StereographicCombing a Porcupine or Studies on Direction Diffusion in Image Processing
Chen Sagiv, Nir Sochen, Ron Kimmel · SIAM Journal on Applied Mathematics · 2004
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 structures of the fiber spaces of interest in this paper are the unit circle S 1 , the unit sphere S 2 , and the unit hypersphere S n . 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 nontrivial geometries in image processing.