On Total Variation Minimization and Surface Evolution Using Parametric Maximum Flows
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Abstract
In a recent paper Boykov et al. (LNCS, Vol. 3953, pp. 409–422, 2006) propose an approach for computing curve and surface evolution using a variational approach and the geo-cuts method of Boykov and Kolmogorov (International conference on computer vision, pp. 26–33, 2003). We recall in this paper how this is related to well-known approaches for mean curvature motion, introduced by Almgren et al. (SIAM Journal on Control and Optimization 31(2):387–438, 1993) and Luckhaus and Sturzenhecker (Calculus of Variations and Partial Differential Equations 3(2):253–271, 1995), and show how the corresponding problems can be solved with sub-pixel accuracy using Parametric Maximum Flow techniques. This provides interesting algorithms for computing crystalline curvature motion, possibly with a forcing term.
Keywords
Crystalline and anisotropic mean curvature flow Variational approaches Total variation Submodular functions Max-flow/min-cut Parametric max-flow algorithmsReferences
- Ahuja, R. K., Magnanti, T. L., & Orlin, J. B. (1993). Network flows. Theory, algorithms, and applications. Englewood Cliffs: Prentice-Hall. Google Scholar
- Allard, W. K. (2007). Total variation regularization for image denoising, I. Geometric theory. SIAM Journal on Mathematical Analysis, 39(4), 1150–1190. zbMATHCrossRefMathSciNetGoogle Scholar
- Almgren, R. (1993). Variational algorithms and pattern formation in dendritic solidification. Journal of Computational Physics, 106(2), 337–354. zbMATHMathSciNetGoogle Scholar
- Almgren, F., Taylor, J. E., & Wang, L.-H. (1993). Curvature-driven flows: a variational approach. SIAM Journal on Control and Optimization, 31(2), 387–438. zbMATHCrossRefMathSciNetGoogle Scholar
- Alter, F., Caselles, V., & Chambolle, A. (2005). A characterization of convex calibrable sets in ℝN. Mathematische Annalen, 332(2), 329–366. zbMATHCrossRefMathSciNetGoogle Scholar
- Babenko, M. A., Derryberry, J., Goldberg, A. V., Tarjan, R. E., & Zhou, Y. (2007). Experimental evaluation of parametric max-flow algorithms. In Lecture notes in computer science (Vol. 4252, pp. 256–269). Berlin: Springer. Google Scholar
- Beck, A., & Teboulle, M. (2009). A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM Journal on Imaging Sciences, 2(1), 183–202. CrossRefMathSciNetGoogle Scholar
- Bellettini, G., & Paolini, M. (1996). Anisotropic motion by mean curvature in the context of Finsler geometry. Hokkaido Mathematical Journal, 25(3), 537–566. zbMATHMathSciNetGoogle Scholar
- Bellettini, G., Novaga, M., & Paolini, M. (1999). Facet-breaking for three-dimensional crystals evolving by mean curvature. Interfaces and Free Boundaries, 1(1), 39–55. zbMATHMathSciNetGoogle Scholar
- Bellettini, G., Caselles, V., Chambolle, A., & Novaga, M. (2006). Crystalline mean curvature flow of convex sets. Archive for Rational Mechanics and Analysis, 179(1), 109–152. zbMATHCrossRefMathSciNetGoogle Scholar
- Boros, E., & Hammer, P. L. (2002). Pseudo-boolean optimization. Discrete Applied Mathematics, 123(1–3), 155–225. zbMATHMathSciNetGoogle Scholar
- Bouchitté, G. (1998). Recent convexity arguments in the calculus of variations. In Lecture notes from the 3rd int. summer school on the calculus of variations, Pisa. Google Scholar
- Boykov, Y., & Kolmogorov, V. (2003). Computing geodesics and minimal surfaces via graph cuts. In International conference on computer vision, pp. 26–33. Google Scholar
- Boykov, Y., & Kolmogorov, V. (2004). An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision. IEEE Transactions on Pattern Analysis and Machine Intelligence, 26(9), 1124–1137. CrossRefGoogle Scholar
- Boykov, Y., Veksler, O., & Zabih, R. (2001). Fast approximate energy minimization via graph cuts. IEEE Transactions on Pattern Analysis and Machine Intelligence, 23(11), 1222–1239. CrossRefGoogle Scholar
- Boykov, Y., Kolmogorov, V., Cremers, D., & Delong, A. (2006). An integral solution to surface evolution PDEs via Geo-Cuts. In A. Leonardis, H. Bischof, & A. Pinz (Eds.), Lecture notes in computer science : Vol. 3953. European conference on computer vision (ECCV) (pp. 409–422). Graz, Austria, May 2006. Berlin: Springer. Google Scholar
- Caselles, V., & Chambolle, A. (2006). Anisotropic curvature-driven flow of convex sets. Nonlinear Analysis, 65(8), 1547–1577. zbMATHCrossRefMathSciNetGoogle Scholar
- Caselles, V., Chambolle, A., & Novaga, M. (2007). The discontinuity set of solutions of the TV denoising problem and some extensions. Multiscale Modeling and Simulation, 6(3), 879–894. zbMATHCrossRefMathSciNetGoogle Scholar
- Chambolle, A. (2004). An algorithm for mean curvature motion. Interfaces and Free Boundaries, 6(2), 195–218. zbMATHMathSciNetGoogle Scholar
- Chambolle, A. (2005). Total variation minimization and a class of binary MRF models. In Lecture notes in computer science. Energy minimization methods in computer vision and pattern recognition (pp. 136–152). Berlin: Springer. CrossRefGoogle Scholar
- Chambolle, A., & Novaga, M. (2007). Approximation of the anisotropic mean curvature flow. Mathematical Models and Methods in the Applied Sciences, 17(6), 833–844. zbMATHCrossRefMathSciNetGoogle Scholar
- Chambolle, A., & Novaga, M. (2008). Implicit time discretization of the mean curvature flow with a discontinuous forcing term. Interfaces and Free Boundaries, 10(3), 283–300. zbMATHMathSciNetGoogle Scholar
- Chan, T. F., & Esedoḡlu, S. (2005). Aspects of total variation regularized L 1 function approximation. SIAM Journal on Applied Mathematics, 65(5), 1817–1837 (electronic). zbMATHCrossRefMathSciNetGoogle Scholar
- Cohen, L. D. (1991). On active contour models and balloons. CVGIP: Image Understanding, 53(2), 211–218. zbMATHCrossRefGoogle Scholar
- Combettes, P. L., & Wajs, V. R. (2005). Signal recovery by proximal forward-backward splitting. Multiscale Modeling and Simulation, 4(4), 1168–1200 (electronic). zbMATHCrossRefMathSciNetGoogle Scholar
- Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (2001). Introduction to algorithms. Cambridge: MIT Press. zbMATHGoogle Scholar
- Cunningham, W. H. (1985). On submodular function minimization. Combinatoria, 5, 185–192. zbMATHCrossRefMathSciNetGoogle Scholar
- Darbon, J. (2005) Total variation minimization with L 1 data fidelity as a contrast invariant filter. In Proceedings of the 4th international symposium on image and signal processing and analysis (ISPA 2005), Zagreb, Croatia, September 2005. Google Scholar
- Darbon, J., & Sigelle, M. (2006). Image restoration with discrete constrained total variation part i: fast and exact optimization. Journal of Mathematical Imaging and Vision, 26(3), 261–276. CrossRefMathSciNetGoogle Scholar
- Eisner, M. J., & Severance, D. G. (1976). Mathematical techniques for efficient record segmentation in large shared databases. Journal of the ACM, 23(4), 619–635. zbMATHCrossRefMathSciNetGoogle Scholar
- Ekeland, I., & Témam, R. (1999). Classics in applied mathematics : Vol. 28. Convex analysis and variational problems. Philadelphia: SIAM. (English ed., translated from the French). zbMATHGoogle Scholar
- Federer, H. (1969). Geometric measure theory. New York: Springer. zbMATHGoogle Scholar
- Freedman, D., & Drineas, P. (2005). Energy minimization via graph cuts: settling what is possible. In IEEE computer society conference on computer vision and pattern recognition (CVPR) (pp. 939–946). Google Scholar
- Gallo, G., Grigoriadis, M. D., & Tarjan, R. E. (1989). A fast parametric maximum flow algorithm and applications. SIAM Journal on Computing, 18(1), 30–55. zbMATHCrossRefMathSciNetGoogle Scholar
- Giusti, E. (1984). Monographs in mathematics : Vol. 80. Minimal surfaces and functions of bounded variation. Basel: Birkhäuser. zbMATHGoogle Scholar
- Goldberg, A. V., & Tarjan, R. E. (1986). A new approach to the maximum flow problem. In STOC’86: proceedings of the eighteenth annual ACM symposium on theory of computing (pp. 136–146). New York: ACM Press. CrossRefGoogle Scholar
- Goldfarb, D., & Yin, Y. (2007). Parametric maximum flow algorithms for fast total variation minimization. Technical report, Rice University (2007). Google Scholar
- Greig, D. M., Porteous, B. T., & Seheult, A. H. (1989). Exact maximum a posteriori estimation for binary images. Journal of the Royal Statistical Society Series B, 51, 271–279. Google Scholar
- Grötschel, M., Lovász, L., & Schrijver, A. (1981). The ellipsoid method and its consequences in combinatorial optimization. Combinatoria, 1, 169–197. zbMATHCrossRefGoogle Scholar
- Hochbaum, D. S. (2001). An efficient algorithm for image segmentation, Markov random fields and related problems. Journal of the ACM, 48(4), 686–701 (electronic). zbMATHCrossRefMathSciNetGoogle Scholar
- Hochbaum, D. S. (2005). Complexity and algorithms for convex network optimization and other nonlinear problems. A Quarterly Journal of Operations Research, 3(3), 171–216. zbMATHMathSciNetGoogle Scholar
- Iwata, S., Fleischer, L., & Fujishige, S. (2000). A combinatorial, strongly polynomial-time algorithm for minimizing submodular functions. In STOC’00: Proceedings of the thirty-second annual ACM symposium on theory of computing (pp. 97–106). New York: ACM. CrossRefGoogle Scholar
- Juan, O., & Boykov, Y. (2006). Active graph cuts. In Proceedings of the IEEE Computer Society conference on computer vision and pattern recognition (CVPR), (pp. 1023–1029). Google Scholar
- Kholi, P., & Torr, P. (2005). Efficient solving dynamic Markov random fields using graph cuts. In Proceedings of the 10th international conference on computer vision (pp. 922–929). Google Scholar
- Kolmogorov, V., & Zabih, R. (2002). What energy functions can be minimized via graph cuts? In European conference on computer vision (Vol. 3, pp. 65–81), May 2002. Google Scholar
- Kolmogorov, V., & Zabih, R. (2004). What energy functions can be minimized via graph cuts? IEEE Transactions on Pattern Analysis and Machine Intelligence, 2(26), 147–159. CrossRefGoogle Scholar
- Kolmogorov, V., Boykov, Y., & Rother, C. (2007). Applications of parametric maxflow in computer vision. In Proceedings of the IEEE 11th international conference on computer vision (ICCV 2007) (pp. 1–8). Google Scholar
- Lee, J. (2004). A first course in combinatorial optimization. Cambridge: Cambridge University Press. zbMATHGoogle Scholar
- Lovász, L. (1983). Submodular functions and convexity. In Mathematical programming: the state of the art (pp. 235–257). Bonn, 1982. Berlin: Springer. Google Scholar
- Luckhaus, S., & Sturzenhecker, T. (1995). Implicit time discretization for the mean curvature flow equation. Calculus of Variations and Partial Differential Equations, 3(2), 253–271. zbMATHMathSciNetCrossRefGoogle Scholar
- McCormick, S. T. (1996) Fast algorithms for parametric scheduling come from extensions to parametric maximum flow. In Proceedings of the twenty-eighth annual ACM symposium on theory of computing (pp. 319–328). Google Scholar
- Murota, K. (2003). SIAM monographs on discrete mathematics and applications. Discrete convex analysis. Philadelphia: SIAM. zbMATHGoogle Scholar
- Nesterov, Y. (2004). Introductory lectures on convex optimization: a basic course. Dordrecht: Kluwer. zbMATHGoogle Scholar
- Nesterov, Y. (2007) Gradient methods for minimizing composite objective function. Technical report, CORE discussion paper 2007/76, Catholic University of Louvain (2007). Google Scholar
- Paolini, M., & Pasquarelli, F. (2000). Numerical simulation of crystalline curvature flow in 3D by interface diffusion. In GAKUTO international series. Mathematical sciences and applications : Vol. 14. Free boundary problems: theory and applications, II (pp. 376–389). Chiba, 1999. Tokyo: Gakkōtosho. Google Scholar
- Picard, J. C., & Ratliff, H. D. (1975). Minimum cuts and related problems. Networks, 5(4), 357–370. zbMATHCrossRefMathSciNetGoogle Scholar
- Rockafellar, R. T. (1997). Princeton landmarks in mathematics. Convex analysis. Princeton: Princeton University Press (Reprint of the 1970 original Princeton paperbacks). zbMATHGoogle Scholar
- Rouy, E., & Tourin, A. (1992). A viscosity solutions approach to shape-from-shading. SIAM Journal on Numerical Analysis, 29(3), 867–884. zbMATHCrossRefMathSciNetGoogle Scholar
- Schrijver, A. (2000). A combinatorial algorithm minimizing submodular functions in strongly polynomial time. Journal of Combinatorial Theory (B), 80, 436–355. MathSciNetGoogle Scholar
- Sethian, J. A. (1999). Fast marching methods. SIAM Review, 41(2), 199–235 (electronic). zbMATHCrossRefMathSciNetGoogle Scholar
- Tsitsiklis, J. N. (1995). Efficient algorithms for globally optimal trajectories. IEEE Transactions on Automatic Control, 40(9), 1528–1538. zbMATHCrossRefMathSciNetGoogle Scholar