Abstract
We follow recent work by Schoenemann et al. [25] for expressing curvature regularity as a linear program. While the original formulation focused on binary segmentation, we address several multi-label problems, including segmentation, denoising and inpainting, all cast as a single linear program.
Our multi-label segmentation introduces a “curvature Potts model” and combines a well-known Potts model relaxation [14] with the above work. For inpainting, we improve on [25] by grouping intensities into bins. Finally, we address the problem of denoising with absolute differences in the data term.
Furthermore, we explore alternative solving strategies, including higher order Markov Random Fields, min-sum diffusion and a combination of augmented Lagrangians and an accelerated first order scheme to solve the linear programs.
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
Preview
Unable to display preview. Download preview PDF.
References
Amini, A., Weymouth, T., Jain, R.: Using dynamic programming for solving variational problems in vision. IEEE Transactions on Pattern Analysis and Machine Intelligence (PAMI) 12(9), 855–867 (1990)
Aujol, J.-F., Gilboa, G., Chan, T., Osher, S.: Structure-texture image decomposition modeling, algorithms, and parameter selection. International Journal on Computer Vision (IJCV) 67(1), 111–136 (2006)
Bertsekas, D.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)
Bhusnurmath, A., Taylor, C.: Graph cuts via l 1 norm minimization. IEEE Transactions on Pattern Analysis and Machine Intelligence (PAMI) 30(10), 1866–1871 (2008)
Bornemann, F., März, T.: Fast image inpainting based on coherence transport. Journal on Mathematical Imaging and Vision 28(3), 259–278 (2007)
Brito-Loeza, C., Chen, K.: Multigrid algorithm for higher order denoising. SIAM Journal for Imageing Science 3(3), 363–389 (2010)
Chan, T., Esedoglu, S.: Aspects of total variation regularized l1 function approximation. SIAM Journal of Applied Mathematics 65(5), 1817–1837 (2004)
Chan, T., Kang, S., Shen, J.: Euler’s elastica and curvature based inpaintings. SIAM Journal of Applied Mathematics 2, 564–592 (2002)
Dantzig, G., Thapa, M.: Linear Programming 1: Introduction. Series in Operations Research. Springer, Heidelberg (1997)
El-Zehiry, N., Grady, L.: Fast global optimization of curvature. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR), San Francisco, California (June 2010)
Esedoglu, S., March, R.: Segmentation with depth but without detecting junctions. Journal on Mathematical Imaging and Vision 18, 7–15 (2003)
Greig, D., Porteous, B., Seheult, A.: Exact maximum a posteriori estimation for binary images. Journal of the Royal Statistical Society, Series B 51(2), 271–279 (1989)
Hammer, P., Hansen, P., Simeone, B.: Roof duality, complementation and persistency in quadratic 0-1 optimization. Mathematical Programming 28(2), 121–155 (1984)
Kleinberg, J., Tardos, E.: Approximation algorithms for classification problems with pairwise relationships: metric labeling and Markov Random Fields. In: Symposium on Foundations of Computer Science (1999)
Masnou, S.: Disocclusion: A variational approach using level lines. IEEE Transactions on Image Processing (TIP) 11, 68–76 (2002)
Michelot, C.: A finite algorithm for finding the projection of a point onto the canonical simplex of ℝn. Journal on Optimization Theory and Applications 50(1) (July 1986)
Nesterov, Y.: Introductory lectures on convex optimization. Applied Optimization. Kluwer Academic Publishers, Dordrecht (2004)
Nikolova, M., Esedoglu, S., Chan, T.: Algorithms for finding global minimizers of image segmentation and denoising models. SIAM Journal of Applied Mathematics 66(5), 1632–1648 (2006)
Nitzberg, M., Mumford, D., Shiota, T.: Filtering, segmentation and depth.In: LNCS, vol. 662. Springer, Heidelberg (1993)
Parent, P., Zucker, S.: Trace inference, curvature consistency, and curve detection. IEEE Transactions on Pattern Analysis and Machine Intelligence (PAMI) 11(8), 823–839 (1989)
Pock, T., Chambolle, A., Bischof, H., Cremers, D.: A convex relaxation approach for computing minimal partitions. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR), Miami, Florida (June 2009)
Potts, R.: Some generalized order-disorder transformation. Proceedings of the Cambridge Philosophical Society 48, 106–109 (1952)
Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)
Schoenemann, T., Cremers, D.: Introducing curvature into globally optimal image segmentation: Minimum ratio cycles on product graphs. In: IEEE International Conference on Computer Vision (ICCV), Rio de Janeiro, Brazil (October 2007)
Schoenemann, T., Kahl, F., Cremers, D.: Curvature regularity for region-based image segmentation and inpainting: A linear programming relaxation. In: IEEE International Conference on Computer Vision (ICCV), Kyoto, Japan (September 2009)
Schoenemann, T., Kahl, F., Masnou, S., Cremers, D.: A linear framework for region-based image segmentation and inpainting involving curvature regularity. Technical report, ArXiv report (February 2011)
Strandmark, P., Kahl, F.: Curvature regularization for curves and surfaces in a global optimization framework. In: International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition, St. Petersburg, Russia (July 2011)
Tai, X.-C., Hahn, J., Chung, G.: A fast algorithm for Euler’s elastica model using augmented Lagrangian method. Technical report, UCLA CAM report (July 2010)
Tschumperlé, D.: Fast anisotropic smoothing of multi-valued images using curvature-preserving PDE’s. International Journal on Computer Vision (IJCV) 68(1), 65–82 (2006)
Werner, T.: High-arity interactions, polyhedral relaxations, and cutting plane algorithm for soft constraint optimisation (MAP-MRF). In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR), Anchorage, Alaska ((June 2008)
Yuan, J., Bae, E., Tai, X.-C., Boykov, Y.: A continuous max-flow approach to Potts model. In: European Conference on Computer Vision (ECCV), Iraklion, Greece (September 2010)
Zach, C., Gallup, D., Frahm, J.-M., Niethammer, M.: Fast global labeling for real-time stereo using multiple plane sweeps. In: Vision, Modeling and Visualization Workshop (VMV), Konstanz, Germany (October 2008)
Zhu, W., Chan, T.: Image denoising and mean curvature. Technical report, Courant Institute, New York, Preprint (November 2006), Preprint, http://www.cims.nyu.edu/~wzhu/meancurvature_06Nov30.pdf
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Schoenemann, T., Kuang, Y., Kahl, F. (2011). Curvature Regularity for Multi-label Problems - Standard and Customized Linear Programming. In: Boykov, Y., Kahl, F., Lempitsky, V., Schmidt, F.R. (eds) Energy Minimization Methods in Computer Vision and Pattern Recognition. EMMCVPR 2011. Lecture Notes in Computer Science, vol 6819. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23094-3_12
Download citation
DOI: https://doi.org/10.1007/978-3-642-23094-3_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-23093-6
Online ISBN: 978-3-642-23094-3
eBook Packages: Computer ScienceComputer Science (R0)