Geometry, Imaging and Computing

Volume 2 (2015)

Number 3

A class of nonlocal variational problems on a vector bundle for color image local contrast reduction/enhancement

Pages: 187 – 236

DOI: https://dx.doi.org/10.4310/GIC.2015.v2.n3.a2

Authors

Thomas Batard (Department of Information and Communications Technologies, University Pompeu Fabra, Barcelona, Spain)

Marcelo Bertalmío (Department of Information and Communications Technologies, University Pompeu Fabra, Barcelona, Spain)

Abstract

We extend two existing variational models from the Euclidean space to a vector bundle over a Riemannian manifold. The Euclidean models, dedicated to regularize or enhance some color image features, are based on the concept of nonlocal gradient operator acting on a function of the Euclidean space. We then extend these models by generalizing this operator to a vector bundle over a Riemannian manifold with the help of the parallel transport map associated to some class of covariant derivatives. Through the dual formulations of the proposed models, we obtain the expressions of their solutions, which exhibit the functional spaces that describe the image features. Finally, for a well-chosen covariant derivative and its nonlocal extension, the proposed models perform local contrast modification (reduction or enhancement) and experiments show that they preserve more the aspect of the original image than the Euclidean models do while modifying equally its contrast.

Keywords

nonlocal variational model, vector bundle, covariant derivative, image contrast reduction/enhancement, primal-dual algorithm

2010 Mathematics Subject Classification

Primary 49M29, 53C05, 68U10. Secondary 90C25, 90C26.

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Published 13 May 2016