Angular channels in a multidimensional wavelet transform

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Abstract

Given a subgroup S of GL (n), let G be the semidirect product of S with ℝn. The wavelet transform is defined for functions in L2 (ℝn) by using the action of G on this space. The standard properties of the wavelet transform and its inverse are quickly and easily derived in this formalism. In particular, the admissibility condition for the wavelet is expressed in terms of an integral over S. The notion of orthogonal wavelet channels is defined, and the wavelet transform is decomposed in terms of them. Other operators on L2 (ℝn) can also be analyzed in terms of their mixing of wavelet channels. For n = 2 and n = 3, details are given for the expansion of an arbitrary wavelet transform in terms of angular wavelet channels. An example is provided for n = 2. The correspondence between angular channels and the spherical harmonic decomposition of the Fourier transform of the wavelet transform is also outlined.

Original languageEnglish (US)
Pages (from-to)80-102
Number of pages23
JournalSIAM Journal on Mathematical Analysis
Volume32
Issue number1
Publication statusPublished - 2000

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Keywords

  • Channel models
  • Orthogonal functions
  • Signal reconstruction
  • Spherical harmonics
  • Unitary representations of locally compact groups
  • Wavelet transform

ASJC Scopus subject areas

  • Mathematics(all)
  • Analysis
  • Applied Mathematics

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