Wavelet transform on manifolds: Old and new approaches

Abstract

Given a two-dimensional smooth manifold M and a bijective projection p from M on a fixed plane (or a subset of that plane), we explore systematically how a wavelet transform (WT) on M may be generated from a plane WT by the inverse projection p −1 . Examples where the projection maps the whole manifold onto a plane include the two-sphere, the upper sheet of the two-sheeted hyperboloid and the paraboloid. When no such global projection is available, the construction may be performed locally, i.e., around a given point on M. We apply this procedure both to the Continuous WT, already treated in the literature, and to the Discrete WT. Finally, we discuss the case of a WT on a graph, for instance, the graph defined by linking the elements of a discrete set of points on the manifold.

Unified heat kernel regression for diffusion, kernel smoothing and wavelets on manifolds and its application to mandible growth modeling in CT images

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“…Furthermore, such basis functions are only orthonormal for data defined on the sphere and result in a less parsimonious representation for data defined on other surfaces compared to the intrinsic LB-eigenfunction expansion (Seo and Chung, 2011). To remedy the limitations of spherical wavelets, the diffusion wavelet transform on graph data structures has been proposed (Antoine et al, 2010; Coifman and Maggioni, 2006; Hammond et al, 2011; Kim et al, 2012). …”

Continuous and discrete Mexican hat wavelet transforms on manifolds

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Wavelet analysis on developable surface base on area preserving projection

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