Advertisement

Measuring Uncertainty in Graph Cut Solutions – Efficiently Computing Min-marginal Energies Using Dynamic Graph Cuts

  • Pushmeet Kohli
  • Philip H. S. Torr
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3952)

Abstract

In recent years the use of graph-cuts has become quite popular in computer vision. However, researchers have repeatedly asked the question whether it might be possible to compute a measure of uncertainty associated with the graph-cut solutions. In this paper we answer this particular question by showing how the min-marginals associated with the label assignments in a MRF can be efficiently computed using a new algorithm based on dynamic graph cuts. We start by reporting the discovery of a novel relationship between the min-marginal energy corresponding to a latent variable label assignment, and the flow potentials of the node representing that variable in the graph used in the energy minimization procedure. We then proceed to show how the min-marginal energy can be computed by minimizing a projection of the energy function defined by the MRF. We propose a fast and novel algorithm based on dynamic graph cuts to efficiently minimize these energy projections. The min-marginal energies obtained by our proposed algorithm are exact, as opposed to the ones obtained from other inference algorithms like loopy belief propagation and generalized belief propagation. We conclude by showing how min-marginals can be used to compute a confidence measure for label assignments in labelling problems such as image segmentation.

Keywords

Energy Function Image Segmentation Markov Random Field Graph Construction Multiple Label 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Kolmogorov, V.: Convergent tree-reweighted message passing for energy minimization. In: AISTATS 2005, pp. 182–189 (2005)Google Scholar
  2. 2.
    Wainwright, M.J., Willsky, T.S.J., Map, A.S.: estimation via agreement on (hyper)trees: Message-passing and linear-programming approaches. Technical Report UCB/CSD-03-1269 (2003)Google Scholar
  3. 3.
    Kolmogorov, V., Zabih, R.: What energy functions can be minimized via graph cuts? In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) ECCV 2002. LNCS, vol. 2352, pp. 65–81. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  4. 4.
    Boykov, Y., Veksler, O., Zabih, R.: Markov random fields with efficient approximations. In: CVPR 1998, pp. 648–655 (1998)Google Scholar
  5. 5.
    Boykov, Y., Kolmogorov, V.: An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision. PAMI 26, 1124–1137 (2004)CrossRefMATHGoogle Scholar
  6. 6.
    Kohli, P., Torr, P.: Eficiently solving dynamic markov random fields using graph cuts. In: ICCV (2005)Google Scholar
  7. 7.
    Chiang, Y.J., Tamassia, R.: Dynamic algorithms in computational geometry. Technical Report CS-91-24 (1991)Google Scholar
  8. 8.
    Boykov, Y., Jolly, M.: Interactive graph cuts for optimal boundary and region segmentation of objects in n-d images. In: ICCV 2001, pp. I:105–112 (2001)Google Scholar
  9. 9.
    Weiss, Y., Freeman, W.T.: On the optimality of solutions of the max-product belief-propagation algorithm in arbitrary graphs. IEEE Transactions on Information Theory (2001)Google Scholar
  10. 10.
    Yanover, C., Weiss, Y.: Finding the m most probable configurations in arbitrary graphical models. In: Advances in Neural Information Processing Systems 16, MIT Press, Cambridge (2004)Google Scholar
  11. 11.
    Dawid, P.: Applications of a general propagation algorithm for probabilistic expert systems. Statistics and Computing 2, 25–36 (1992)CrossRefGoogle Scholar
  12. 12.
    Nilsson, D.: An efficient algorithm for finding the m most probable configurations in bayesian networks. Statistics and Computing 8, 159–173 (1998)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ford, L., Fulkerson, D.: Flows in Networks. Princeton University Press, Princeton (1962)MATHGoogle Scholar
  14. 14.
    Ishikawa, H.: Exact optimization for markov random fields with convex priors. PAMI 25, 1333–1336 (2003)CrossRefGoogle Scholar
  15. 15.
    Bray, M., Kohli, P., Torr, P.: poseCut: Simultaneous segmentation and 3D pose estimation of humans using dynamic graph-cuts. In: Leonardis, A., Bischof, H., Pinz, A. (eds.) ECCV 2006. LNCS, vol. 3952, pp. 642–655. Springer, Heidelberg (2006)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Pushmeet Kohli
    • 1
  • Philip H. S. Torr
    • 1
  1. 1.Department of ComputingOxford Brookes UniversityOxfordUK

Personalised recommendations