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 -