Direction diffusion

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 regularize directional data and to obtain multiscale representations of it. The basic idea is to apply and extend results from the theory of harmonic maps in liquid crystals. This theory deals with the regularization of vectorial data, while satisfying the 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 L₁ norm. 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. We present a number of theoretical results, open questions, and examples for gradient vectors, optical flow, and color images.