StereographicCombing a Porcupine or Studies on Direction Diffusion in Image Processing

This paper addresses the problem of feature enhancement in noisy images, when the feature is known to be constrained to a manifold. As an example, we approach the direction denoising problem in a general dimension via the geometric Beltrami framework for image processing. The spatial-direction space is a fiber bundle in which the spatial part is the base manifold and the direction space is the fiber. The feature (direction) field is represented accordingly as a section of the spatial-feature fiber bundle. The resulting Beltrami flow is a selective smoothing process that respects the bundle’s structure i.e. the feature constraint. Direction diffusion is treated as a canonical example of a non-Euclidean feature space. The structure of the fiber spaces of interest in this paper is the unit circle S1, the unit sphere S2, and the unit hyper-sphere Sn. Applications to color analysis are discussed and numerical experiments demonstrate again the benefits of the Beltrami framework in comparison to other feature enhancement schemes for non-trivial geometries in image processing.