Advertisement

Discretized Convex Relaxations for the Piecewise Smooth Mumford-Shah Model

  • Christopher ZachEmail author
  • Christian Häne
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10746)

Abstract

The Mumford-Shah model for image formation is an important, but also difficult energy functional. In this work we focus on several approaches based on convex relaxation operating on a discretized image domain. Existing methods typically use discretized intensity labels, but in this work we propose to retain the continuous label structure. To this end we employ a recently proposed framework for a new convex relaxation of the Mumford-Shah functional. Numerical results illustrate the performance of the various approaches.

Keywords

Mumford-Shah energy Convex relaxations 

References

  1. 1.
    Alberti, G., Bouchitté, G., Maso, G.D.: The calibration method for the Mumford-Shah functional and free-discontinuity problems. Calc. Var. Partial Differ. Eqn. 16(3), 299–333 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ambrosio, L., Tortorelli, V.: Approximation of functionals depending on jumps by elliptic functionals via \(\varGamma \)-convergence. Commun. Pure Appl. Math. 43, 999–1036 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Blake, A., Zisserman, A.: Visual Reconstruction. MIT Press, Cambridge (1987)Google Scholar
  4. 4.
    Boykov, Y., Kolmogorov, V.: Computing geodesics and minimal surfaces via graph cuts. In: Proceedings of ICCV, pp. 26–33 (2003)Google Scholar
  5. 5.
    Chan, T.F., Vese, L.: Active contours without edges. IEEE Trans. Image Process. 10(2), 266–277 (2001)CrossRefzbMATHGoogle Scholar
  6. 6.
    Combettes, P.L.: Perspective functions: properties, constructions, and examples. Set-Valued Variational Anal. 1–18 (2016)Google Scholar
  7. 7.
    Felzenszwalb, P.F., Huttenlocher, D.P.: Efficient belief propagation for early vision. In: Proceedings of CVPR, pp. 261–268 (2004)Google Scholar
  8. 8.
    Geman, D., Reynolds, G.: Constrained restoration and the recovery of discontinuities. IEEE Trans. Pattern Anal. Mach. Intell. 14(3), 367–383 (1992)CrossRefGoogle Scholar
  9. 9.
    Geman, S., Geman, D.: Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741 (1984)CrossRefzbMATHGoogle Scholar
  10. 10.
    Globerson, A., Jaakkola, T.: Fixing max-product: convergent message passing algorithms for MAP LP-relaxations. In: NIPS (2007)Google Scholar
  11. 11.
    Hazan, T., Shashua, A.: Norm-prodcut belief propagtion: primal-dual message-passing for LP-relaxation and approximate-inference. IEEE Trans. Inf. Theory 56(12), 6294–6316 (2010)CrossRefzbMATHGoogle Scholar
  12. 12.
    Mumford, D., Shah, J.: Optimal approximation by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42, 577–685 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Pock, T., Chambolle, A.: Diagonal preconditioning for first order primal-dual algorithms in convex optimization. In: Proceedings of ICCV, pp. 1762–1769 (2011)Google Scholar
  14. 14.
    Pock, T., Cremers, D., Bischof, H., Chambolle, A.: An algorithm for minimizing the piecewise smooth Mumford-Shah functional. In: Proceedings of ICCV (2009)Google Scholar
  15. 15.
    Strekalovskiy, E., Goldluecke, B., Cremers, D.: Tight convex relaxations for vector-valued labeling problems. In: Proceedings of ICCV (2011)Google Scholar
  16. 16.
    Werner, T.: A linear programming approach to max-sum problem: a review. IEEE Trans. Pattern Anal. Mach. Intell. 29(7), 1165–1179 (2007)CrossRefGoogle Scholar
  17. 17.
    Zach, C.: Dual decomposition for joint discrete-continuous optimization. In: AISTATS (2013)Google Scholar
  18. 18.
    Zach, C., Häne, C., Pollefeys, M.: What is optimized in convex relaxations for multi-label problems: connecting discrete and continuously-inspired MAP inference. IEEE Trans. Pattern Anal. Mach. Intell. (2013)Google Scholar
  19. 19.
    Zach, C., Kohli, P.: A convex discrete-continuous approach for Markov random fields. In: Proceedings of ECCV (2012)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Toshiba Research EuropeCambridgeUK
  2. 2.University of CaliforniaBerkeleyUSA

Personalised recommendations