Wavelets on Manifolds

  • Syed Twareque Ali
  • Jean-Pierre Antoine
  • Jean-Pierre Gazeau
Chapter
Part of the Theoretical and Mathematical Physics book series (TMP)

Abstract

In this chapter, we discuss the construction of wavelets related to other groups than similitude groups. The first, and most important, case is that of wavelets on the two-sphere \({\mathbb{S}}^{2}\). We start with the continuous approach, based on the use of stereographic dilations, i.e., dilations obtained by lifting to \({\mathbb{S}}^{2}\) ordinary dilations on a tangent plane by an inverse stereographic projection. Next we describe briefly a number of techniques for obtaining discrete wavelets on \({\mathbb{S}}^{2}\). Then we extend the analysis to wavelets on other manifolds, such as conic sections, a torus, general surfaces of revolution or graphs.

Keywords

Microwave Manifold Covariance Convolution Radon 

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Syed Twareque Ali
    • 1
  • Jean-Pierre Antoine
    • 2
  • Jean-Pierre Gazeau
    • 3
    • 4
  1. 1.Department of Mathematics and StatisticsConcordia UniversityMontréalCanada
  2. 2.Institut de Recherche en Mathématique et Physique (IRMP)Université Catholique de LouvainLouvain-la-NeuveBelgium
  3. 3.Astroparticules et Cosmologie (APC, UMR 7164)Université Paris DiderotSorbonne Paris Cité ParisFrance
  4. 4.Centro Brasileiro de Pesquisas Fisicas (CBPF)Rio de JaneiroBrasil

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