A Fast Anisotropic Mumford-Shah Functional Based Segmentation

  • J. F. Garamendi
  • N. Malpica
  • E. Schiavi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5524)


Digital (binary) image segmentation is a critical step in most image processing protocols, especially in medical imaging where accurate and fast segmentation and classification are a challenging issue. In this paper we present a fast relaxation algorithm to minimize an anistropic Mumford-Shah energy functional for piecewise constant approximation of corrupted data. The algorithm is tested with synthetic phantoms and some CT images of the abdomen. Our results are finally compared with manual segmentations in order to validate the proposed model.


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  1. 1.
    Kass, M., Witkin, A., Terzopoulos, D.: Snakes: Active Contour Models. International Journal of Computer Vision 1(4), 321–331 (1988)CrossRefzbMATHGoogle Scholar
  2. 2.
    Caselles, V., Catté, F., Coll, B., Dibos, F.: A geometric Model for Active Contours in Image ProcessingGoogle Scholar
  3. 3.
    Chan, T.F., Vese, L.A.: Active Contours Without Edges. IEEE Transactions on Image Processing 10(2), 266–277 (2001)CrossRefzbMATHGoogle Scholar
  4. 4.
    Badshah, N., Chen, K.: Multigrid method for the Chan-Vese model in variational segmentation. Commun. Comput. Phys. 4, 294–316 (2008)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Grady, L., Alvino, C.: Reformulating and optimizing the Mumford-Shah functional on a graph - A faster, lower energy solution. In: Forsyth, D., Torr, P., Zisserman, A. (eds.) ECCV 2008, Part I. LNCS, vol. 5302, pp. 248–261. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chambolle, A.: An Algorithm for Total Variation Minimization and Applications. Journal of Mathematical Imaging and Vision 20(1-2), 89–97 (2004)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems. In: Oxford Mathematical Monographs, pp. xviii+434. The Clarendon Press/ Oxford University Press, New York (2000)Google Scholar
  9. 9.
    Osher, S., Paragios, N.: Geometric Level Set Methods in Imaging, Vision, and Graphics. Springer, Heidelberg (2003)zbMATHGoogle Scholar
  10. 10.
    Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42, 577–685 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Trottenberg, U., Oosterlee, C., Schüller, A.: Multigrid. Academic Press, London (2001)zbMATHGoogle Scholar
  12. 12.
    Garamendi, J.F., Malpica, N., Schiavi, E., Gaspar, F.J.: ROF based segmentation of the liver in ct images. In: Congreso Anual de la Sociedad Española de Ingenieria Biomedica (2008)Google Scholar
  13. 13.
    Garamendi, J.F., Malpica, N., Martel, J., Schiavi, E.: Automatic Segmentation of the Liver in CT Using Level Sets Without Edges. In: Martí, J., Benedí, J.M., Mendonça, A.M., Serrat, J. (eds.) IbPRIA 2007. LNCS, vol. 4477, pp. 161–168. Springer, Heidelberg (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • J. F. Garamendi
    • 1
  • N. Malpica
    • 1
  • E. Schiavi
    • 1
  1. 1.Universidad Rey Juan Carlos, MóstolesMadridSpain

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