Wavelets onS³andSO(3)-Their construction, relation to each other and Radon transform of wavelets onSO(3)

Abstract

The paper at hand is concerned with creating a flexible wavelet theory on the three sphere S 3 and the rotation group SO(3). The theory of zonal functions and reproducing kernels will be used to develop conditions for an admissible wavelet. After explaining some preliminaries on group actions and some basics on approximation theory, we will prove reconstruction formulas of linear and bilinear wavelet transformed L 2 -functions on S 3 . Moreover, specific examples will be constructed and visualized. Second, we deal with the construction of wavelets on the rotation group SO(3). It will be shown that the Radon transform of a wavelet packet on SO(3) gives a wavelet packet on S 2 for every fixed detection direction.

Continuous wavelet transforms on n-dimensional spheres

“…This definition is used e.g. in [7,8,9,22,21,20,2]. However, condition (14) is necessary neither for the approximation property (16) nor for the definition of spherical wavelets.…”

Geometric Space–Frequency Analysis on Manifolds

“…Here is a very brief account of related work, mostly by other authors. The papers [3], [8], [15], [23]- [27], [41], [45], [47], [52], [62], [63], [70], [77], [84], [85], [86], [148], contain a number of results about frames, wavelets, and Besov spaces on Riemannian manifolds, on Lie groups of polynomial growth, on metric-measure spaces, and on quasi-metric measure spaces. One can say that most of these papers generalize and further develop ideas which are rooted in the classical Littlewood-Paley theory and/or Calderon reproducing formula.…”

On the uncertainty product of spherical functions

“…This concept -generalized to the n-dimensional case -was also studied by the author of the present paper. I showed in [19] that directional derivatives of some zonal wavelets satisfy a slightly modified definition of wavelets derived from approximate identities (see [11,10,9,1,8,3,2] for the origins of this concept and [18] for a comprehensive survey). In [20] I proposed further relaxation on the constraints on wavelets derived from approximate identities and showed that directional derivatives of a wide class of functions are wavelets according to the new definition.…”