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Approximate Envelope Minimization for Curvature Regularity

  • Stefan Heber
  • Rene Ranftl
  • Thomas Pock
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7585)

Abstract

We propose a method for minimizing a non-convex function, which can be split up into a sum of simple functions. The key idea of the method is the approximation of the convex envelopes of the simple functions, which leads to a convex approximation of the original function. A solution is obtained by minimizing this convex approximation. Cost functions, which fulfill such a splitting property are ubiquitous in computer vision, therefore we explain the method based on such a problem, namely the non-convex problem of binary image segmentation based on Euler’s Elastica.

Keywords

Curvature segmentation convex conjugate convex envelope 

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References

  1. 1.
    Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM Journal on Imaging Sciences 2(1), 183 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bruckstein, A.M., Netravali, A.N., Richardson, T.J.: Epi-convergence of discrete elastica. Applicable Analysis 79, 137–171 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    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
  4. 4.
    Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. Journal of Mathematical Imaging and Vision 40(1), 120–145 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    El-Zehiry, N.Y., Grady, L.: Fast global optimization of curvature. In: CVPR, pp. 3257–3264 (2010)Google Scholar
  6. 6.
    Greig, D.M., Porteous, B.T., Seheult, A.H.: Exact maximum a posteriori estimation for binary images. Journal of the Royal Statistics Society 51(Series B), 271–279 (1989)Google Scholar
  7. 7.
    Hammer, P.L., Hansen, P., Simeone, B.: Roof duality, complementation and persistency in quadratic 0-1 optimization. Mathematical Programming 28(2), 121–155 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Ishikawa, H.: Exact optimization for markov random fields with convex priors. IEEE Transactions on Pattern Analysis and Machine Intelligence 25(10), 1333–1336 (2003)CrossRefGoogle Scholar
  9. 9.
    Ishikawa, H.: Transformation of general binary MRF minimization to the first-order case. IEEE Transactions on Pattern Analysis and Machine Intelligence 33, 1234–1249 (2011)CrossRefGoogle Scholar
  10. 10.
    Kahl, F., Strandmark, P.: Generalized roof duality for pseudo-boolean optimization. In: ICCV (2011)Google Scholar
  11. 11.
    Kanizsa, G.: Organization in Vision. Praeger, New York (1979)Google Scholar
  12. 12.
    Kolmogorov, V., Zabih, R.: What energy functions can be minimized via graph cuts. IEEE Transactions on Pattern Analysis and Machine Intelligence 26(2), 147–159 (2004)CrossRefGoogle Scholar
  13. 13.
    Komodakis, N., Paragios, N., Tziritas, G.: MRF energy minimization and beyond via dual decomposition. IEEE Transactions on Pattern Analysis and Machine Intelligence 33, 531–552 (2011)CrossRefGoogle Scholar
  14. 14.
    Komodakis, N., Tziritas, G.: A new framework for approximate labeling via graph cuts. In: ICCV, pp. 1018–1025 (2005)Google Scholar
  15. 15.
    Mumford, D.: Elastica and computer vision. In: Algebraic Geometry and Its Applications, pp. 491–506 (1994)Google Scholar
  16. 16.
    Pock, T., Cremers, D., Bischof, H., Chambolle, A.: Global solutions of variational models with convex regularization. SIAM Journal on Imaging Sciences 3(4), 1122–1145 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Rother, C., Kolmogorov, V., Lempitsky, V., Szummer, M.: Optimizing binary MRFs via extended roof duality. In: CVPR, pp. 1–8 (2007)Google Scholar
  18. 18.
    Schlesinger, D., Flach, B.: Transforming an arbitrary minsum problem into a binary one. Technical Report TUD-FI06-01, Dresden University of Technology (2006)Google Scholar
  19. 19.
    Schlesinger, M.: Syntactic analysis of two-dimensional visual signals in noisy conditions. Kibernetika 4, 113–130 (1976) (in Russian)Google Scholar
  20. 20.
    Schoenemann, T., Kahl, F., Cremers, D.: Curvature regularity for region-based image segmentation and inpainting: A linear programming relaxation. In: ICCV, September 29-October 2, pp. 17–23 (2009)Google Scholar
  21. 21.
    Schoenemann, T., Kahl, F., Masnou, S., Cremers, D.: A linear framework for region-based image segmentation and inpainting involving curvature penalization. International Journal of Computer Vision (to appear, 2012)Google Scholar
  22. 22.
    Werner, T.: A linear programming approach to max-sum problem: A review. IEEE Transactions on Pattern Analysis and Machine Intelligence 29(7), 1165–1179 (2007)CrossRefGoogle Scholar
  23. 23.
    Werner, T.: Revisiting the linear programming relaxation approach to gibbs energy minimization and weighted constraint satisfaction. IEEE Trans. Pattern Anal. Mach. Intell. 32(8), 1474–1488 (2010)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Stefan Heber
    • 1
  • Rene Ranftl
    • 1
  • Thomas Pock
    • 1
  1. 1.Institute for Computer Graphics and VisionGraz University of TechnologyGrazAustria

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