Osiris Wavelets in Three Dimensions

✍ Scribed by Guy Battle

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 690 KB
• Volume: 286
• Category: Article
• ISSN: 0003-4916
• DOI: 10.1006/aphy.2000.6081

✦ Synopsis

We extend to three dimensions an unusual wavelet construction that we first introduced in a two-dimensional context (in ``Wavelet Transforms and Time Frequency Signal Analysis'' (L. Debnath, Ed.), Birkha user, Basel, in press). This is part of an on-going program to analyze critical behavior in classical equilibrium statistical mechanics. Following Golner's general idea of using an incomplete multiscale set of functions (1973, Phys. Rev. B 8, 339) to obtain more realistic modeling that is still hierarchical, we introduce a wavelet set whose mother wavelets are continuous, piecewise-linear functions supported in the unit cube. Such a wavelet set is necessarily incomplete, but in three dimensions we have packed seven mother wavelets into the unit cube four based on the 8 sub-cubes, and three based on the 12 octahedra that intersect adjacent sub-cubes. The generated wavelet set is not Sobolevorthogonal, but we derive a positive lower bound on the multi-scale Sobolev overlap matrix.