Geometry Motivated Variational Segmentation for Color Images

  • Alexander Brook
  • Ron Kimmel
  • Nir A. Sochen
Conference paper
Part of the Lecture Notes in Computer Science 2106 book series (LNCS, volume 2106)


We propose image enhancement, edge detection, and segmentation models for the multi-channel case, motivated by the philosophy of processing images as surfaces, and generalizing the Mumford-Shah functional. Refer to for color ?gures.


Color Image Lower Semicontinuity Jump Point Elliptic Approximation Numerical Minimization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    R. Alicandro, A. Braides, and J. Shah. Free-discontinuity problems via functionals involving the L1-norm of the gradient and their approximation. Interfaces and Free Boundaries, 1(1):17–37, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    L. Ambrosio, N. Fusco, and D. Pallara. Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. Oxford University Press, 2000.Google Scholar
  3. [3]
    L. Ambrosio and V.M. Tortorelli. Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Comm. Pure Appl. Math., 43(8):999–1036, 1990.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    L. Ambrosio and V.M. Tortorelli. On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B (7), 6(1):105–123, 1992.MathSciNetzbMATHGoogle Scholar
  5. [5]
    C. Ballester and M. González. Texture segmentation by variational methods. In ICAOS’ 96. 12th International Conference on Analysis and Optimization of Systems. Images, Wavelets and PDEs, pages 187–193. Springer Verlag, 1996.Google Scholar
  6. [6]
    G. Bellettini and A. Coscia. Discrete approximation of a free discontinuity problem. Numer. Funct. Anal. Optim., 15(3-4):201–224, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    P. Blomgren and T.F. Chan. Color TV: total variation methods for restoration of vector-valued images. IEEE Trans. Image Processing, 7(3):304–309, 1998.CrossRefGoogle Scholar
  8. [8]
    A. Bonnet. On the regularity of edges in image segmentation. Ann. Inst. H. Poincaré Anal. Non Linéaire, 13(4):485–528, 1996.MathSciNetzbMATHGoogle Scholar
  9. [9]
    B. Bourdin and A. Chambolle. Implementation of an adaptive finite-element approximation of the Mumford-Shah functional. Numer. Math., 85(4):609–646, 2000.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    A. Braides. Approximation of free-discontinuity problems. Lecture Notes in Math., 1694. Springer-Verlag, Berlin, 1998.zbMATHGoogle Scholar
  11. [11]
    T. Chan and J. Shen. Variational restoration of non-flat image features: models and algorithms. Technical Report 99-20, UCLA CAM, 1999.Google Scholar
  12. [12]
    T.F. Chan, B.Y. Sandberg, and L.A. Vese. Active contours without edges for vector-valued images. J. Visual Communication Image Representation, 11(2):130–141, 2000.CrossRefGoogle Scholar
  13. [13]
    T.F. Chan and L.A. Vese. An active contour model without edges. In M. Nielsen, P. Johansen, O.F. Olsen, and J. Weickert, editors, Scale-Space Theories in Computer Vision, Second International Conference, Scale-Space’99, volume 1682 of Lecture Notes in Comp. Sci., pages 141–151. Springer, 1999.Google Scholar
  14. [14]
    T.F. Chan and L.A. Vese. Image segmentation using level sets and the piecewiseconstant Mumford-Ssah model. Technical Report 00-14, UCLA CAM, 2000.Google Scholar
  15. [15]
    G. Congedo and I. Tamanini. On the existence of solution to a problem in multidimensional segmentation. Ann. Inst. Henri Poincaré Anal. Non Linéaire, 8(2):175–195, 1991.MathSciNetzbMATHGoogle Scholar
  16. [16]
    A. Cumani. Edge detection in multispectral images. CVGIP: Graphical Models and Image Processing, 53(1):40–51, 1991.zbMATHCrossRefGoogle Scholar
  17. [17]
    G. Dal Maso. An introduction to Γ-convergence. Birkhäuser Boston Inc., Boston, MA, 1993.CrossRefGoogle Scholar
  18. [18]
    E. De Giorgi and L. Ambrosio. New functionals in the calculus of variations. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 82(2):199–210 (1989), 1988.MathSciNetzbMATHGoogle Scholar
  19. [19]
    E. De Giorgi, M. Carriero, and A. Leaci. Existence theorem for a minimum problem with free discontinuity set. Arch. Rational Mech. Anal., 108(3):195–218, 1989.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    E. De Giorgi and T. Franzoni. Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 58(6):842–850, 1975.MathSciNetzbMATHGoogle Scholar
  21. [21]
    S. Di Zenzo. A note on the gradient of a multi-image. Computer Vision, Graphics, and Image Processing, 33(1):116–125, 1986.zbMATHCrossRefGoogle Scholar
  22. [22]
    A.I. El-Fallah and G.E. Ford. The evolution of mean curvature in image filtering. In Proceedings of IEEE International Conference on Image Processing, volume 1, pages 298–302, 1994.Google Scholar
  23. [23]
    M. Focardi. On the variational approximation of free-discontinuity problems in the vectorial case. Technical report, SNS, Pisa, 1999.Google Scholar
  24. [24]
    R. Kimmel, R. Malladi, and N. Sochen. Images as embedded maps and minimal surfaces: Movies, color, texture, and volumetric medical images. Int. J. Computer Vision, 39(2):111–129, 2000.zbMATHCrossRefGoogle Scholar
  25. [25]
    R. Kimmel and N. Sochen. Geometric-variational approach for color image enhancement and segmentation. In M. Nielsen, P. Johansen, O.F. Olsen, and J. Weickert, editors, Scale-space theories in computer vision, volume 1682 of Lecture Notes in Comp. Sci., pages 294–305. Springer, 1999.Google Scholar
  26. [26]
    T.S. Lee, D. Mumford, and A. Yuille. Texture segmentation by minimizing vectorvalued energy functionals: the coupled-membrane model. In G. Sandini, editor, Computer vision-ECCV’ 92, volume 558 of LNCS, pages 165–173. Springer Verlag, 1992.CrossRefGoogle Scholar
  27. [27]
    J.-M. Morel and S. Solimini. Variational methods in image segmentation. Birkhäuser Boston Inc., Boston, MA, 1995. With seven image processing experiments.CrossRefGoogle Scholar
  28. [28]
    D. Mumford and J. Shah. Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math., 42(5):577–685, 1989.MathSciNetzbMATHCrossRefGoogle Scholar
  29. [29]
    M. Negri. The anisotropy introduced by the mesh in the finite element approximation of the Mumford-Shah functional. Numer. Funct. Anal. Optim., 20(9-10):957–982, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [30]
    G. Sapiro and D.L. Ringach. Anisotropic difusion of multivalued images with application to color filtering. IEEE Trans. Image Processing, 5(11):1582–1586, Oct. 1996.CrossRefGoogle Scholar
  31. [31]
    N. Sochen, R. Kimmel, and R. Malladi. A general framework for low level vision. IEEE Trans. Image Processing, 7(3):310–318, 1998.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [32]
    B. Tang, G. Sapiro, and V. Caselles. Difusion of general data on non-flat manifolds via harmonic maps theory: The direction difusion case. Int. J. Computer Vision, 36(2):149–161, 2000.CrossRefGoogle Scholar
  33. [33]
    A. Tsai, A. Yezzi, and A.S. Willsky. Curve evolution, boundary value stochastic processes, the Mumford-Shah problem, and missing data applications. In IEEE International Conference on Image Processing, volume 3, pages 588–591. IEEE, 2000.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Alexander Brook
    • 1
  • Ron Kimmel
    • 2
  • Nir A. Sochen
    • 3
  1. 1.Dept. of MathematicsTechnionIsrael
  2. 2.Dept. of Computer ScienceTechnionIsrael
  3. 3.Dept. of Applied MathematicsTel-Aviv UniversityIsrael

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