On Total Variation Minimization and Surface Evolution Using Parametric Maximum Flows

Open Access
Article

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 algorithms 

References

  1. Ahuja, R. K., Magnanti, T. L., & Orlin, J. B. (1993). Network flows. Theory, algorithms, and applications. Englewood Cliffs: Prentice-Hall. Google Scholar
  2. Allard, W. K. (2007). Total variation regularization for image denoising, I. Geometric theory. SIAM Journal on Mathematical Analysis, 39(4), 1150–1190. MATHCrossRefMathSciNetGoogle Scholar
  3. Almgren, R. (1993). Variational algorithms and pattern formation in dendritic solidification. Journal of Computational Physics, 106(2), 337–354. MATHMathSciNetGoogle Scholar
  4. 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. MATHCrossRefMathSciNetGoogle Scholar
  5. Alter, F., Caselles, V., & Chambolle, A. (2005). A characterization of convex calibrable sets in ℝN. Mathematische Annalen, 332(2), 329–366. MATHCrossRefMathSciNetGoogle Scholar
  6. 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
  7. 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
  8. Bellettini, G., & Paolini, M. (1996). Anisotropic motion by mean curvature in the context of Finsler geometry. Hokkaido Mathematical Journal, 25(3), 537–566. MATHMathSciNetGoogle Scholar
  9. 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. MATHMathSciNetGoogle Scholar
  10. 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. MATHCrossRefMathSciNetGoogle Scholar
  11. Boros, E., & Hammer, P. L. (2002). Pseudo-boolean optimization. Discrete Applied Mathematics, 123(1–3), 155–225. MATHMathSciNetGoogle Scholar
  12. 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
  13. Boykov, Y., & Kolmogorov, V. (2003). Computing geodesics and minimal surfaces via graph cuts. In International conference on computer vision, pp. 26–33. Google Scholar
  14. 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
  15. 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
  16. 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
  17. Caselles, V., & Chambolle, A. (2006). Anisotropic curvature-driven flow of convex sets. Nonlinear Analysis, 65(8), 1547–1577. MATHCrossRefMathSciNetGoogle Scholar
  18. 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. MATHCrossRefMathSciNetGoogle Scholar
  19. Chambolle, A. (2004). An algorithm for mean curvature motion. Interfaces and Free Boundaries, 6(2), 195–218. MATHMathSciNetGoogle Scholar
  20. 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
  21. Chambolle, A., & Novaga, M. (2007). Approximation of the anisotropic mean curvature flow. Mathematical Models and Methods in the Applied Sciences, 17(6), 833–844. MATHCrossRefMathSciNetGoogle Scholar
  22. 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. MATHMathSciNetGoogle Scholar
  23. 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). MATHCrossRefMathSciNetGoogle Scholar
  24. Cohen, L. D. (1991). On active contour models and balloons. CVGIP: Image Understanding, 53(2), 211–218. MATHCrossRefGoogle Scholar
  25. Combettes, P. L., & Wajs, V. R. (2005). Signal recovery by proximal forward-backward splitting. Multiscale Modeling and Simulation, 4(4), 1168–1200 (electronic). MATHCrossRefMathSciNetGoogle Scholar
  26. Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (2001). Introduction to algorithms. Cambridge: MIT Press. MATHGoogle Scholar
  27. Cunningham, W. H. (1985). On submodular function minimization. Combinatoria, 5, 185–192. MATHCrossRefMathSciNetGoogle Scholar
  28. 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
  29. 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
  30. 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. MATHCrossRefMathSciNetGoogle Scholar
  31. Ekeland, I., & Témam, R. (1999). Classics in applied mathematics : Vol. 28. Convex analysis and variational problems. Philadelphia: SIAM. (English ed., translated from the French). MATHGoogle Scholar
  32. Federer, H. (1969). Geometric measure theory. New York: Springer. MATHGoogle Scholar
  33. 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
  34. 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. MATHCrossRefMathSciNetGoogle Scholar
  35. Giusti, E. (1984). Monographs in mathematics : Vol. 80. Minimal surfaces and functions of bounded variation. Basel: Birkhäuser. MATHGoogle Scholar
  36. 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
  37. Goldfarb, D., & Yin, Y. (2007). Parametric maximum flow algorithms for fast total variation minimization. Technical report, Rice University (2007). Google Scholar
  38. 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
  39. Grötschel, M., Lovász, L., & Schrijver, A. (1981). The ellipsoid method and its consequences in combinatorial optimization. Combinatoria, 1, 169–197. MATHCrossRefGoogle Scholar
  40. 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). MATHCrossRefMathSciNetGoogle Scholar
  41. 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. MATHMathSciNetGoogle Scholar
  42. 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
  43. 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
  44. 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
  45. 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
  46. 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
  47. 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
  48. Lee, J. (2004). A first course in combinatorial optimization. Cambridge: Cambridge University Press. MATHGoogle Scholar
  49. Lovász, L. (1983). Submodular functions and convexity. In Mathematical programming: the state of the art (pp. 235–257). Bonn, 1982. Berlin: Springer. Google Scholar
  50. 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. MATHMathSciNetCrossRefGoogle Scholar
  51. 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
  52. Murota, K. (2003). SIAM monographs on discrete mathematics and applications. Discrete convex analysis. Philadelphia: SIAM. MATHGoogle Scholar
  53. Nesterov, Y. (2004). Introductory lectures on convex optimization: a basic course. Dordrecht: Kluwer. MATHGoogle Scholar
  54. Nesterov, Y. (2007) Gradient methods for minimizing composite objective function. Technical report, CORE discussion paper 2007/76, Catholic University of Louvain (2007). Google Scholar
  55. 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
  56. Picard, J. C., & Ratliff, H. D. (1975). Minimum cuts and related problems. Networks, 5(4), 357–370. MATHCrossRefMathSciNetGoogle Scholar
  57. Rockafellar, R. T. (1997). Princeton landmarks in mathematics. Convex analysis. Princeton: Princeton University Press (Reprint of the 1970 original Princeton paperbacks). MATHGoogle Scholar
  58. Rouy, E., & Tourin, A. (1992). A viscosity solutions approach to shape-from-shading. SIAM Journal on Numerical Analysis, 29(3), 867–884. MATHCrossRefMathSciNetGoogle Scholar
  59. Schrijver, A. (2000). A combinatorial algorithm minimizing submodular functions in strongly polynomial time. Journal of Combinatorial Theory (B), 80, 436–355. MathSciNetGoogle Scholar
  60. Sethian, J. A. (1999). Fast marching methods. SIAM Review, 41(2), 199–235 (electronic). MATHCrossRefMathSciNetGoogle Scholar
  61. Tsitsiklis, J. N. (1995). Efficient algorithms for globally optimal trajectories. IEEE Transactions on Automatic Control, 40(9), 1528–1538. MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.CMAPEcole Polytechnique, CNRSPalaiseauFrance
  2. 2.Mathematics DepartmentUCLALos AngelesUSA

Personalised recommendations