Diffusion of General Data on Non-Flat Manifolds viaHarmonic Maps Theory: The Direction Diffusion Case

  • Authors:

  • Bei Tang;Guillermo Sapiro;Vicent Caselles

  • Affiliations:

  • Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455, USA&semi/ Escola Superior Politecnica, Universitat Pompeu Fabra, 08002 Barcelona, Spain;Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455, USA&semi/ Escola Superior Politecnica, Universitat Pompeu Fabra, 08002 Barcelona, Spain. guille@ece.umn.edu;Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455, USA&semi/ Escola Superior Politecnica, Universitat Pompeu Fabra, 08002 Barcelona, Spain

  • Venue:
  • International Journal of Computer Vision
  • Year:
  • 2000

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Abstract

In a number of disciplines, directional dataprovides a fundamental source of information. A novel framework forisotropic and anisotropic diffusion of directions is presented inthis paper. The framework can be applied both to denoise directionaldata and to obtain multiscale representations of it. The basic ideais to apply and extend results from the theory of harmonic maps, andin particular, harmonic maps in liquid crystals. This theory dealswith the regularization of vectorial data, while satisfying theintrinsic unit norm constraint of directional data. We show thecorresponding variational and partial differential equationsformulations for isotropic diffusion, obtained from an L_2norm, and edge preserving diffusion, obtained from an Lnorm in generaland an L_1 norm in particular. In contrast with previousapproaches, the framework is valid for directions in any dimensions,supports non-smooth data, and gives both isotropic and anisotropicformulations. In addition, the framework of harmonic maps heredescribed can be used to diffuse and analyze general image datadefined on general non-flat manifolds, that is, functions between twogeneral manifolds. We present a number of theoretical results, openquestions, and examples for gradient vectors, optical flow, and colorimages.