English

Real Clifford Algebra Cl(n,0), n=2,3(mod 4) Wavelet Transform

Rings and Algebras 2013-06-10 v1 Representation Theory

Abstract

We show how for n=2,3(mod4)n=2,3 (\mod 4) continuous Clifford (geometric) algebra (GA) ClnCl_n-valued admissible wavelets can be constructed using the similitude group SIM(n)SIM(n). We strictly aim for real geometric interpretation, and replace the imaginary unit i\Ci \in \C therefore with a GA blade squaring to 1-1. Consequences due to non-commutativity arise. We express the admissibility condition in terms of a ClnCl_{n} Clifford Fourier Transform and then derive a set of important properties such as dilation, translation and rotation covariance, a reproducing kernel, and show how to invert the Clifford wavelet transform. As an example, we introduce Clifford Gabor wavelets. We further invent a generalized Clifford wavelet uncertainty principle.

Cite

@article{arxiv.1306.1615,
  title  = {Real Clifford Algebra Cl(n,0), n=2,3(mod 4) Wavelet Transform},
  author = {Eckhard Hitzer},
  journal= {arXiv preprint arXiv:1306.1615},
  year   = {2013}
}

Comments

4 pages

R2 v1 2026-06-22T00:29:39.542Z