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

  • Rita Ferreira
  • Irene Fonseca
  • M. Luísa Mascarenhas
Article
  • 115 Downloads

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.

Mathematics Subject Classification

49J45 26B30 94A08 

Notes

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.

References

  1. 1.
    Alberti, G.: Rank one property for derivatives of functions with bounded variation. Proc. R. Soc. Edinb. Sect. A 123(2), 239–274 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alicandro, R., Esposito, A.C., Leone, C.: Relaxation in BV of integral functionals defined on Sobolev functions with values in the unit sphere. J. Convex Anal. 14(1), 69–98 (2007)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Ambrosio, L., Dal Maso, G.: On the relaxation in \({\text{ BV }}(\Omega; {R}^m)\) of quasi-convex integrals. J. Funct. Anal. 109(1), 76–97 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. Oxford University Press, New York (2000)zbMATHGoogle Scholar
  5. 5.
    Ambrosio, L., Mortola, S., Tortorelli, V.M.: Functionals with linear growth defined on vector valued BV functions. J. Math. Pures Appl. (9) 70(3), 269–323 (1991)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Aubert, G., Aujol, J.-F.: Modeling very oscillating signals. Application to image processing. Appl. Math. Optim. 51(2), 163–182 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing. Partial Differential Equations and the Calculus of Variations. Foreword by Olivier Faugeras, 2nd edn. Springer, New York (2006)zbMATHGoogle Scholar
  8. 8.
    Aujol, J.-F., Aubert, G., Blanc-Féraud, L., Chambolle, A.: Image decomposition into a bounded variation component and an oscillating component. J. Math. Imaging Vis. 22(1), 71–88 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Babadjian, J.-F., Millot, V.: Homogenization of variational problems in manifold valued \(BV\)-spaces. Calc. Var. Partial Differ. Equ. 36(1), 7–47 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Babadjian, J.-F., Millot, V.: Homogenization of variational problems in manifold valued Sobolev spaces. ESAIM Control Optim. Calc. Var. 16(4), 833–855 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bethuel, F.: The approximation problem for Sobolev maps between two manifolds. Acta Math. 167(3–4), 153–206 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bethuel, F., Zheng, X.M.: Density of smooth functions between two manifolds in Sobolev spaces. J. Funct. Anal. 80(1), 60–75 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Brézis, H., Coron, J.-M., Lieb, E.H.: Estimations d’énergie pour des applications de \({{\bf R}}^3\) à valeurs dans \(S^2\). C. R. Acad. Sci. Paris Sér. I Math. 303(5), 207–210 (1986)MathSciNetGoogle Scholar
  14. 14.
    Carita, G., Fonseca, I., Leoni, G.: Relaxation in \(\text{ SBV }_{p } (\varOmega ; S^{d-1})\). Calc. Var. Partial Differ. Equ. 42(1–2), 211–255 (2011)CrossRefzbMATHGoogle Scholar
  15. 15.
    Chan, T., Marquina, A., Mulet, P.: High-order total variation-based image restoration. SIAM J. Sci. Comput. 22(2), 503–516 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Chan, T.F., Esedoglu, S., Park, F., Yip, A.M.: Total variation image restoration: overview and recent developments. In: Handbook of Mathematical Models in Computer Vision, pp. 17–31. Springer, New York (2006)Google Scholar
  17. 17.
    Chan, T.F., Kang, S.H., Shen, J.: Total variation denoising and enhancement of color images based on the CB and HSV color models. J. Vis. Commun. Image Represent. 12(4), 422–435 (2001)CrossRefGoogle Scholar
  18. 18.
    Dacorogna, B., Fonseca, I., Malý, J., Trivisa, K.: Manifold constrained variational problems. Calc. Var. Partial Differ. Equ. 9(3), 185–206 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Dal Maso, G., Fonseca, I., Leoni, G., Morini, M.: A higher order model for image restoration: the one-dimensional case. SIAM J. Math. Anal. 40(6), 2351–2391 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Dal Maso, G.: An Introduction to \(\Gamma \)-Convergence. Progress in Nonlinear Differential Equations and Their Applications, vol. 8. Birkhäuser Boston Inc, Boston (1993)Google Scholar
  21. 21.
    De Giorgi, E.: Su una teoria generale della misura \((r-1)\)-dimensionale in uno spazio ad \(r\) dimensioni. Ann. Mat. Pura Appl. 4(36), 191–213 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Duval, V., Aujol, J.-F., Vese, L.A.: Mathematical modeling of textures: application to color image decomposition with a projected gradient algorithm. J. Math. Imaging Vis. 37(3), 232–248 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ericksen, J.L.: Liquid crystals with variable degree of orientation. Arch. Ration. Mech. Anal. 113(2), 97–120 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ferreira, R., Fonseca, I., Luísa Mascarenhas, M.: A chromaticity-brightness model for color images denoising in a Meyer’s “\(u + v\)” framework. Preprint http://arxiv.org/archive/math and http://www.math.cmu.edu/cna/Publications/publications.php (2016)
  25. 25.
    Fonseca, I., Leoni, G., Müller, S.: \({\cal{A}}\)-quasiconvexity: weak-star convergence and the gap. Ann. Inst. H. Poincaré Anal. Non Linéaire 21(2), 209–236 (2004)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Fonseca, I., Malý, J.: Relaxation of multiple integrals below the growth exponent. Ann. Inst. H. Poincaré Anal. Non Linéaire 14(3), 309–338 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Fonseca, I., Müller, S.: Relaxation of quasiconvex functionals in \({\rm BV}(\Omega,{ R}^p)\) for integrands \(f(x, u,\nabla u)\). Arch. Ration. Mech. Anal. 123(1), 1–49 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Giusti, E.: Minimal Surfaces and Functions of Bounded Variation, Volume 80 of Monographs in Mathematics. Birkhäuser Verlag, Basel (1984)CrossRefGoogle Scholar
  29. 29.
    Hang, F., Lin, F.: Topology of Sobolev mappings. II. Acta Math. 191(1), 55–107 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Hardt, R., Kinderlehrer, D., Lin, F.-H.: Existence and partial regularity of static liquid crystal configurations. Commun. Math. Phys. 105(4), 547–570 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Hardt, R., Lin, F.-H.: Mappings minimizing the \(L^p\) norm of the gradient. Commun. Pure Appl. Math. 40(5), 555–588 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Kang, S.H., March, R.: Variational models for image colorization via chromaticity and brightness decomposition. IEEE Trans. Image Process. 16(9), 2251–2261 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Meyer, Y.: Oscillating patterns in image processing and nonlinear evolution equations, volume 22 of University Lecture Series. American Mathematical Society, Providence, RI, 2001. The fifteenth Dean Jacqueline B. Lewis memorial lectures (2001)Google Scholar
  34. 34.
    Mingione, G., Mucci, D.: Integral functionals and the gap problem: sharp bounds for relaxation and energy concentration. SIAM J. Math. Anal. 36(5), 1540–1579 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Mucci, D.: Relaxation of isotropic functionals with linear growth defined on manifold constrained Sobolev mappings. ESAIM Control Optim. Calc. Var. 15(2), 295–321 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Osher, S., Solé, A., Vese, L.: Image decomposition and restoration using total variation minimization and the \(H^{-1}\) norm. Multiscale Model. Simul. 1(3), 349–370 (2003). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60(1–4), 259–268 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Tang, B., Sapiro, G., Caselles, V.: Color image enhancement via chromaticity diffusion. IEEE Trans. Image Process. 10(5), 701–707 (2001)CrossRefzbMATHGoogle Scholar
  39. 39.
    Tikhonov, A.N., Arsenin, V.Y.: Solutions of ill-posed problems. Translation editor Frity John (1977)Google Scholar
  40. 40.
    Vese, L.A., Osher, S.J.: Modeling textures with total variation minimization and oscillating patterns in image processing. J. Sci. Comput. 19(1–3), 553–572 (2003). Special issue in honor of the sixtieth birthday of Stanley OsherMathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Virga, E.G.: Variational Theories for Liquid Crystals. Applied Mathematics and Mathematical Computation, vol. 8. Chapman & Hall, London (1994)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Rita Ferreira
    • 1
  • Irene Fonseca
    • 2
  • M. Luísa Mascarenhas
    • 3
  1. 1.CEMSE Division & KAUST SRI, Center for Uncertainty Quantification in Computational Science and EngineeringKing Abdullah University of Science and Technology (KAUST)ThuwalSaudi Arabia
  2. 2.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA
  3. 3.Departamento de MatemáticaC.M.A. - F.C.T./U.N.L.CaparicaPortugal

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