Skip to main content
SpringerLink
Log in

Comparison of Discrete and Continuous Wavelet Transforms

  • Living reference work entry
  • First Online:
Encyclopedia of Complexity and Systems Science

Our purpose is to outline a number of direct links between the two cases of wavelet analysis: continuous and discrete. The theme of the first is perhaps best known, for example, the creation of compactly supported wavelets in L 2(ℝn) with suitable properties such as localization, vanishing moments, and differentiability. The second (discrete) deals with computation, with sparse matrices, and with algorithms for encoding digitized information such as speech and images. This is centered on constructive approaches to subdivision filters, their matrix representation (by sparse matrices), and corresponding fast algorithms. For both approaches, we outline computational transforms; but our emphasis is on effective and direct links between computational analysis of discrete filters on the one side and on continuous wavelets on the other. By the latter, we include both L 2(ℝn) analysis and fractal analysis. To facilitate the discussion of the interplay between discrete (used by engineers) and...

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

Access this chapter

Log in via an institution

Institutional subscriptions

Bibliography

  • Aubert G, Kornprobst P (2006) Mathematical problems in image processing. Springer, New York

    MATH  Google Scholar 

  • Baggett L, Jorgensen P, Merrill K, Packer J (2005) A non-MRA Cr frame wavelet with rapid decay. Acta Appl Math 89:251–270

    Article  MATH  MathSciNet  Google Scholar 

  • Baladi V (2000) Positive transfer operators and decay of correlations, vol 16, Advanced series in nonlinear dynamics. World Scientific, River Edge

    MATH  Google Scholar 

  • Bratelli O, Jorgensen P (2002) Wavelets through a looking glass: the world of the spectrum. Birkhäuser, Boston

    Book  Google Scholar 

  • Braverman M (2006) Parabolic Julia sets are polynomial time computable. Nonlinearity 19(6):1383–1401

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Braverman M, Yampolsky M (2006) Non-computable Julia sets. J Am Math Soc 19(3):551–578 (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  • Bredies K, Lorenz DA, Maass P (2006) An optimal control problem in medical image processing

    Google Scholar 

  • Daubechies I (1992) Ten lectures on wavelets, vol 61, CBMS-NSF regional conference series in applied mathematics. Society for Industrial and Applied Mathematics, Philadelphia

    Book  MATH  Google Scholar 

  • Daubechies I (1993) Wavelet transforms and orthonormal wavelet bases. Proc Sympos Appl Math

    Google Scholar 

  • Devaney RL, Look DM (2006) A criterion for Sierpinski curve Julia sets. Topol Proc 30(1):163–179, Spring topology and dynamical systems conference

    MATH  MathSciNet  Google Scholar 

  • Devaney RL, Rocha MM, Siegmund S (2007) Rational maps with generalized Sierpinski gasket Julia sets. Topol Appl 154(1):11–27

    Article  MATH  MathSciNet  Google Scholar 

  • Dutkay DE (2004) The spectrum of the wavelet Galerkin operator. Integral Equ Oper Theory 50:477–487

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Dutkay DE, Jorgensen PET (2005) Wavelet constructions in non-linear dynamics. Electron Res Announc Am Math Soc 11:21–23

    Article  MATH  MathSciNet  Google Scholar 

  • Dutkay DE, Jorgensen PET (2006a) Wavelets on fractals. Rev Mat Iberoamericana 22:131–180

    Article  MATH  MathSciNet  Google Scholar 

  • Dutkay DE, Jorgensen PET (2006b) Hilbert spaces built on a similarity and on dynamical renormalization. J Math Phys 47:053504

    Article  ADS  MathSciNet  Google Scholar 

  • Dutkay DE, Jorgensen PET (2006c) Iterated function systems, Ruelle operators, and invariant projective measures. Math Comput 75:1931

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • DE Dutkay, K Roysland (2007) The algebra of harmonic functions for a matrix-valued transfer operator. arXiv:math/0611539

    Google Scholar 

  • Dutkay DE, Roysland K (2007) Covariant representations for matrix-valued transfer operators. arXiv:math/0701453

    Google Scholar 

  • Dutkay DE, Picioroaga G, M-S Song (2012) Orthonormal bases generated by Cuntz algebras. arXiv:1212.4134

    Google Scholar 

  • Heil C, Walnut DF (eds) (2006) Fundamental papers in wavelet theory. Princeton University Press, Princeton

    MATH  Google Scholar 

  • Hutchinson JE (1981) Fractals and self-similarity. Indiana Univ Math J 30(5):713–747

    Article  MATH  MathSciNet  Google Scholar 

  • Daubechies I, Lagarias JC (1992) Two-scale difference equations. II. Local regularity, infinite products of matrices and fractals. SIAM J Math Anal

    Google Scholar 

  • Jorgensen PET (2003) Matrix factorizations, algorithms, wavelets. Not Am Math Soc 50:880–895

    MATH  Google Scholar 

  • Jorgensen PET (2006a) Analysis and probability: wavelets, signals, fractals, vol 234, Graduate texts in mathematics. Springer, New York

    Google Scholar 

  • Jorgensen T (2006b) Certain representations of the Cuntz relations, and a question on wavelets decompositions. Contemp Math 414:165–188

    Article  Google Scholar 

  • Liu F (2006) Diffusion filtering in image processing based on wavelet transform. Sci China Ser F 49:1–25

    Article  MathSciNet  Google Scholar 

  • Milnor J (2004) Pasting together Julia sets: a worked out example of mating. Exp Math 13(1):55–92

    Article  MATH  MathSciNet  Google Scholar 

  • Petersen CL, Zakeri S (2004) On the Julia set of a typical quadratic polynomial with a Siegel disk. Ann Math 159(1):1–52

    Article  MATH  MathSciNet  Google Scholar 

  • Schipp F, Wade WR, Simon P (1990) Walsh series. Adam Hilger Ltd., Bristol. An introduction to dyadic harmonic analysis, With the collaboration of J. Pál

    Google Scholar 

  • Skodras A, Christopoulos C, Ebrahimi T (2001) Jpeg 2000 still image compression standard. IEEE Signal Process Mag 18:36–58

    Article  ADS  Google Scholar 

  • Song M-S (2006) Wavelet image compression. PhD thesis, The University of Iowa

    Google Scholar 

  • Song M-S (2006b) Wavelet image compression. In: Operator theory, operator algebras, and applications, vol 414, Contemporary mathematics. American Mathematical Society, Providence, pp 41–73

    Chapter  Google Scholar 

  • Strang G (1997) Wavelets from filter banks. Springer, New York

    Google Scholar 

  • Strang G (2000) Signal processing for everyone. Lecture notes in mathematics, Springer, vol 1739

    Google Scholar 

  • Strang G, Nguyen T (1996) Wavelets and filter banks. Wellesley-Cambridge Press, Wellesley

    MATH  Google Scholar 

  • Usevitch BE (2001) A tutorial on modern lossy wavelet image compression: foundations of JPEG 2000. IEEE Signal Process Mag 18:22–35

    Article  ADS  Google Scholar 

  • Walker JS (1999) A primer on wavelets and their scientific applications. Chapman & Hall/CRC, Boca Raton

    Book  MATH  Google Scholar 

Download references

Acknowledgments

We thank Professors Dorin Dutkay, Gabriel Picioroaga, and Judy Packer for the helpful discussions.

Author information

Authors and Affiliations

  1. Department of Mathematics, The University of Iowa, 14 MacLean Hall, 52242, Iowa City, IA, USA

    Palle E. T. Jorgensen

  2. Department of Mathematics and Statistics, Southern Illinois University, Science Building, Box 1653, 62026, Edwardsville, IL, USA

    Myung-Sin Song

Authors
  1. Palle E. T. Jorgensen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Myung-Sin Song
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Palle E. T. Jorgensen .

Editor information

Editors and Affiliations

  1. Ramtech Ltd., Larkspur, California, USA

    Robert A. Meyers

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this entry

Cite this entry

Jorgensen, P.E.T., Song, MS. (2013). Comparison of Discrete and Continuous Wavelet Transforms. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-3-642-27737-5_77-2

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-27737-5_77-2

  • Received:

  • Accepted:

  • Published:

  • Publisher Name: Springer, New York, NY

  • Online ISBN: 978-3-642-27737-5

  • eBook Packages: Springer Reference Physics and AstronomyReference Module Physical and Materials ScienceReference Module Chemistry, Materials and Physics

Publish with us

Policies and ethics

Search

Navigation