Abstract
We propose a spatially continuous formulation of Ishikawa’s discrete multi-label problem. We show that the resulting non-convex variational problem can be reformulated as a convex variational problem via embedding in a higher dimensional space. This variational problem can be interpreted as a minimal surface problem in an anisotropic Riemannian space. In several stereo experiments we show that the proposed continuous formulation is superior to its discrete counterpart in terms of computing time, memory efficiency and metrication errors.
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Greig, D., Porteous, B., Seheult, A.: Exact maximum a posteriori estimation for binary images. J. Royal Statistics Soc. 51(Series B), 271–279 (1989)
Kolmogorov, V., Zabih, R.: What energy functions can be minimized via graph cuts. IEEE Trans. Pattern Anal. Mach. Intell. 26(2), 147–159 (2004)
Boykov, Y., Veksler, O., Zabih, R.: Fast approximate energy minimization via graph cuts. IEEE Trans. Pattern Anal. Mach. Intell. 23(11), 1222–1239 (2001)
Veksler, O.: Efficient Graph-based Energy Minimization Methods in Computer Vision. PhD thesis, Cornell University (July 1999)
Schlesinger, D., Flach, B.: Transforming an arbitrary minsum problem into a binary one. Technical Report TUD-FI06-01, Dresden University of Technology (2006)
Werner, T.: A linear programming approach to max-sum problem: A review. IEEE Trans. Pattern Anal. Mach. Intell. 29(7), 1165–1179 (2007)
Ishikawa, H.: Exact optimization for markov random fields with convex priors. IEEE Trans. Pattern Anal. Mach. Intell. 25(10), 1333–1336 (2003)
Ford, L., Fulkerson, D.: Flows in Networks. Princeton University Press, Princeton (1962)
Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)
Chan, T., Esedoglu, S., Nikolova, M.: Algorithms for finding global minimizers of image segmentation and denoising models. SIAM Journal of Applied Mathematics 66(5), 1632–1648 (2006)
Fleming, W., Rishel, R.: An integral formula for total gradient variation. Arch. Math. 11, 218–222 (1960)
Vogel, C., Oman, M.: Iteration methods for total variation denoising. SIAM J. Sci. Comp. 17, 227–238 (1996)
Chan, T., Golub, G., Mulet, P.: A nonlinear primal-dual method for total variation-based image restoration. SIAM J. Sci. Comp. 20(6), 1964–1977 (1999)
Carter, J.: Dual Methods for Total Variation-based Image Restoration. PhD thesis, UCLA, Los Angeles, CA (2001)
Chambolle, A.: An algorithm for total variation minimizations and applications. J. Math. Imaging Vis. (2004)
Chambolle, A.: Total variation minimization and a class of binary MRF models. Energy Minimization Methods in Computer Vision and Pattern Recognition, 136–152 (2005)
Rockafellar, R.: Augmented lagrangians and applications of the proximal point algorithm in convex programming. Math. of Oper Res 1, 97–116 (1976)
Bresson, X., Esedoglu, S., Vandergheynst, P., Thiran, J., Osher, S.: Fast global minimization of the active contour/snake model. J. Math. Imaging Vis. 28(2), 151–167 (2007)
Appleton, B., Talbot, H.: Globally minimal surfaces by continuous maximal flows. IEEE Trans. Pattern Anal. Mach. Intell. 28(1), 106–118 (2006)
Scharstein, D., Szeliski, R.: A taxonomy and evaluation of dense two-frame stereo correspondence algorithms. Int. J. Comp. Vis. 47(1-3), 7–42 (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Pock, T., Schoenemann, T., Graber, G., Bischof, H., Cremers, D. (2008). A Convex Formulation of Continuous Multi-label Problems. In: Forsyth, D., Torr, P., Zisserman, A. (eds) Computer Vision – ECCV 2008. ECCV 2008. Lecture Notes in Computer Science, vol 5304. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-88690-7_59
Download citation
DOI: https://doi.org/10.1007/978-3-540-88690-7_59
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-88689-1
Online ISBN: 978-3-540-88690-7
eBook Packages: Computer ScienceComputer Science (R0)