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Efficient Global Minimization for the Multiphase Chan-Vese Model of Image Segmentation

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Energy Minimization Methods in Computer Vision and Pattern Recognition (EMMCVPR 2009)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 5681))

Abstract

The Mumford-Shah model is an important variational image segmentation model. A popular multiphase level set approach, the Chan-Vese model, was developed for this model by representing the phases by several overlapping level set functions. Recently, exactly the same model was also formulated by using binary level set functions. In both approaches, the gradient descent equations had to be solved numerically, a procedure which is slow and has the potential of getting stuck in a local minima. In this work, we develop an efficient and global minimization method for the binary level set representation of the multiphase Chan-Vese model based on graph cuts. If the average intensity values of the different phases are sufficiently evenly distributed, the discretized energy function becomes submodular. Otherwise, a novel method for minimizing nonsubmodular functions is proposed with particular emphasis on this energy function.

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References

  1. Mumford, D., Shah, J.: Optimal approximation by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42, 577–685 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  2. Dervieux, A., Thomasset, F.: A finite element method for the simulation of a Rayleigh-Taylor instability. In: Approximation methods for Navier-Stokes problems (Proc. Sympos., Univ. Paderborn, Paderborn, 1979). Lecture Notes in Math., vol. 771, pp. 145–158. Springer, Berlin (1980)

    Chapter  Google Scholar 

  3. Osher, S., Sethian, J.: Fronts propagating with curvature dependent speed: algorithms based on hamilton-jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chan, T., Vese, L.: Active contours without edges. IEEE Image Proc. 10, 266–277 (2001)

    Article  MATH  Google Scholar 

  5. Vese, L.A., Chan, T.F.: A new multiphase level set framework for image segmentation via the mumford and shah model. International Journal of Computer Vision 50, 271–293 (2002)

    Article  MATH  Google Scholar 

  6. Geman, S., Geman, D.: Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. In: Readings in uncertain reasoning, pp. 452–472. Morgan Kaufmann Publishers Inc., San Francisco (1990)

    Google Scholar 

  7. Boykov, Y., Veksler, O., Zabih, R.: Fast approximate energy minimization via graph cuts. In: ICCV (1), pp. 377–384 (1999)

    Google Scholar 

  8. Greig, D.M., Porteous, B.T., Seheult, A.H.: Exact maximum a posteriori estimation for binary images. Journal of the Royal Statistical Society, Series B, 271–279 (1989)

    Google Scholar 

  9. Boykov, Y., Kolmogorov, V.: An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision. In: Energy Minimization Methods in Computer Vision and Pattern Recognition, pp. 359–374 (2001)

    Google Scholar 

  10. Kolmogorov, V., Zabih, R.: What energy functions can be minimized via graph cuts? IEEE Transactions on Pattern Analysis and Machine Intelligence 26(2), 147–159 (2004)

    Article  MATH  Google Scholar 

  11. Komodakis, N., Tziritas, G., Paragios, N.: Fast, approximately optimal solutions for single and dynamic mrfs. In: IEEE Conference on Computer Vision and Pattern Recognition, 2007. CVPR 2007, June 17-22, pp. 1–8 (2007)

    Google Scholar 

  12. Boykov, Y., Kolmogorov, V.: Computing geodesics and minimal surfaces via graph cuts. In: ICCV 2003: Proceedings of the Ninth IEEE International Conference on Computer Vision, Washington, DC, USA, p. 26. IEEE Computer Society Press, Los Alamitos (2003)

    Google Scholar 

  13. Boykov, Y., Kolmogorov, V., Cremers, D., Delong, A.: An integral solution to surface evolution pdes via geo-cuts. In: Leonardis, A., Bischof, H., Pinz, A. (eds.) ECCV 2006. LNCS, vol. 3953, pp. 409–422. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  14. Darbon, J., Sigelle, M.: Image restoration with discrete constrained total variation part i: Fast and exact optimization. J. Math. Imaging Vis. 26(3), 261–276 (2006)

    Article  Google Scholar 

  15. Darbon, J., Sigelle, M.: Image restoration with discrete constrained total variation part ii: Levelable functions, convex priors and non-convex cases. J. Math. Imaging Vis. 26(3), 277–291 (2006)

    Article  Google Scholar 

  16. Darbon, J.: A note on the discrete binary mumford-shah model. In: Gagalowicz, A., Philips, W. (eds.) MIRAGE 2007. LNCS, vol. 4418, pp. 283–294. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  17. Zehiry, N.E., Xu, S., Sahoo, P., Elmaghraby, A.: Graph cut optimization for the mumford-shah model. In: Proceedings of the Seventh IASTED International Conference visualization, imaging and image processing, pp. 182–187. Springer, Heidelberg (2007)

    Google Scholar 

  18. Pock, T., Chambolle, A., Bischof, H., Cremers, D.: A convex relaxation approach for computing minimal partitions. In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Miami, Florida (to appear, 2009)

    Google Scholar 

  19. Lellmann, J., Kappes, J., Yuan, J., Becker, F., Schnorr, C.: Convex multi-class image labeling by simplex-constrained total variation. In: SSVM 2009, pp. 150–162 (2009)

    Google Scholar 

  20. Lie, J., Lysaker, M., Tai, X.: A binary level set model and some applications to mumford-shah image segmentation. IEEE Transactions on Image Processing 15(5), 1171–1181 (2006)

    Article  MATH  Google Scholar 

  21. Bresson, X., Chan, T.: Non-local unsupervised variational image segmentation models (2008)

    Google Scholar 

  22. Lie, J., Lysaker, M., Tai, X.: A variant of the level set method and applications to image segmentation. Math. Comp. 75(255), 1155–1174 (2006) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  23. Nikolova, M., Esedoglu, S., Chan, T.F.: Algorithms for finding global minimizers of image segmentation and denoising models. SIAM Journal on Applied Mathematics 66(5), 1632–1648 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ford, L., Fulkerson, D.: Flows in networks. Princeton University Press, Princeton (1962)

    MATH  Google Scholar 

  25. Bae, E., Tai, X.C.: Efficient global optimization for the multiphase chan-vese model of image segmentation by graph cuts. UCLA, Applied Mathematics, CAM-report-09-53 (June 2009)

    Google Scholar 

  26. Kolmogorov, V., Rother, C.: Minimizing nonsubmodular functions with graph cuts-a review. IEEE Trans. Pattern Anal. Mach. Intell. 29(7), 1274–1279 (2007)

    Article  Google Scholar 

  27. http://www.eecs.berkeley.edu/Research/Projects/CS/vision/grouping/segbench/

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Bergen, Norway

    Egil Bae & Xue-Cheng Tai

  2. Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore

    Xue-Cheng Tai

Authors
  1. Egil Bae
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  2. Xue-Cheng Tai
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Editor information

Editors and Affiliations

  1. Institut für Informatik III, Rö, Universität Bonn, merstraße 164, 53117, Bonn, Germany

    Daniel Cremers

  2. Computer Science Department, University of Western Ontario, N6A 5A5, London, ON, Canada

    Yuri Boykov

  3. Microsoft Research, J.J. Thomson Avenue, CB3 OFB, Cambridge, United Kingdom

    Andrew Blake

  4. Institut für Informatik III, Universität Bonn, Römerstraße 164, 53117, Bonn, Germany

    Frank R. Schmidt

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© 2009 Springer-Verlag Berlin Heidelberg

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Bae, E., Tai, XC. (2009). Efficient Global Minimization for the Multiphase Chan-Vese Model of Image Segmentation. In: Cremers, D., Boykov, Y., Blake, A., Schmidt, F.R. (eds) Energy Minimization Methods in Computer Vision and Pattern Recognition. EMMCVPR 2009. Lecture Notes in Computer Science, vol 5681. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03641-5_3

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  • DOI: https://doi.org/10.1007/978-3-642-03641-5_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-03640-8

  • Online ISBN: 978-3-642-03641-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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