K-Smallest Spanning Tree Segmentations

  • Christoph Straehle
  • Sven Peter
  • Ullrich Köthe
  • Fred A. Hamprecht
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8142)

Abstract

Real-world images often admit many different segmentations that have nearly the same quality according to the underlying energy function. The diversity of these solutions may be a powerful uncertainty indicator. We provide the crucial prerequisite in the context of seeded segmentation with minimum spanning trees (i.e. edge-weighted watersheds). Specifically, we show how to efficiently enumerate the k smallest spanning trees that result in different segmentations; and we prove that solutions are indeed found in the correct order. Experiments show that about half of the trees considered by our algorithm represent unique segmentations. This redundancy is orders of magnitude lower than can be achieved by just enumerating the k-smallest MSTs, making the algorithm viable in practice.

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References

  1. 1.
    Batra, D., Yadollahpour, P., Guzman-Rivera, A., Shakhnarovich, G.: Diverse M-best solutions in markov random fields. In: Fitzgibbon, A., Lazebnik, S., Perona, P., Sato, Y., Schmid, C. (eds.) ECCV 2012, Part V. LNCS, vol. 7576, pp. 1–16. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  2. 2.
    Blake, A., Kohli, P., Rother, C.: Markov random fields for vision and image processing. MIT Press (2011)Google Scholar
  3. 3.
    Boykov, Y., Jolly, M.: Interactive graph cuts for optimal boundary and region segmentation of objects in ND images. In: ICCV (2001)Google Scholar
  4. 4.
    Brendel, W., Todorovic, S.: Segmentation as maximumweight independent set. In: NIPS, vol. 4 (2010)Google Scholar
  5. 5.
    Briggman, K.L., Denk, W., et al.: Towards neural circuit reconstruction with volume electron microscopy techniques. Current Opinion in Neurobiology (2006)Google Scholar
  6. 6.
    Couprie, C., Grady, L., Najman, L., Talbot, H.: Power watershed: A unifying graph-based optimization framework. IEEE PAMI (2010)Google Scholar
  7. 7.
    Cousty, J., Bertrand, G., Najman, L., Couprie, M.: Watershed cuts: Minimum spanning forests and the drop of water principle. IEEE PAMI, 1362–1374 (2009)Google Scholar
  8. 8.
    Falcão, A.X., Stolfi, J., Lotufo, R.A.: The image foresting transform: Theory, algorithms, and applications. IEEE PAMI 26 (2004)Google Scholar
  9. 9.
    Fromer, M., Globerson, A.: An LP view of the M-best MAP problem. NIPS (2009)Google Scholar
  10. 10.
    Gabow, H.: Two algorithms for generating weighted spanning trees in order. SIAM Journal on Computing (1977)Google Scholar
  11. 11.
    Grady, L.: Random walks for image segmentation. IEEE PAMI 28 (2006)Google Scholar
  12. 12.
    Meyer, F.: Minimum spanning forests for morphological segmentation. In: Mathematical Morphology and its Applications to Image Processing, pp. 77–84. Springer (1994)Google Scholar
  13. 13.
    Meyer, F., Beucher, S.: Morphological segmentation. Journal of Visual Communication and Image Representation 1(1), 21–46 (1990)CrossRefGoogle Scholar
  14. 14.
    Nilsson, D.: An efficient algorithm for finding the M most probable configurations in probabilistic expert systems. Statistics and Computing 8(2), 159–173 (1998)CrossRefGoogle Scholar
  15. 15.
    Rollon, N.E., Dechter, R.: Inference schemes for M-best solutions for soft CSPs. In: Proceedings of Workshop on Preferences and Soft Constraints, vol. 2 (2011)Google Scholar
  16. 16.
    Seroussi, B., Golmard, J.: An algorithm directly finding the K most probable configurations in bayesian networks. International Journal of Approximate Reasoning 11(3), 205–233 (1994)CrossRefGoogle Scholar
  17. 17.
    Straehle, C.N., Köthe, U., Knott, G., Hamprecht, F.A.: Carving: Scalable interactive segmentation of neural volume electron microscopy images. In: Fichtinger, G., Martel, A., Peters, T. (eds.) MICCAI 2011, Part I. LNCS, vol. 6891, pp. 653–660. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  18. 18.
    Tzeng, W.J., Wu, F.: Spanning trees on hypercubic lattices and nonorientable surfaces. Applied Mathematics Letters 13 (2000)Google Scholar
  19. 19.
    Vincent, L., Soille, P.: Watersheds in digital spaces: an efficient algorithm based on immersion simulations. IEEE PAMI (1991)Google Scholar
  20. 20.
    Yanover, C., Weiss, Y.: Finding the AI most probable configurations using loopy belief propagation. In: NIPS (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Christoph Straehle
    • 1
  • Sven Peter
    • 1
  • Ullrich Köthe
    • 1
  • Fred A. Hamprecht
    • 1
  1. 1.HCI, University of HeidelbergGermany

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