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.
Chapter PDF
Similar content being viewed by others
References
Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM Journal on Imaging Sciences 2(1), 183 (2009)
Bruckstein, A.M., Netravali, A.N., Richardson, T.J.: Epi-convergence of discrete elastica. Applicable Analysis 79, 137–171 (2001)
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)
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)
El-Zehiry, N.Y., Grady, L.: Fast global optimization of curvature. In: CVPR, pp. 3257–3264 (2010)
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)
Hammer, P.L., Hansen, P., Simeone, B.: Roof duality, complementation and persistency in quadratic 0-1 optimization. Mathematical Programming 28(2), 121–155 (1984)
Ishikawa, H.: Exact optimization for markov random fields with convex priors. IEEE Transactions on Pattern Analysis and Machine Intelligence 25(10), 1333–1336 (2003)
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)
Kahl, F., Strandmark, P.: Generalized roof duality for pseudo-boolean optimization. In: ICCV (2011)
Kanizsa, G.: Organization in Vision. Praeger, New York (1979)
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)
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)
Komodakis, N., Tziritas, G.: A new framework for approximate labeling via graph cuts. In: ICCV, pp. 1018–1025 (2005)
Mumford, D.: Elastica and computer vision. In: Algebraic Geometry and Its Applications, pp. 491–506 (1994)
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)
Rother, C., Kolmogorov, V., Lempitsky, V., Szummer, M.: Optimizing binary MRFs via extended roof duality. In: CVPR, pp. 1–8 (2007)
Schlesinger, D., Flach, B.: Transforming an arbitrary minsum problem into a binary one. Technical Report TUD-FI06-01, Dresden University of Technology (2006)
Schlesinger, M.: Syntactic analysis of two-dimensional visual signals in noisy conditions. Kibernetika 4, 113–130 (1976) (in Russian)
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)
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)
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)
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Heber, S., Ranftl, R., Pock, T. (2012). Approximate Envelope Minimization for Curvature Regularity. In: Fusiello, A., Murino, V., Cucchiara, R. (eds) Computer Vision – ECCV 2012. Workshops and Demonstrations. ECCV 2012. Lecture Notes in Computer Science, vol 7585. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33885-4_29
Download citation
DOI: https://doi.org/10.1007/978-3-642-33885-4_29
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-33884-7
Online ISBN: 978-3-642-33885-4
eBook Packages: Computer ScienceComputer Science (R0)