A class of generalized Laplacians on vector bundles devoted to multi-channel image processing

Thomas Batard*, Nir Sochen

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


In the context of fibre bundles theory, there exist some differential operators of order 2, called generalized Laplacians, acting on sections of vector bundles over Riemannian manifolds, and generalizing the Laplace-Beltrami operator. Such operators are determined by covariant derivatives on vector bundles. In this paper, we construct a class of generalized Laplacians, devoted to multi-channel image processing, from the construction of optimal covariant derivatives. The key idea is to consider an image as a section of an associate bundle, that is a vector bundle related to a principal bundle through a group representation. In this context, covariant derivatives are determined by connection 1-forms on principal bundles. We construct optimal connection 1-forms by the minimization of a variational problem on principal bundles. From the heat equations of the generalized Laplacians induced by the corresponding optimal covariant derivatives, we obtain diffusions whose behaviors depend of the choice of the group representation. We provide experiments on color images.

Original languageEnglish
Pages (from-to)517-543
Number of pages27
JournalJournal of Mathematical Imaging and Vision
Issue number3
StatePublished - Mar 2014


  • Differential geometry
  • Generalized Laplacian
  • Heat equation
  • Multi-channel image processing
  • Variational methods


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