Journal of Mathematical Imaging and Vision

, Volume 35, Issue 1, pp 1–22

Multiphase Image Segmentation and Modulation Recovery Based on Shape and Topological Sensitivity

Article

Abstract

Topological sensitivity analysis is performed for the piecewise constant Mumford-Shah functional. Topological and shape derivatives are combined in order to derive an algorithm for image segmentation with fully automatized initialization. Segmentation of 2D and 3D data is presented. Further, a generalized Mumford-Shah functional is proposed and numerically investigated for the segmentation of images modulated due to, e.g., coil sensitivities.

Keywords

Image processing k-means clustering Modulation recovery Mumford-Shah functional Piecewise constant reconstruction Segmentation Shape and topological sensitivity 

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References

  1. 1.
    Chan, T.F., Vese, L.A.: A level set algorithm for minimizing the Mumford-Shah functional in image processing. In: Proceedings of the IEEE Workshop on Variational and Level Set Methods (VLSM ’01), pp. 161–168. Vancouver, BC, Canada, July 2001 Google Scholar
  2. 2.
    Delfour, M., Zolesio, J.-P.: Shapes and Geometries. Analysis, Differential Calculus and Optimization. SIAM Advances in Design and Control. SIAM, Philadelphia (2001) MATHGoogle Scholar
  3. 3.
    Giaquinta, M.: Introduction to Regularity Theory for Nonlinear Elliptic Systems. Birkhäuser, Basel (1993) MATHGoogle Scholar
  4. 4.
    Hartigan, J.A.: Clustering Algorithms. Wiley Series in Probability and Mathematical Statistics. Wiley, New York (1975) MATHGoogle Scholar
  5. 5.
    Hartigan, J.A., Wong, M.A.: A k-means clustering algorithm. J. R. Stat. Soc., Ser. C, Appl. Stat. 28, 100–108 (1979) MATHGoogle Scholar
  6. 6.
    He, L., Osher, S.: Solving the Chan-Vese model by a multiphase level set algorithm based on the topological derivative. In: Scale Space Variational Methods in Computer Vision. Lecture Notes in Computer Science, pp. 777–788. Springer, Berlin (2007) CrossRefGoogle Scholar
  7. 7.
    Henrot, A., Pierre, M.: In: Variation et Optimisation de Formes: Une Analyse Géométrique. Mathématiques et Applications, vol. 48. Springer, Berlin (2005) MATHGoogle Scholar
  8. 8.
    Hintermüller, M., Ring, W.: A second-order shape optimization approach for image segmentation. SIAM J. Appl. Math. 64, 442–467 (2003) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Keeling, S.L., Bammer, R.: A variational approach to magnetic resonance coil sensitivity estimation. Appl. Math. Comput. 158, 53–82 (2004) CrossRefMathSciNetGoogle Scholar
  10. 10.
    MacQueen, J.: Some methods for classification and analysis of multivariate observations. In: Proc. Fifth Berkeley Sympos. Math. Statist. and Probability, Berkeley, CA, 1965/1966. Statistics, vol. I, pp. 281–297. Univ. California Press, Berkeley (1967) Google Scholar
  11. 11.
    Morel, J.-M., Solimini, S.: Variational Methods in Image Segmentation. Progress in Nonlinear Differential Equations and Their Applications, vol. 14. Birkhäuser Boston, Boston (1995) Google Scholar
  12. 12.
    Mumford, D., Shah, J.: Optimal approximations by piece-wise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42, 577–685 (1989) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Springer, Berlin (2004) Google Scholar
  14. 14.
    Osher, S., Sethian, J.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulation. J. Comput. Phys. 79, 12–49 (1988) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Peng, D., Merriman, B., Osher, S., Zhao, H., Kang, M.: A PDE-based fast local level set method. J. Comput. Phys. 155, 410–438 (1999) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Sethian, J.A.: Level Set Methods and Fast Marching Methods, 2nd edn. Cambridge University Press, Cambridge (1999) MATHGoogle Scholar
  17. 17.
    Sokołowski, J., Żochowski, A.: On the topological derivative in shape optimization. SIAM J. Control Optim. 37, 1251–1272 (1999) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Sokołowski, J., Zolesio, J.-P.: Introduction to Shape Optimization. Springer Series in Computational Mathematics, vol. 16. Springer, Berlin (1992) MATHGoogle Scholar
  19. 19.
    Vese, L.A., Chan, T.F.: A multiphase level set framework for image segmentation using the Mumford and Shah model. Int. J. Comput. Vis. 50, 271–293 (2002) MATHCrossRefGoogle Scholar
  20. 20.
    Wright, S.J.: Primal-Dual Interior-Point Methods. SIAM, Philadelphia (1997) MATHGoogle Scholar
  21. 21.
    Zhao, H.K., Chan, T., Merriman, B., Osher, S.: A variational level set approach to multi-phase motion. J. Comput. Phys. 122, 179–195 (1996) CrossRefMathSciNetGoogle Scholar
  22. 22.
    Ziemer, W.P.: Weakly Differentiable Functions. Graduate Texts in Mathematics. Springer, New York (1989) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Humboldt-Universität zu BerlinBerlinGermany
  2. 2.Department of Mathematics and Scientific ComputingUniversity of GrazGrazAustria

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