Stereographic combing a porcupine or studies on direction diffusion in image processing

Nir A. Sochen*, Chen Sagiv, Ron Kimmel

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

9 Scopus citations


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 S1, the unit sphere S2, and the unit hypersphere 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 nontrivial geometries in image processing.

Original languageEnglish
Pages (from-to)1477-1508
Number of pages32
JournalSIAM Journal on Applied Mathematics
Issue number5
StatePublished - 2004


  • Anisotropic diffusion
  • Beltrami framework
  • Constrained optimization
  • Orientation diffusion


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