Segmentation with Non-linear Regional Constraints via Line-Search Cuts

  • Lena Gorelick
  • Frank R. Schmidt
  • Yuri Boykov
  • Andrew Delong
  • Aaron Ward
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7572)


This paper is concerned with energy-based image segmentation problems. We introduce a general class of regional functionals defined as an arbitrary non-linear combination of regional unary terms. Such (high-order) functionals are very useful in vision and medical applications and some special cases appear in prior art. For example, our general class of functionals includes but is not restricted to soft constraints on segment volume, its appearance histogram, or shape.

Our overall segmentation energy combines regional functionals with standard length-based regularizers and/or other submodular terms. In general, regional functionals make the corresponding energy minimization NP-hard. We propose a new greedy algorithm based on iterative line search. A parametric max-flow technique efficiently explores all solutions along the direction (line) of the steepest descent of the energy. We compute the best “step size”, i.e. the globally optimal solution along the line. This algorithm can make large moves escaping weak local minima, as demonstrated on many real images.


  1. 1.
    Chan, T., Esedoḡlu, S., Nikolova, M.: Algorithms for finding global minimizers of image segmentation and denoising models. SIAM Appl. Math. 66, 1632–1648 (2006)zbMATHCrossRefGoogle Scholar
  2. 2.
    Chambolle, A.: Total Variation Minimization and a Class of Binary MRF Models. In: Rangarajan, A., Vemuri, B.C., Yuille, A.L. (eds.) EMMCVPR 2005. LNCS, vol. 3757, pp. 136–152. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  3. 3.
    Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. of Math. Imaging and Vision 40, 120–145 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Kolmogorov, V., Zabih, R.: What Energy Functions Can Be Optimized via Graph Cuts. IEEE Trans. on Pattern Analysis and Machine Intelligence (TPAMI) 26, 147–159 (2004)CrossRefGoogle Scholar
  5. 5.
    Boykov, Y., Kolmogorov, V.: An Experimental Comparison of Min-Cut/Max-Flow Algorithms for Energy Minimization in Vision. IEEE Trans. on Pattern Analysis and Machine Intelligence (TPAMI) 29, 1124–1137 (2004)CrossRefGoogle Scholar
  6. 6.
    Boykov, Y., Jolly, M.P.: Interactive Graph Cuts for Optimal Boundary and Region Segmentation of Objects in N-D Images. In: Int. Conf. on Computer Vision, ICCV (2001)Google Scholar
  7. 7.
    Rother, C., Kolmogorov, V., Blake, A.: GrabCut: Interactive Foreground Extraction using Iterated Graph Cuts. In: ACM SIGGRAPH (2004)Google Scholar
  8. 8.
    Vese, L.A., Chan, T.F.: A Multiphase Level Set Framework for Image Segmentation Using the Mumford and Shah Model. Int. Journal of Computer Vision (IJCV) 50, 271–293 (2002)zbMATHCrossRefGoogle Scholar
  9. 9.
    Freedman, D., Zhang, T.: Active contours for tracking distributions. IEEE Trans. on Pattern Analysis and Machine Intelligence (PAMI) 13 (2004)Google Scholar
  10. 10.
    Rother, C., Kolmogorov, V., Minka, T., Blake, A.: Cosegmentation of Image Pairs by Histogram Matching - Incorporating a Global Constraint into MRFs. In: Computer Vision and Pattern Recognition, CVPR (2006)Google Scholar
  11. 11.
    Ayed, I., Chen, H., Punithakumar, K., Ross, I., Shuo, L.: Graph cut segmentation with a global constraint: Recovering region distribution via a bound of the Bhattacharyya measure. In: Computer Vision and Pattern Recognition, CVPR (2010)Google Scholar
  12. 12.
    Kim, J., Kolmogorov, V., Zabih, R.: Visual correspondence using energy minimization and mutual information. In: Int. Conf. on Comp. Vision, ICCV (2003)Google Scholar
  13. 13.
    Werner, T.: High-arity interactions, polyhedral relaxations, and cutting plane algorithm for soft constraint optimisation (MAP-MRF). In: Computer Vision and Pattern Recognition, CVPR (2008)Google Scholar
  14. 14.
    Woodford, O.J., Rother, C., Kolmogorov, V.: A global perspective on MAP inference for low-level vision. In: Int. Conf. on Computer Vision, ICCV (2009)Google Scholar
  15. 15.
    Kolmogorov, V., Boykov, Y., Rother, C.: Applications of Parametric Maxflow in Computer Vision. In: Int. Conf. on Computer Vision, ICCV (2007)Google Scholar
  16. 16.
    Chambolle, A., Darbon, J.: On total variation minimization and surface evolution using parametric maximum flows. Int. Journal of Comp. Vision (IJCV) 84 (2009)Google Scholar
  17. 17.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge Univ. Press (2004)Google Scholar
  18. 18.
    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, Part III. LNCS, vol. 3953, pp. 409–422. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  19. 19.
    Boykov, Y., Kolmogorov, V.: Computing Geodesics and Minimal Surfaces via Graph Cuts. In: Int. Conf. on Computer Vision, ICCV (2003)Google Scholar
  20. 20.
    Gorelick, L., Galun, M., Sharon, E., Basri, R., Brandt, A.: Shape Representation and Classification Using the Poisson Equation. IEEE Trans. on Pattern Analysis and Machine Intelligence (TPAMI) 28 (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Lena Gorelick
    • 1
  • Frank R. Schmidt
    • 2
  • Yuri Boykov
    • 1
  • Andrew Delong
    • 1
  • Aaron Ward
    • 1
  1. 1.University of Western OntarioCanada
  2. 2.Université Paris EstFrance

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