BIOMEDICAL IMAGING GROUP (BIG)
Laboratoire d'imagerie biomédicale (LIB)

Generalized L-Spline Wavelet Bases

I. Khalidov, T. Blu, M. Unser

Proceedings of the SPIE Optics and Photonics 2005 Conference on Mathematical Methods: Wavelet XI, San Diego CA, USA, July 31-August 3, 2005, vol. 5914, pp. 59140F-1–59140F-8.

We build wavelet-like functions based on a parametrized family of pseudo-differential operators Lv that satisfy some admissibility and scalability conditions. The shifts of the generalized B-splines, which are localized versions of the Green function of Lv, generate a family of L-spline spaces. These spaces have the approximation order equal to the order of the underlying operator. A sequence of embedded spaces is obtained by choosing a dyadic scale progression a = 2i. The consecutive inclusion of the spaces yields the refinement equation, where the scaling filter depends on scale. The generalized L-wavelets are then constructed as basis functions for the orthogonal complements of spline spaces. The vanishing moment property of conventional wavelets is generalized to the vanishing null space element property. In spite of the scale dependence of the filters, the wavelet decomposition can be performed using an adapted version of Mallat's filterbank algorithm. 

@INPROCEEDINGS(http://bigwww.epfl.ch/publications/khalidov0503.html,
AUTHOR="Khalidov, I. and Blu, T. and Unser, M.",
TITLE="Generalized ${\mathrm{L}}$-Spline Wavelet Bases",
BOOKTITLE="Proceedings of the {SPIE} Conference on Mathematical Imaging:
	{W}avelet {XI}",
YEAR="2005",
editor="",
volume="5914",
series="",
pages="59140F-1--59140F-8",
address="San Diego CA, USA",
month="July 31-August 3,",
organization="",
publisher="",
note="")
© 2005 SPIE. 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 SPIE. 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.