Abstract
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.
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Führ, H. (2012). Continuous Diffusion Wavelet Transforms and Scale Space over Euclidean Spaces and Noncommutative Lie Groups. In: Florack, L., Duits, R., Jongbloed, G., van Lieshout, MC., Davies, L. (eds) Mathematical Methods for Signal and Image Analysis and Representation. Computational Imaging and Vision, vol 41. Springer, London. https://doi.org/10.1007/978-1-4471-2353-8_7
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DOI: https://doi.org/10.1007/978-1-4471-2353-8_7
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