TY - JOUR
T1 - A class of generalized Laplacians on vector bundles devoted to multi-channel image processing
AU - Batard, Thomas
AU - Sochen, Nir
PY - 2014/3
Y1 - 2014/3
N2 - 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.
AB - 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.
KW - Differential geometry
KW - Generalized Laplacian
KW - Heat equation
KW - Multi-channel image processing
KW - Variational methods
UR - http://www.scopus.com/inward/record.url?scp=84899459938&partnerID=8YFLogxK
U2 - 10.1007/s10851-013-0426-7
DO - 10.1007/s10851-013-0426-7
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AN - SCOPUS:84899459938
SN - 0924-9907
VL - 48
SP - 517
EP - 543
JO - Journal of Mathematical Imaging and Vision
JF - Journal of Mathematical Imaging and Vision
IS - 3
ER -