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A chromaticity-brightness model for color images denoising in a Meyer’s “u + v” framework

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 \(\varepsilon \rightarrow 0^+\) 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.

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Notes

  1. 1.

    We refer to Sect. 2.3 for the notation concerning BV functions.

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Acknowledgements

The authors acknowledge the funding of Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the ICTI CMU-Portugal Program in Applied Mathematics and UTACMU/MAT/0005/2009. The authors also thank the Center for Nonlinear Analysis (NSF Grant DMS-0635983), where part of this research was carried out. R. Ferreira was partially supported by the KAUST SRI, Center for Uncertainty Quantification in Computational Science and Engineering and by the Fundação para a Ciência e a Tecnologia through the grant SFRH/BPD/81442/2011. The work of I. Fonseca was partially supported by the National Science Foundation under Grant No. DMS-1411646. The work of L.M. Mascarenhas was partially supported by UID/MAT/00297/ 2013.

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Correspondence to Irene Fonseca.

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Communicated by L. Ambrosio.

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Ferreira, R., Fonseca, I. & Mascarenhas, M.L. A chromaticity-brightness model for color images denoising in a Meyer’s “u + v” framework. Calc. Var. 56, 140 (2017). https://doi.org/10.1007/s00526-017-1223-8

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Mathematics Subject Classification

  • 49J45
  • 26B30
  • 94A08