory. W. A. Benjamin, Inc., New York, 1968.

    Google Scholar 

  3. M. G. Bellanger and J. L. Daguet. TDM-FDM transmultiplexer: Digital polyphase and FFT. IEEE Trans. Commun., 22(9):1199–1204, 1974.

    Google Scholar 

  4. R. E. Blahut. Fast Algorithms for Digital Signal Processing. Addison-Wesley, Reading, MA, 1984.

    Google Scholar 

  5. A. A. M. L. Bruekens and A. W. M. van den Enden. New networks for perfect inversion and perfect reconstruction. IEEE J. Selected Areas Commun., 10(1), 1992.

    Google Scholar 

  6. R. Calderbank, I. Daubechies, W. Sweldens, and B.-L. Yeo. Wavelet transforms that map integers to integers. Appl. Comput. Harmon. Anal., 5(3):332–369, 1998.

    Google Scholar 

  7. J. M. Carnicer, W. Dahmen, and J. M. Peña. Local decompositions of refinable spaces. Appl. Comput. Harmon. Anal., 3:127–153, 1996.

    Google Scholar 

  8. C. K. Chui. An Introduction to Wavelets. Academic Press, San Diego, CA, 1992.

    Google Scholar 

  9. C. K. Chui, L. Montefusco, and L. Puccio, editors. Conference on Wavelets: Theory, Algorithms, and Applications. Academic Press, San Diego, CA, 1994.

    Google Scholar 

  10. C. K. Chui and J. Z. Wang. A cardinal spline approach to wavelets. Proc. Amer. Math. Soc., 113:785–793, 1991.

    Google Scholar 

  11. C. K. Chui and J. Z. Wang. A general framework of compactly supported splines and wavelets. J. Approx. Theory, 71(3):263–304, 1992.

    Google Scholar 

  12. A. Cohen, I. Daubechies, and J. Feauveau. Bi-orthogonal bases of compactly supported wavelets. Comm. Pure Appl. Math., 45:485–560, 1992.

    Google Scholar 

  13. J. M. Combes, A. Grossmann, and Ph. Tchamitchian, editors. Wavelets: Time-Frequency Methods and Phase Space. Inverse problems and theoretical imaging. Springer-Verlag, New York, 1989.

    Google Scholar 

  14. W. Dahmen and C. A. Micchelli. Banded matrices with banded inverses II: Locally finite decompositions of spline spaces. Constr. Approx., 9(2–3):263–281, 1993.

    Google Scholar 

  15. W. Dahmen, S. Prössdorf, and R. Schneider. Multiscale methods for pseudo-differential equations on smooth manifolds. In [9], pages 385–424. 1994.

    Google Scholar 

  16. I. Daubechies. Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math., 41:909–996, 1988.

    Google Scholar 

  17. I. Daubechies. Ten Lectures on Wavelets. CBMS-NSF Regional Conf. Series in Appl. Math., Vol. 61. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1992.

    Google Scholar 

  18. I. Daubechies, A. Grossmann, and Y. Meyer. Painless nonorthogonal expansions. J. Math. Phys., 27(5):1271–1283, 1986.

    Google Scholar 

  19. D. L. Donoho. Interpolating wavelet transforms. Preprint, Department of Statistics, Stanford University, 1992.

    Google Scholar 

  20. R. E. Van Dyck, T. G. Marshall, M. Chine, and N. Moayeri. Wavelet video coding with ladder structures and entropy-constrained quantization. IEEE Trans. Circuits Systems Video Tech., 6(5):483–495, 1996.

    Google Scholar 

  21. M. Frazier and B. Jawerth. Decomposition of Besov spaces. Indiana Univ. Math. J., 34(4):777–799, 1985.

    Google Scholar 

  22. A. Grossmann and J. Morlet. Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM J. Math. Anal., 15(4):723–736, 1984.

    Google Scholar 

  23. A. Harten. Multiresolution representation of data: A general framework. SIAM J. Numer. Anal., 33(3):1205–1256, 1996.

    Google Scholar 

  24. B. Hartley and T. O. Hawkes. Rings, Modules and Linear Algebra. Chapman and Hall, New York, 1983.

    Google Scholar 

  25. C. Herley and M. Vetterli. Wavelets and recursive filter banks. IEEE Trans. Signal Process., 41(8):2536–2556, 1993.

    Google Scholar 

  26. A. K. Jain. Fundamentals of Digital Image Processing. Prentice Hall, 1989.

    Google Scholar 

  27. N. S. Jayant and P. Noll. Digital coding of waveforms. Prentice Hall, Englewood Cliffs, NJ, 1984.

    Google Scholar 

  28. T. A. C. M. Kalker and I. Shah. Ladder Structures for multidimensional linear phase perfect reconstruction filter banks and wavelets. In Proceedings of the SPIE Conference on Visual Communications and Image Processing (Boston), pages 12–20, 1992.

    Google Scholar 

  29. M. Lounsbery, T. D. DeRose, and J. Warren. Multiresolution surfaces of arbitrary topological type. ACM Trans. on Graphics, 16(1):34–73, 1997.

    Google Scholar 

  30. S. G. Mallat. Multifrequency channel decompositions of images and wavelet models. IEEE Trans. Acoust. Speech Signal Process., 37(12):2091–2110, 1989.

    Google Scholar 

  31. S. G. Mallat. Multiresolution approximations and wavelet orthonormal bases of L2(ℜ211D;). Trans. Amer. Math. Soc., 315(1):69–87, 1989.

    Google Scholar 

  32. T. G. Marshall. A fast wavelet transform based upon the Euclidean algorithm. In Conference on Information Science and Systems, Johns Hopkins, MD, 1993.

    Google Scholar 

  33. T. G. Marshall. U-L block-triangular matrix and ladder realizations of subband coders. In Proc. IEEE ICASSP, volume III, pages 177–180, 1993.

    Google Scholar 

  34. Y. Meyer. Ondelettes et Opérateurs, I: Ondelettes, II: Opérateurs de Calderón-Zygmund, III: (with R. Coifman), Opérateurs multilinéaires. Hermann, Paris, 1990. English translation of first volume, Wavelets and Operators, is published by Cambridge University Press, 1993.

    Google Scholar 

  35. F. Mintzer. Filters for distortion-free two-band multirate filter banks. IEEE Trans. Acoust. Speech Signal Process., 33:626–630, 1985.

    Google Scholar 

  36. T. Q. Nguyen and P. P. Vaidyanathan. Two-channel perfect-reconstruction FIR QMF structures which yield linear-phase analysis and synthesis filters. IEEE Trans. Acoust. Speech Signal Process., 37:676–690, 1989.

    Google Scholar 

  37. H.-J. Park. A computational theory of Laurent polynomial rings and multidimensional FIR systems. PhD thesis, University of California, Berkeley, May 1995.

    Google Scholar 

  38. L.-M. Reissell. Wavelet multiresolution representation of curves and surfaces. CVGIP: Graphical Models and Image Processing, 58(2):198–217, 1996.

    Google Scholar 

  39. O. Rioul and P. Duhamel. Fast algorithms for discrete and continuous wavelet transforms. IEEE Trans. Inform. Theory, 38(2):569–586, 1992.

    Google Scholar 

  40. P. Schröder and W. Sweldens. Spherical wavelets: Efficiently representing functions on the sphere. Computer Graphics Proceedings, (SIGGRAPH 95), pages 161–172, 1995.

    Google Scholar 

  41. I. Shah and T. A. C. M. Kalker. On Ladder Structures and Linear Phase Conditions for Bi-Orthogonal Filter Banks. In Proceedings of ICASSP-94, volume 3, pages 181–184, 1994.

    Google Scholar 

  42. M. J. T. Smith and T. P. Barnwell. Exact reconstruction techniques for tree-structured subband coders. IEEE Trans. Acoust. Speech Signal Process., 34(3):434–441, 1986.

    Google Scholar 

  43. G. Strang and T. Nguyen. Wavelets and Filter Banks. Wellesley, Cambridge, 1996.

    Google Scholar 

  44. W. Sweldens. The lifting scheme: A custom-design construction of biorthogonal wavelets. Appl. Comput. Harmon. Anal., 3(2):186–200, 1996.

    Google Scholar 

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

    Google Scholar 

  46. W. Sweldens and P. Schröder. Building your own wavelets at home. In Wavelets in Computer Graphics, pages 15–87. ACM SIGGRAPH Course notes, 1996.

    Google Scholar 

  47. J. Tian and R. O. Wells. Vanishing moments and biorthogonal wavelet systems. In Mathematics in Signal Processing IV. Institute of Mathematics and Its Applications Conference Series, Oxford University Press, 1996.

    Google Scholar 

  48. L. M. G. Tolhuizen, H. D. L. Hollmann, and T. A. C. M. Kalker. On the realizability of bi-orthogonal M-dimensional 2-band filter banks. IEEE Transactions on Signal processing, 1995.

    Google Scholar 

  49. M. Unser, A. Aldroubi, and M. Eden. A family of polynomial spline wavelet transforms. Signal Process., 30:141–162, 1993.

    Google Scholar 

  50. P. P. Vaidyanathan. Theory and design of M-channel maximally decimated quadrature mirror filters with arbitrary M, having perfect reconstruction property. IEEE Trans. Acoust. Speech Signal Process., 35(2):476–492, 1987.

    Google Scholar 

  51. P. P. Vaidyanathan and P.-Q. Hoang. Lattice structures for optimal design and robust implementation of two-band perfect reconstruction QMF banks. IEEE Trans. Acoust. Speech Signal Process., 36:81–94, 1988.

    Google Scholar 

  52. P. P. Vaidyanathan, T. Q. Nguyen, Z. Douganata, and T. Saramäki. Improved technique for design of perfect reconstruction FIR QMF banks with lossless polyphase matrices. IEEE Trans. Acoust. Speech Signal Process., 37(7):1042–1055, 1989.

    Google Scholar 

  53. M. Vetterli. Filter banks allowing perfect reconstruction. Signal Process., 10:219–244, 1986.

    Google Scholar 

  54. M. Vetterli. Running FIR and IIR filtering using multirate filter banks. IEEE Trans. Signal Process., 36:730–738, 1988.

    Google Scholar 

  55. M. Vetterli and D. Le Gall. Perfect reconstruction FIR filter banks: Some properties and factorizations. IEEE Trans. Acoust. Speech Signal Process., 37:1057–1071, 1989.

    Google Scholar 

  56. M. Vetterli and C. Herley. Wavelets and filter banks: Theory and design. IEEE Trans. Acoust. Speech Signal Process., 40(9):2207–2232, 1992.

    Google Scholar 

  57. M. Vetterli and J. Kovaucević. Wavelets and Subband Coding. Prentice Hall, Englewood Cliffs, NJ, 1995.

    Google Scholar 

  58. Y. Wang, M. Orchard, A. Reibman, and V. Vaishampayan. Redundancy rate-distortion analysis of multiple description coding using pairwise correlation transforms. In Proc. IEEE ICIP, volume I, pages 608–611, 1977.

    Google Scholar 

  59. J. W. Woods and S. D. O'Neil. Subband coding of images. IEEE Trans. Acoust. Speech Signal Process., 34(5):1278–1288, 1986.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Roland Klees Roger Haagmans

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer-Verlag

About this chapter

Cite this chapter

Daubechies, I., Sweldens, W. (2000). Factoring wavelet transforms into lifting steps. In: Klees, R., Haagmans, R. (eds) Wavelets in the Geosciences. Lecture Notes in Earth Sciences, vol 90. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0011095

Download citation

  • DOI: https://doi.org/10.1007/BFb0011095

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-66951-7

  • Online ISBN: 978-3-540-46590-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics