Department of Mathematics
 Search | Help | Login | pdf version | printable version

Math @ Duke



Publications [#264926] of Guillermo Sapiro

Papers Published

  1. Tang, B; Sapiro, G; Caselles, V, Diffusion of general data on non-flat manifolds via harmonic maps theory: The direction diffusion case, International Journal of Computer Vision, vol. 36 no. 2 (2000), pp. 149-161 [doi]
    (last updated on 2019/06/26)

    In a number of disciplines, directional data provides a fundamental source of information. A novel framework for isotropic and anisotropic diffusion of directions is presented in this paper. The framework can be applied both to denoise directional data and to obtain multiscale representations of it. The basic idea is to apply and extend results from the theory of harmonic maps, and in particular, harmonic maps in liquid crystals. This theory deals with the regularization of vectorial data, while satisfying the intrinsic unit norm constraint of directional data. We show the corresponding variational and partial differential equations formulations for isotropic diffusion, obtained from an L2 norm, and edge preserving diffusion, obtained from an Lp norm in general and an L1 norm in particular. In contrast with previous approaches, the framework is valid for directions in any dimensions, supports non-smooth data, and gives both isotropic and anisotropic formulations. In addition, the framework of harmonic maps here described can be used to diffuse and analyze general image data defined on general non-flat manifolds, that is, functions between two general manifolds. We present a number of theoretical results, open questions, and examples for gradient vectors, optical flow, and color images.
ph: 919.660.2800
fax: 919.660.2821

Mathematics Department
Duke University, Box 90320
Durham, NC 27708-0320