Factoring wavelet transforms into lifting steps

  • Ingrid Daubechies
  • Wim Sweldens
Article

Abstract

This article is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This decomposition corresponds to a factorization of the polyphase matrix of the wavelet or subband filters into elementary matrices. That such a factorization is possible is well-known to algebraists (and expressed by the formulaSL(n;R[z, z−1])=E(n;R[z, z−1])); it is also used in linear systems theory in the electrical engineering community. We present here a self-contained derivation, building the decomposition from basic principles such as the Euclidean algorithm, with a focus on applying it to wavelet filtering. This factorization provides an alternative for the lattice factorization, with the advantage that it can also be used in the biorthogonal, i.e., non-unitary case. Like the lattice factorization, the decomposition presented here asymptotically reduces the computational complexity of the transform by a factor two. It has other applications, such as the possibility of defining a wavelet-like transform that maps integers to integers.

Math Subject Classifications

42C15 42C05 19-02 

Keywords and Phrases

Wavelet lifting elementary matrix Euclidean algorithm Laurent polynomial 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Aldroubi, A. and Unser, M. (1993). Families of multiresolution and wavelet spaces with optimal properties.Numer. Funct. Anal. Optim.,14, 417–446.MATHMathSciNetGoogle Scholar
  2. [2]
    Bass, H. (1968).Algebraic K-Theory, W. A. Benjamin, New York.Google Scholar
  3. [3]
    Bellanger, M.G. and Daguet, J.L. (1974). TDM-FDM transmultiplexer: Digital polyphase and FFT.IEEE Trans. Commun.,22(9), 1199–1204.CrossRefGoogle Scholar
  4. [4]
    Blahut, R.E. (1984).Fast Algorithms for Digital Signal Processing. Addison-Wesley, Reading, MA.MATHGoogle Scholar
  5. [5]
    Bruekens, A.A.M.L. and van den Enden, A.W.M. (1992). New networks for perfect inversion and perfect reconstruction.IEEE J. Selected Areas Commun.,10(1).Google Scholar
  6. [6]
    Calderbank, R., Daubechies, I., Sweldens, W., and Yeo, B.-L. Wavelet transforms that map integers to integers.Appl. Comput. Harmon. Anal., (to appear).Google Scholar
  7. [7]
    Carnicer, J.M., Dahmen, W., and Peña, J.M. (1996). Local decompositions of refinable spaces.Appl. Comput. Harmon. Anal.,3, 127–153.MATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    Chui, C.K. (1992).An Introduction to Wavelets. Academic Press, San Diego, CA.MATHGoogle Scholar
  9. [9]
    Chui, C.K., Montefusco, L., and Puccio, L., Eds. (1994).Conference on Wavelets: Theory, Algorithms, and Applications. Academic Press, San Diego, CA.Google Scholar
  10. [10]
    Chui, C.K. and Wang, J.Z. (1991). A cardinal spline approach to wavelets.Proc. Amer. Math. Soc.,113, 785–793.MATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    Chui, C.K. and Wang, J.Z. (1992). A general framework of compactly supported splines and wavelets.J. Approx. Theory,71(3), 263–304.MATHMathSciNetCrossRefGoogle Scholar
  12. [12]
    Cohen, A., Daubechies, I., and Feauveau, J. (1992). Bi-orthogonal bases of compactly supported wavelets.Comm. Pure Appl. Math.,45, 485–560.MATHMathSciNetGoogle Scholar
  13. [13]
    Combes, J.M., Grossmann, A., and Tchamitchian, Ph. Eds. (1989).Wavelets: Time-Frequency Methods and Phase Space. Inverse problems and Theoretical Imaging. Springer-Verlag, New York.Google Scholar
  14. [14]
    Dahmen, W. and Micchelli, C.A. (1993). Banded matrices with banded inverses II: Locally finite decompositions of spline spaces.Constr. Approx.,9(2–3), 263–281.MATHMathSciNetCrossRefGoogle Scholar
  15. [15]
    Dahmen, W., Prössdorf, S., and Schneider, R. (1994). Multiscale methods for pseudo-differential equations on smooth manifolds. In [9], 385–424.Google Scholar
  16. [16]
    Daubechies, I. (1988). Orthonormal bases of compactly supported wavelets.Comm. Pure Appl. Math.,41, 909–996.MATHMathSciNetGoogle Scholar
  17. [17]
    Daubechies, I. (1992).Ten Lectures on Wavelet. CBMS-NSF Regional Conf. Series in Appl. Math., vol. 61. Society for Industrial and Applied Mathematics, Philadelphia, PA.Google Scholar
  18. [18]
    Daubechies, I., Grossmann, A., and Meyer, Y. (1986). Painless nonorthogonal expansions.J. Math. Phys.,27(5), 1271–1283.MATHMathSciNetCrossRefGoogle Scholar
  19. [19]
    Donoho, D.L. (1992). Interpolating wavelet transforms. Preprint, Department of Statistics, Stanford University.Google Scholar
  20. [20]
    Van Dyck, R.E., Marshall, T.G., Chine, M. and Moayeri, N. (1996). Wavelet video coding with ladder structures and entropy-constrained quantization.IEEE Trans. Circuits Systems Video Tech.,6(5), 483–495.CrossRefGoogle Scholar
  21. [21]
    Frazier, M. and Jawerth, B. (1985). Decomposition of Besov spaces.Indiana Univ. Math. J.,34 (4), 777–799.MATHMathSciNetCrossRefGoogle Scholar
  22. [22]
    Grossmann, A. and Morlet, J. (1984). Decomposition of Hardy functions into square integrable wavelets of constant shape.SIAM J. Math. Anal.,15(4), 723–736.MATHMathSciNetCrossRefGoogle Scholar
  23. [23]
    Harten, A. (1996). Multiresolution representation of data: A general framework.SIAM J. Numer. Anal.,33 (3), 1205–1256.MATHMathSciNetCrossRefGoogle Scholar
  24. [24]
    Hartley, B. and Hawkes, T.O. (1983).Rings, Modules and Linear Algebra. Chapman and Hall, New York.MATHGoogle Scholar
  25. [25]
    Herley, C. and Vetterli, M. (1993). Wavelets and recursive filter banks.IEEE Trans. Signal Process.,41(8), 2536–2556.MATHCrossRefGoogle Scholar
  26. [26]
    Jain, A.K. (1989).Fundamentals of Digital Image Processing. Prentice Hall, Englewood Cliffs, NJ.MATHGoogle Scholar
  27. [27]
    Jayanat, N.S. and Noll, P. (1984).Digital Coding of Waveforms. Prentice Hall, Englewood Cliffs, NJ.Google Scholar
  28. [28]
    Kalker, T.A.C.M. and Shah, I. (1992). Ladder Structures for multidimensional linear phase perfect reconstruction filter banks and wavelets. InProceedings of the SPIE Conference on Visual Communications and Image Processing (Boston), 12–20.Google Scholar
  29. [29]
    Lounsbery, M., DeRose, T.D., and Warren, J. (1997). Multiresolution surfaces of arbitrary topological type.ACM Trans. on Graphics,16(1), 34–73.CrossRefGoogle Scholar
  30. [30]
    Mallat, S.G. (1989). Multifrequency channel decompositions of images and wavelet models.IEEE Trans. Acoust. Speech Signal Process.,37(12), 2091–2110.CrossRefGoogle Scholar
  31. [31]
    Mallat, S.G. (1989). Multiresolution approximations and wavelet orthonormal bases of L2 (R).Trans. Amer. Math. Soc.,315(1), 69–87.MATHMathSciNetCrossRefGoogle Scholar
  32. [32]
    Marshall, T.G. (1993). A fast wavelet transform based upon the Euclidean algorithm. InConference on Information Science and Systems, Johns Hopkins, Maryland.Google Scholar
  33. [33]
    Marshall, T.G. (1993). U-L block-triangular matrix and ladder realizations of subband coders. InProc. IEEE ICASSP, III: 177–180.Google Scholar
  34. [34]
    Meyer, Y. (1990).Ondelettes et Opérateurs, I:Ondelettes, II:Opérateurs de Calderón-Zygmund, III: (with R. Coifman),Opérateurs multilinéaires. Hermann, Paris. English translation of first volume,Wavelets and Operators, is published by Cambridge University Press, 1993.Google Scholar
  35. [35]
    Mintzer, F. (1985). Filters for distortion-free two-band multirate filter banks.IEEE Trans. Acoust. Speech Signal Process.,33, 626–630.CrossRefGoogle Scholar
  36. [36]
    Nguyen, T.Q. and Vaidyanathan, P.P. (1989). Two-channel perfect-reconstruction FIR QMF structures which yield linear-phase analysis and synthesis filters.IEEE Trans. Acoust. Speech Signal Process.,37, 676–690.CrossRefGoogle Scholar
  37. [37]
    Park, H.-J..A computational theory of Laurent polynomial rings and multidimensional FIR systems. PhD thesis, University of California, Berkeley, May 1995.Google Scholar
  38. [38]
    Reissell, L.-M. (1996). Wavelet multiresolution representation of curves and surfaces.CVGIP: Graphical Models and Image Processing,58(2), 198–217.Google Scholar
  39. [39]
    Rioul, O. and Duhamel, P. (1992). Fast algorithms for discrete and continuous wavelet transforms.IEEE Trans. Inform. Theory,38(2), 569–586.MATHMathSciNetCrossRefGoogle Scholar
  40. [40]
    Schröder, P. and Sweldens, W. (1995). Spherical wavelets: Efficiently representing functions on the sphere.Computer Graphics Proceedings, (SIGGRAPH 95), 161–172.Google Scholar
  41. [41]
    Shah, I. and Kalker, T.A.C.M. (1994). On Ladder Structures and Linear Phase Conditions for Bi-Orthogonal Filter Banks. InProceedings of ICASSP-94,3, 181–184.Google Scholar
  42. [42]
    Smith, M.J.T. and Barnwell, T.P. (1986). Exact reconstruction techniques for tree-structured subband coders.IEEE Trans. Acoust. Speech Signal Process.,34(3), 434–441.CrossRefGoogle Scholar
  43. [43]
    Strang, G. and Nguyen, T. (1996).Wavelets and Filter Banks. Wellesley, Cambridge, MA.Google Scholar
  44. [44]
    Sweldens, W. (1996). The lifting scheme: A custom-design construction of biorthogonal wavelets.Appl. Comput. Harmon. Anal.,3(2), 186–200.MATHMathSciNetCrossRefGoogle Scholar
  45. [45]
    Sweldens, W. (1997). The lifting scheme: A construction of second generation wavelets.SIAM J. Math. Anal.,29(2), 511–546.MathSciNetCrossRefGoogle Scholar
  46. [46]
    Sweldens, W. and Schröder, P. (1996). Building your own wavelets at home. InWavelets in Computer Graphics, 15–87. ACM SIGGRAPH Course notes.Google Scholar
  47. [47]
    Tian, J. and Wells, R.O. (1996). Vanishing moments and biorthogonal wavelets systems. InMathematics in Signal Processing IV. Institute of Mathematics and Its Applications Conference Series, Oxford University Press.Google Scholar
  48. [48]
    Tolhuizen, L.M.G., Hollmann, H.D.L., and Kalker, T.A.C.M. (1995). On the realizability of bi-orthogonal M-dimensional 2-band filter banks.IEEE Trans. Signal Process.Google Scholar
  49. [49]
    Unser, M., Aldroubi, A., and Eden, M. (1993). A family of polynomial spline wavelet transforms.Signal Process.,30, 141–162.MATHCrossRefGoogle Scholar
  50. [50]
    Vaidyanathan, P.P. (1987). 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.MATHCrossRefGoogle Scholar
  51. [51]
    Vaidyanathan, P.P. and Hoang, P.-Q. (1988). Lattice structures for optimal design and robust implementation of two-band perfect reconstruction QMF banks.IEEE Trans. Acoust. Speech Signal Process.,36, 81–94.CrossRefGoogle Scholar
  52. [52]
    Vaidyanathan, P.P., Nguyen, T.Q., Doĝanata, Z., and Saramäki, T. (1989). Improved technique for design of perfect reconstruction FIR QMF banks with lossless polyphase matrices.IEEE Trans. Acoust. Speech Signal Process.,37(7), 1042–1055.CrossRefGoogle Scholar
  53. [53]
    Vetterli, M. (1986). Filter banks allowing perfect reconstruction.Signal Process.,10, 219–244.MathSciNetCrossRefGoogle Scholar
  54. [54]
    Vetterli, M. (1988) Running FIR and IIR filtering using multirate filter banks.IEEE Trans. Signal Process.,36, 730–738.MATHCrossRefGoogle Scholar
  55. [55]
    Vetterli, M. and Le Gall, D. (1989). Perfect reconstruction FIR filter banks: Some properties and factorizations.IEEE Trans. Acoust. Speech Signal Process.,37, 1057–1071.CrossRefGoogle Scholar
  56. [56]
    Vetterli, M. and Herley, C. (1992). Wavelets and filter banks: Theory and design.IEEE Trans. Acoust. Speech Signal Process.,40(9), 2207–2232.MATHGoogle Scholar
  57. [57]
    Vetterli, M. and Kovačević, J. (1995).Wavelets and Subband Coding. Prentice Hall, Englewood Cliffs, NJ.MATHGoogle Scholar
  58. [58]
    Wang, Y., M. Orchard, M., Reibman, A., and Vaishampayan, V. (1997). Redundancy rate-distortion analysis of multiple description coding using pairwise correlation transforms. InProc. IEEE ICIP, I, 608–611.Google Scholar
  59. [59]
    Woods, J.W. and O'Neil, S.D. (1986). Subband coding of images.IEEE Trans. Acoust. Speech Signal Process. 34(5), 1278–1288.CrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston 1998

Authors and Affiliations

  • Ingrid Daubechies
    • 1
  • Wim Sweldens
    • 2
  1. 1.Program for Applied and Computational MathematicsPrinceton UniversityPrinceton
  2. 2.Bell LaboratoriesLucent TechnologiesMurray Hill

Personalised recommendations