Continuous Diffusion Wavelet Transforms and Scale Space over Euclidean Spaces and Noncommutative Lie Groups

Scale space representations over Euclidean space are closely related to continuous wavelet transforms. E.g., taking the time derivative of a Gaussian scale space representation yields a continuous wavelet transform with respect to the Mexican Hat wavelet. Departing from this observation, we introduce a continuous diffusion wavelet transform associated to any diffusion semigroup acting on an abstract Hilbert space. Under quite general assumptions this wavelet transform is norm-preserving, thus giving rise to wavelet-type inversion formulae. For the diffusion associated to the heat equation on the Heisenberg group ℍ, we recover the so-called Mexican Hat wavelet previously obtained by Mayeli. Our approach provides an alternative short proof of admissibility for the Mexican Hat wavelet on ℍ, as well as a new family of wavelets for ℍ and more general homogeneous Lie groups.