Advertisement

What Energy Functions Can Be Minimized via Graph Cuts?

  • Vladimir Kolmogorov
  • Ramin Zabih
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2352)

Abstract

In the last few years, several new algorithms based on graph cuts have been developed to solve energy minimization problems in computer vision. Each of these techniques constructs a graph such that the minimum cut on the graph also minimizes the energy. Yet because these graph constructions are complex and highly specific to a particular energy function, graph cuts have seen limited application to date. In this paper we characterize the energy functions that can be minimized by graph cuts. Our results are restricted to energy functions with binary variables. However, our work generalizes many previous constructions, and is easily applicable to vision problems that involve large numbers of labels, such as stereo, motion, image restoration and scene reconstruction. We present three main results: a necessary condition for any energy function that can be minimized by graph cuts; a sufficient condition for energy functions that can be written as a sum of functions of up to three variables at a time; and a general-purpose construction to minimize such an energy function. Researchers who are considering the use of graph cuts to optimize a particular energy function can use our results to determine if this is possible, and then follow our construction to create the appropriate graph.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ravindra K. Ahuja, Thomas L. Magnanti, and James B. Orlin. Network Flows: Theory, Algorithms, and Applications. Prentice Hall, 1993.Google Scholar
  2. 2.
    Amir Amini, Terry Weymouth, and Ramesh Jain. Using dynamic programming for solving variational problems in vision. IEEE Transactions on Pattern Analysis and Machine Intelligence, 12(9):855–867, September 1990.Google Scholar
  3. 3.
    Stephen Barnard. Stochastic stereo matching over scale. International Journal of Computer Vision, 3(1):17–32, 1989.CrossRefMathSciNetGoogle Scholar
  4. 4.
    S. Birchfield and C. Tomasi. Multiway cut for stereo and motion with slanted surfaces. In International Conference on Computer Vision, pages 489–495, 1999.Google Scholar
  5. 5.
    Yuri Boykov and Marie-Pierre Jolly. Interactive organ segmentation using graph cuts. In Medical Image Computing and Computer-Assisted Intervention, pages 276–286, 2000.Google Scholar
  6. 6.
    Yuri Boykov and Marie-Pierre Jolly. Interactive graph cuts for optimal boundary and region segmentation of objects in N-D images. In International Conference on Computer Vision, pages I:105–112, 2001.Google Scholar
  7. 7.
    Yuri Boykov, Olga Veksler, and Ramin Zabih. Markov Random Fields with efficient approximations. In IEEE Conference on Computer Vision and Pattern Recognition, pages 648–655, 1998.Google Scholar
  8. 8.
    Yuri Boykov, Olga Veksler, and Ramin Zabih. Fast approximate energy minimization via graph cuts. IEEE Transactions on Pattern Analysis and Machine Intelligence, 23(11):1222–1239, November 2001.Google Scholar
  9. 9.
    L. Ford and D. Fulkerson. Flows in Networks. Princeton University Press, 1962.Google Scholar
  10. 10.
    S. Geman and D. Geman. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6:721–741, 1984.zbMATHCrossRefGoogle Scholar
  11. 11.
    A. Goldberg and R. Tarjan. A new approach to the maximum flow problem. Journal of the Association for Computing Machinery, 35(4):921–940, October 1988.Google Scholar
  12. 12.
    D. Greig, B. Porteous, and A. Seheult. Exact maximum a posteriori estimation for binary images. Journal of the Royal Statistical Society, Series B, 51(2):271–279, 1989.Google Scholar
  13. 13.
    H. Ishikawa and D. Geiger. Occlusions, discontinuities, and epipolar lines in stereo. In European Conference on Computer Vision, pages 232–248, 1998.Google Scholar
  14. 14.
    H. Ishikawa and D. Geiger. Segmentation by grouping junctions. In IEEE Conference on Computer Vision and Pattern Recognition, pages 125–131, 1998.Google Scholar
  15. 15.
    Junmo Kim, John Fish, Andy Tsai, Cindy Wible, Ala Willsky, and William Wells. Incorporating spatial priors into an information theoretic approach for fMRI data analysis. In Medical Image Computing and Computer-Assisted Intervention, pages 62–71, 2000.Google Scholar
  16. 16.
    Vladimir Kolmogorov and Ramin Zabih. Visual correspondence with occlusions using graph cuts. In International Conference on Computer Vision, pages 508–515, 2001.Google Scholar
  17. 17.
    Vladimir Kolmogorov and Ramin Zabih. What energy functions can be minimized via graph cuts? Technical report CUCS-TR2001-1857, Cornell Computer Science Department, November 2001.Google Scholar
  18. 18.
    Vladimir Kolmogorov and Ramin Zabih. Multi-camera scene reconstruction via graph cuts. In European Conference on Computer Vision, 2002.Google Scholar
  19. 19.
    S. Li. Markov Random Field Modeling in Computer Vision. Springer-Verlag, 1995.Google Scholar
  20. 20.
    S. Roy. Stereo without epipolar lines: A maximum flow formulation. International Journal of Computer Vision, 1(2):1–15, 1999.Google Scholar
  21. 21.
    S. Roy and I. Cox. A maximum-flow formulation of the n-camera stereo correspondence problem. In International Conference on Computer Vision, 1998.Google Scholar
  22. 22.
    Daniel Scharstein and Richard Szeliski. A taxonomy and evaluation of dense two-frame stereo correspondence algorithms. Technical Report 81, Microsoft Research, 2001. To appear in IJCV. An earlier version appears in CVPR 2001 Workshop on Stereo Vision.Google Scholar
  23. 23.
    Dan Snow, Paul Viola, and Ramin Zabih. Exact voxel occupancy with graph cuts. In IEEE Conference on Computer Vision and Pattern Recognition, pages 345–352, 2000.Google Scholar
  24. 24.
    Richard Szeliski and Ramin Zabih. An experimental comparison of stereo algorithms. In B. Triggs, A. Zisserman, and R. Szeliski, editors, Vision Algorithms: Theory and Practice, number 1883 in LNCS, pages 1–19, Corfu, Greece, September 1999. Springer-Verlag.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Vladimir Kolmogorov
    • 1
  • Ramin Zabih
    • 1
  1. 1.Computer Science DepartmentCornell UniversityIthacaUSA

Personalised recommendations