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A Fast Projection Method for Connectivity Constraints in Image Segmentation

  • Jan Stühmer
  • Daniel Cremers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8932)

Abstract

We propose to solve an image segmentation problem with connectivity constraints via projection onto the constraint set. The constraints form a convex set and the convex image segmentation problem with a total variation regularizer can be solved to global optimality in a primal-dual framework. Efficiency is achieved by directly computing the update of the primal variable via a projection onto the constraint set, which results in a special quadratic programming problem similar to the problems studied as isotonic regression methods in statistics, which can be solved with O(n logn) complexity. We show that especially for segmentation problems with long range connections this method is by orders of magnitudes more efficient, both in iteration number and runtime, than solving the dual of the constrained optimization problem. Experiments validate the usefulness of connectivity constraints for segmenting thin structures such as veins and arteries in medical image analysis.

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References

  1. 1.
    Bai, X., Sapiro, G.: A geodesic framework for fast interactive image and video segmentation and matting. In: IEEE 11th International Conference on Computer Vision, ICCV 2007, pp. 1–8. IEEE (2007)Google Scholar
  2. 2.
    Barlow, R., Brunk, H.: The isotonic regression problem and its dual. Journal of the American Statistical Association 67(337), 140–147 (1972)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Benmansour, F., Cohen, L.: Tubular structure segmentation based on minimal path method and anisotropic enhancement. International Journal of Computer Vision 92, 192–210 (2011), http://dx.doi.org/10.1007/s11263-010-0331-0 CrossRefGoogle Scholar
  4. 4.
    Boykov, Y., Veksler, O., Zabih, R.: Fast approximate energy minimization via graph cuts. IEEE Transactions on Pattern Analysis and Machine Intelligence 23(11), 1222–1239 (2001)CrossRefGoogle Scholar
  5. 5.
    Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120–145 (2011), http://dx.doi.org/10.1007/s10851-010-0251-1 CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Chen, C., Freedman, D., Lampert, C.H.: Enforcing topological constraints in random field image segmentation. In: Proc. International Conference on Computer Vision and Pattern Recognition, pp. 2089–2096 (2011)Google Scholar
  7. 7.
    Criminisi, A., Sharp, T., Blake, A.: GeoS: Geodesic image segmentation. In: Forsyth, D., Torr, P., Zisserman, A. (eds.) ECCV 2008, Part I. LNCS, vol. 5302, pp. 99–112. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  8. 8.
    Dijkstra, E.: A note on two problems in connexion with graphs. Numerische Mathematik 1, 269–271 (1959), http://dx.doi.org/10.1007/BF01386390 CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    El-Zehiry, N.Y., Grady, L.: Fast global optimization of curvature. In: CVPR, pp. 3257–3264. IEEE (2010)Google Scholar
  10. 10.
    Gulshan, V., Rother, C., Criminisi, A., Blake, A., Zisserman, A.: Geodesic star convexity for interactive image segmentation. In: Proc. International Conference on Computer Vision and Pattern Recognition, pp. 3129–3136. IEEE (2010)Google Scholar
  11. 11.
    Han, X., Xu, C., Prince, J.L.: A topology preserving level set method for geometric deformable models. IEEE Transactions on Pattern Analysis and Machine Intelligence 25(6), 755–768 (2003)CrossRefGoogle Scholar
  12. 12.
    Nowozin, S., Lampert, C.H.: Global connectivity potentials for random field models. In: IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2009, pp. 818–825. IEEE (2009)Google Scholar
  13. 13.
    Pardalos, P.M., Xue, G.: Algorithms for a class of isotonic regression problems. Algorithmica 23(3), 211–222 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Pock, T., Cremers, D., Bischof, H., Chambolle, A.: An algorithm for minimizing the piecewise smooth mumford-shah functional. In: IEEE International Conference on Computer Vision (ICCV), Kyoto, Japan (2009)Google Scholar
  15. 15.
    Stühmer, J., Schröder, P., Cremers, D.: Tree shape priors with connectivity constraints using convex relaxation on general graphs. In: IEEE International Conference on Computer Vision (ICCV), Sydney, Australia (December 2013)Google Scholar
  16. 16.
    Vicente, S., Kolmogorov, V., Rother, C.: Graph cut based image segmentation with connectivity priors. In: Proc. International Conference on Computer Vision and Pattern Recognition (2008)Google Scholar
  17. 17.
    Zeng, Y., Samaras, D., Chen, W., Peng, Q.: Topology cuts: A novel min-cut/max-flow algorithm for topology preserving segmentation in n–d images. Computer Vision and Image Understanding 112(1), 81–90 (2008)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Jan Stühmer
    • 1
  • Daniel Cremers
    • 1
  1. 1.Department of Computer ScienceTechnische Universität MünchenGermany

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