Biomedical Imaging Group
Logo EPFL
    • Splines Tutorials
    • Splines Art Gallery
    • Wavelets Tutorials
    • Image denoising
    • ERC project: FUN-SP
    • Sparse Processes - Book Preview
    • ERC project: GlobalBioIm
    • The colored revolution of bioimaging
    • Deconvolution
    • SMLM
    • One-World Seminars: Representer theorems
    • A Unifying Representer Theorem
Follow us on Twitter.
Join our Github.
Masquer le formulaire de recherche
Menu
BIOMEDICAL IMAGING GROUP (BIG)
Laboratoire d'imagerie biomédicale (LIB)
  1. School of Engineering STI
  2. Institute IEM
  3.  LIB
  4.  Wavelet Theory Demystified
  • Laboratory
    • Laboratory
    • Laboratory
    • People
    • Jobs and Trainees
    • News
    • Events
    • Seminars
    • Resources (intranet)
    • Twitter
  • Research
    • Research
    • Researchs
    • Research Topics
    • Talks, Tutorials, and Reviews
  • Publications
    • Publications
    • Publications
    • Database of Publications
    • Talks, Tutorials, and Reviews
    • EPFL Infoscience
  • Code
    • Code
    • Code
    • Demos
    • Download Algorithms
    • Github
  • Teaching
    • Teaching
    • Teaching
    • Courses
    • Student projects
  • Splines
    • Teaching
    • Teaching
    • Splines Tutorials
    • Splines Art Gallery
    • Wavelets Tutorials
    • Image denoising
  • Sparsity
    • Teaching
    • Teaching
    • ERC project: FUN-SP
    • Sparse Processes - Book Preview
  • Imaging
    • Teaching
    • Teaching
    • ERC project: GlobalBioIm
    • The colored revolution of bioimaging
    • Deconvolution
    • SMLM
  • Machine Learning
    • Teaching
    • Teaching
    • One-World Seminars: Representer theorems
    • A Unifying Representer Theorem

Wavelet Theory Demystified

M. Unser, T. Blu

IEEE Transactions on Signal Processing, vol. 51, no. 2, pp. 470-483, February 2003.


In this paper, we revisit wavelet theory starting from the representation of a scaling function as the convolution of a B-spline (the regular part of it) and a distribution (the irregular or residual part). This formulation leads to some new insights on wavelets and makes it possible to rederive the main results of the classical theory—including some new extensions for fractional orders—in a self-contained, accessible fashion. In particular, we prove that the B-spline component is entirely responsible for five key wavelet properties: order of approximation, reproduction of polynomials, vanishing moments, multiscale differentiation property, and smoothness (regularity) of the basis functions. We also investigate the interaction of wavelets with differential operators giving explicit time domain formulas for the fractional derivatives of the basis functions. This allows us to specify a corresponding dual wavelet basis and helps us understand why the wavelet transform provides a stable characterization of the derivatives of a signal. Additional results include a new peeling theory of smoothness, leading to the extended notion of wavelet differentiability in the Lp-sense and a sharper theorem stating that smoothness implies order.

@ARTICLE(http://bigwww.epfl.ch/publications/unser0301.html,
AUTHOR="Unser, M. and Blu, T.",
TITLE="Wavelet Theory Demystified",
JOURNAL="{IEEE} Transactions on Signal Processing",
YEAR="2003",
volume="51",
number="2",
pages="470--483",
month="February",
note="")

© 2003 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from IEEE. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.
  • Laboratory
  • Research
  • Publications
    • Database of Publications
    • Talks, Tutorials, and Reviews
    • EPFL Infoscience
  • Code
  • Teaching
Logo EPFL, Ecole polytechnique fédérale de Lausanne
Emergencies: +41 21 693 3000 Services and resources Contact Map Webmaster email

Follow EPFL on social media

Follow us on Facebook. Follow us on Twitter. Follow us on Instagram. Follow us on Youtube. Follow us on LinkedIn.
Accessibility Disclaimer Privacy policy

© 2022 EPFL, all rights reserved