Symmetry-adapted wavelet analysis

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. 1. HOW TO DEFINE AN ADAPTED WAVELET TRANSFORM? 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 efficient 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 finite energy signals living on a manifold Y , i.e. s ∈ L(Y, dμ) ≡ H. For instance, Y could be space IR, the 2-sphere S, space-time IR× IR or IR × IR, etc. Such signals may be measured with a probe ψ, that is, a linear functional over signals, which here reduces to a scalar product: s 7→ 〈ψ|s〉, ψ ∈ 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 〈ψ|U(g)s〉, g ∈ G (active point of view), or on probes, measuring instead 〈U(g)ψ|s〉 (passive point of view). By the global invariance of Y , the two should be equivalent, i.e. 〈U(g)ψ|U(g)s〉 = 〈ψ|s〉, ∀ g ∈ G. (1.1) 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. (2) U is irreducible. (3) U is square integrable, i.e. there exists at least one nonzero vector ψ ∈ H (called admissible) such that ∫ G |〈U(g)ψ|s〉| dg <∞, ∀s ∈ H. (1.2) If Hψ denotes the isotropy subgroup of ψ (up to a phase) and Γ = G/Hψ carries a G-invariant measure ν, condition (1.2) may be replaced by the following one: ∫ Γ |〈U(g)ψ|s〉| dν(γ) <∞, ∀s ∈ H (γ ≡ gHψ), (1.3) since the integrand does not really depend on g, but only on its left coset γ ≡ gHψ. Equivalently, one may replace in (1.3) U(g) by U(σ(γ)), where σ : Γ = G/Hψ → G is an arbitrary section (indeed the integrand 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 fixed admissible vector ψ ∈ H (the analyzing wavelet) and a section σ : Γ → G. Then the wavelets are the vectors ψγ = U(σ(γ))ψ ∈ H (γ ∈ Γ), and the corresponding continuous wavelet transform (CWT) is defined as the linear map Wψ : H → L(Γ, dν) given by (Wψs)(γ) ≡ Sψ(γ) = 〈ψγ |s〉. The CWT has the following properties: (i) Energy conservation: ∫ Γ |Sψ(γ)| dν(γ) = ‖s‖H ≡ ∫ Y |s(y)| dμ(y), (1.4) i.e. Wψ is an isometry; hence its range, that is, the space of wavelet transforms, is a closed subspace Hψ of L(Γ, dν); (ii) By (i), Wψ may be inverted on its range by the transposed map, which gives the reconstruction formula: