Accepted

Inserted: 7 dec 2017
Last Updated: 7 dec 2017

Journal: Calc. Var. Partial Differential Equations
Volume: 56
Year: 2017
Doi: https://doi.org/10.1007/s00526-017-1223-8

Abstract:

A variational model for imaging segmentation and denoising color images is proposed. The model combines Meyer's ``u+v" decomposition with a chromaticity-brightness framework and is expressed by a minimization of energy integral functionals depending on a small parameter $\varepsilon >0$. The asymptotic behavior as $\varepsilon\to0^+$ is characterized, and convergence of infima, almost minimizers, and energies are established. In particular, an integral representation of the lower semicontinuous envelope, with respect to the $L^1$-norm, of functionals with linear growth and defined for maps taking values on a certain compact manifold is provided. This study escapes the realm of previous results since the underlying manifold has boundary, and the integrand and its recession function fail to satisfy hypotheses commonly assumed in the literature. The main tools are $\Gamma$-convergence and relaxation techniques.

Keywords: relaxation, $\Gamma$-convergence, $BV$ functions, imaging denoising, color images, Meyer's $G$-norm, chromaticity-brightness decomposition, manifold constraints