Variational Image Segmentation and Cosegmentation with the Wasserstein Distance

  • Paul Swoboda
  • Christoph Schnörr
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8081)


We present novel variational approaches for segmenting and cosegmenting images. Our supervised segmentation approach extends the classical Continuous Cut approach by a global appearance-based data term enforcing closeness of aggregated appearance statistics to a given prior model. This novel data term considers non-spatial, deformation-invariant statistics with the help of the Wasserstein distance in a single global model. The unsupervised cosegmentation model also employs the Wasserstein distance for finding the common object in two images. We introduce tight convex relaxations for both presented models together with efficient algorithmic schemes for computing global minimizers. Numerical experiments demonstrate the effectiveness of our models and the convex relaxations.


Wasserstein distance (co)segmentation convex relaxation 


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  1. 1.
    Achanta, R., Shaji, A., Smith, K., Lucchi, A., Fua, P., Susstrunk, S.: SLIC Superpixels. Technical report, EPFL (June 2010)Google Scholar
  2. 2.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems (Oxford Mathematical Monographs). Oxford University Press, USA (2000)Google Scholar
  3. 3.
    Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers. Found. Trends Mach. Learning 3(1), 1–122 (2010)zbMATHCrossRefGoogle Scholar
  4. 4.
    Wang, J., Gelautz, M., Kohli, P., Rott, P., Rhemann, C., Rother, C.: Alpha matting evaluation websiteGoogle Scholar
  5. 5.
    Chambolle, A., Cremers, D., Pock, T.: A Convex Approach to Minimal Partitions. SIAM J. Imag. Sci. 5(4), 1113–1158 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Chambolle, A., Pock, T.: A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging. Journal of Mathematical Imaging and Vision 40(1), 120–145 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Chan, T., Esedoglu, S., Ni, K.: Histogram Based Segmentation Using Wasserstein Distances. In: Sgallari, F., Murli, A., Paragios, N. (eds.) SSVM 2007. LNCS, vol. 4485, pp. 697–708. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    Chan, T.F., Vese, L.A.: Active contours without edges. IEEE Trans. Imag. Proc. 10(2), 266–277 (2001)zbMATHCrossRefGoogle Scholar
  9. 9.
    Chan, T.F., Esedoglu, S., Nikolova, M.: Algorithms for Finding Global Minimizers of Image Segmentation and Denoising Models. SIAM J. Appl. Math. 66(5), 1632–1648 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Eckstein, J., Bertsekas, D.P.: On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators. Mathematical Programming 55, 293–318 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Gilboa, G., Osher, S.: Nonlocal Operators with Applications to Image Processing. Multiscale Modeling & Simulation 7(3), 1005–1028 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Lellmann, J., Schnörr, C.: Continuous Multiclass Labeling Approaches and Algorithms. SIAM J. Imag. Sci. 4(4), 1049–1096 (2011)zbMATHCrossRefGoogle Scholar
  13. 13.
    MacQueen, J.: Some methods for classification and analysis of multivariate observations. In: Proc. 5th Berkeley Symp. Math. Stat. Probab., Univ. Calif. 1965/1966, vol. 1, pp. 281–297 (1967)Google Scholar
  14. 14.
    Michelot, C.: A finite algorithm for finding the projection of a point onto the canonical simplex of rn. J. Optim. Theory Appl. 50(1), 195–200 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Peyré, G., Fadili, J., Rabin, J.: Wasserstein Active Contours. Technical report, Preprint Hal-00593424 (2011)Google Scholar
  16. 16.
    Pock, T., Schoenemann, T., Graber, G., Bischof, H., Cremers, D.: A Convex Formulation of Continuous Multi-label Problems. In: Forsyth, D., Torr, P., Zisserman, A. (eds.) ECCV 2008, Part III. LNCS, vol. 5304, pp. 792–805. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  17. 17.
    Raguet, H., Fadili, J., Peyré, G.: Generalized Forward-Backward Splitting. Technical report, Preprint Hal-00613637 (2011)Google Scholar
  18. 18.
    Rother, C., Minka, T., Blake, A., Kolmogorov, V.: Cosegmentation of image pairs by histogram matching - incorporating a global constraint into mrfs. In: CVPR, pp. 993–1000. IEEE, Washington, DC (2006)Google Scholar
  19. 19.
    Vicente, S., Kolmogorov, V., Rother, C.: Cosegmentation revisited: Models and optimization. In: Daniilidis, K., Maragos, P., Paragios, N. (eds.) ECCV 2010, Part II. LNCS, vol. 6312, pp. 465–479. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  20. 20.
    Vicente, S., Rother, C., Kolmogorov, V.: Object cosegmentation. In: CVPR, pp. 2217–2224. IEEE (2011)Google Scholar
  21. 21.
    Villani, C.: Optimal Transport: Old and New, 1st edn. Grundlehren der mathematischen Wissenschaften. Springer (November 2008)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Paul Swoboda
    • 1
  • Christoph Schnörr
    • 1
  1. 1.Image and Pattern Analysis Group & HCI, Dept. of Mathematics and Computer ScienceUniversity of HeidelbergGermany

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