We build wavelet-like functions based on a parametrized family of pseudo-differential operators L_v^→ that satisfy some admissibility and scalability conditions. The shifts of the generalized B-splines, which are localized versions of the Green function of L_v^→, 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.

Date Published: 17 September 2005
PDF: 8 pages
Proc. SPIE 5914, Wavelets XI, 59140F (17 September 2005); doi: 10.1117/12.617123

Published in SPIE Proceedings Vol. 5914:
Wavelets XI
Manos Papadakis; Andrew F. Laine; Michael A. Unser, Editor(s)