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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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)
Ambrosio, L., Tortorelli, V.: Approximation of functionals depending on jumps by elliptic functionals via \(\varGamma \)-convergence. Commun. Pure Appl. Math. 43, 999–1036 (1990)
Blake, A., Zisserman, A.: Visual Reconstruction. MIT Press, Cambridge (1987)
Boykov, Y., Kolmogorov, V.: Computing geodesics and minimal surfaces via graph cuts. In: Proceedings of ICCV, pp. 26–33 (2003)
Chan, T.F., Vese, L.: Active contours without edges. IEEE Trans. Image Process. 10(2), 266–277 (2001)
Combettes, P.L.: Perspective functions: properties, constructions, and examples. Set-Valued Variational Anal. 1–18 (2016)
Felzenszwalb, P.F., Huttenlocher, D.P.: Efficient belief propagation for early vision. In: Proceedings of CVPR, pp. 261–268 (2004)
Geman, D., Reynolds, G.: Constrained restoration and the recovery of discontinuities. IEEE Trans. Pattern Anal. Mach. Intell. 14(3), 367–383 (1992)
Geman, S., Geman, D.: Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741 (1984)
Globerson, A., Jaakkola, T.: Fixing max-product: convergent message passing algorithms for MAP LP-relaxations. In: NIPS (2007)
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)
Mumford, D., Shah, J.: Optimal approximation by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42, 577–685 (1989)
Pock, T., Chambolle, A.: Diagonal preconditioning for first order primal-dual algorithms in convex optimization. In: Proceedings of ICCV, pp. 1762–1769 (2011)
Pock, T., Cremers, D., Bischof, H., Chambolle, A.: An algorithm for minimizing the piecewise smooth Mumford-Shah functional. In: Proceedings of ICCV (2009)
Strekalovskiy, E., Goldluecke, B., Cremers, D.: Tight convex relaxations for vector-valued labeling problems. In: Proceedings of ICCV (2011)
Werner, T.: A linear programming approach to max-sum problem: a review. IEEE Trans. Pattern Anal. Mach. Intell. 29(7), 1165–1179 (2007)
Zach, C.: Dual decomposition for joint discrete-continuous optimization. In: AISTATS (2013)
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)
Zach, C., Kohli, P.: A convex discrete-continuous approach for Markov random fields. In: Proceedings of ECCV (2012)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Zach, C., Häne, C. (2018). Discretized Convex Relaxations for the Piecewise Smooth Mumford-Shah Model. In: Pelillo, M., Hancock, E. (eds) Energy Minimization Methods in Computer Vision and Pattern Recognition. EMMCVPR 2017. Lecture Notes in Computer Science(), vol 10746. Springer, Cham. https://doi.org/10.1007/978-3-319-78199-0_36
Download citation
DOI: https://doi.org/10.1007/978-3-319-78199-0_36
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-78198-3
Online ISBN: 978-3-319-78199-0
eBook Packages: Computer ScienceComputer Science (R0)