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 Families
  • 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

Families of Multiresolution and Wavelet Spaces with Optimal Properties

A. Aldroubi, M. Unser

Numerical Functional Analysis and Optimization, vol. 14, no. 5/6, pp. 417-446, October-December 1993.


Under suitable conditions, if the sampling functions φ1 and φ2 generate the multiresolutions V(j)(φ1) and V(j)(φ2), then their convolution φ1*φ2 also generates a multiresolution V(j)(φ1*φ2). Moreover, if p is an appropriate convolution operator from ℓ2 into itself and if φ is a scaling function generating the multiresolution V(j)(φ), then p*φ is a scaling function generating the same multiresolution V(j)(φ) = V(j)(p*φ). Using these two properties, we group the scaling and wavelet functions into equivalent classes and consider various equivalent basis functions of the associated function spaces. We use the n-fold convolution product to construct sequences of multiresolution and wavelet spaces V(j)(φn) and W(j)(φn) with increasing regularity. We discuss the link between multiresolution analysis and Shannon's sampling theory. We then show that the interpolating and orthogonal pre- and post-filters associated with the multiresolution sequence V(0)(φn) asymptotically converge to the ideal lowpass filter of Shannon. We also prove that the filters associated with the sequence of wavelet spaces W(0)(φn) converge to the ideal bandpass filter. Finally, we construct the basic wavelet sequences ψbn and show that they tend to Gabor functions. This provides wavelets that are nearly time-frequency optimal. The theory is illustrated with the example of polynomial splines.

@ARTICLE(http://bigwww.epfl.ch/publications/aldroubi9302.html,
AUTHOR="Aldroubi, A. and Unser, M.",
TITLE="Families of Multiresolution and Wavelet Spaces with Optimal
	Properties",
JOURNAL="Numerical Functional Analysis and Optimization",
YEAR="1993",
volume="14",
number="4/5",
pages="417--446",
month="October-December",
note="")

© 1993 Informa Ltd.. 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 Informa Ltd.. 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

© 2023 EPFL, all rights reserved