Symmetry-adapted Wavelet Analysis 1. How to Define an Adapted Wavelet Transform?

We review the construction of continuous wavelet transforms adapted to a given symmetry. Then we discuss in detail successively spatial wavelets, wavelets on the sphere and space-time wavelets. As it is well-known 1], wavelet analysis comes in two versions: the continuous one, used mostly for signal or image analysis, and the discrete one, originating from multiresolution analysis and particularly eecient in reconstruction and data compression. Now, if the signal possesses certain symmetry properties, it is natural to build these into the wavelet transform (WT) itself, and this clearly requires the use of the continuous approach. The aim of this talk is to show how a WT adapted to a given symmetry may be derived systematically from the symmetry group itself. Consider the class of nite energy signals living on a could be space IR n , the 2-sphere S 2 , space-time IR IR or IR 2 IR, etc. Such signals may be measured with a probe , that is, a linear functional over signals, which here reduces to a scalar product: s 7 ! hjsi; 2 H. Suppose there is a group G of transformations acting (transitively) on Y. Then we may let it act linearly either on signals, s 7 ! U (g)s, that is, one evaluates hjU(g)si; g 2 G (active point of view), or on probes, measuring instead hU(g)jsi (passive point of view). In other words, U should be a unitary representation of G in the space H of signals. In order to get a wavelet analysis on Y , adapted to the symmetry group G, three conditions must be met: (1) G contains dilations of some kind. one nonzero vector 2 H (called admissible) such that Z G jhU(g)jsij 2 dg < 1; 8s 2 H: (1.2) If H denotes the isotropy subgroup of (up to a phase) and ? = G=H carries a G-invariant measure , condition (1.2) may be replaced by the following one: Z ? jhU(g)jsij 2 dd() < 1; 8s 2 H (gH); (1.3) since the integrand does not really depend on g, but only on its left coset gH. Equivalently, one may replace in (1.3) U (g) by U (()), where : ? = G=H ! G is an arbitrary section (indeed the inte-grand does not depend on the choice of section). Under these three conditions, a G-adapted wavelet analysis on Y may be constructed as follows 2]. Choose a …