Advertisement

International Journal of Computer Vision

, Volume 98, Issue 1, pp 103–121 | Cite as

Completely Convex Formulation of the Chan-Vese Image Segmentation Model

  • Ethan S. Brown
  • Tony F. Chan
  • Xavier BressonEmail author
Article

Abstract

The active contours without edges model of Chan and Vese (IEEE Transactions on Image Processing 10(2):266–277, 2001) is a popular method for computing the segmentation of an image into two phases, based on the piecewise constant Mumford-Shah model. The minimization problem is non-convex even when the optimal region constants are known a priori. In (SIAM Journal of Applied Mathematics 66(5):1632–1648, 2006), Chan, Esedoḡlu, and Nikolova provided a method to compute global minimizers by showing that solutions could be obtained from a convex relaxation. In this paper, we propose a convex relaxation approach to solve the case in which both the segmentation and the optimal constants are unknown for two phases and multiple phases. In other words, we propose a convex relaxation of the popular K-means algorithm. Our approach is based on the vector-valued relaxation technique developed by Goldstein et  al. (UCLA CAM Report 09-77, 2009) and Brown et al. (UCLA CAM Report 10-43, 2010). The idea is to consider the optimal constants as functions subject to a constraint on their gradient. Although the proposed relaxation technique is not guaranteed to find exact global minimizers of the original problem, our experiments show that our method computes tight approximations of the optimal solutions. Particularly, we provide numerical examples in which our method finds better solutions than the method proposed by Chan et al. (SIAM Journal of Applied Mathematics 66(5):1632–1648, 2006), whose quality of solutions depends on the choice of the initial condition.

Keywords

Image segmentation Chan-Vese model Convex relaxation Level set method Vector-valued functional lifting K-means 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alberti, G., Bouchitté, G., & Dal Maso, G. (1999). The calibration method for the Mumford-Shah functional. Comptes Rendus de L’Académie Des Sciences. Series 1, Mathematics, 329(3), 249–254. zbMATHCrossRefGoogle Scholar
  2. Ambrosio, L., Fusco, N., & Pallara, D. (2000). Functions of bounded variation and free discontinuity problems. Oxford: Clarendon Press. zbMATHGoogle Scholar
  3. Arrow, K., Hurwicz, L., & Uzawa, H. (1958). Stanford mathematical studies in the social sciences: Vol. II. Studies in linear and non-linear programming. With contributions by H. B. Chenery, S. M. Johnson, S. Karlin, T. Marschak, R. M. Solow. Stanford: Stanford University Press. Google Scholar
  4. Bae, E., & Tai, X.-C. (2009). Graph cut optimization for the piecewise constant level set method applied to multiphase image segmentation. In International conference on scale space and variational methods in computer vision (pp. 1–13). CrossRefGoogle Scholar
  5. Bae, E., Yuan, J., & Tai, X.-C. (2009). Global minimization for continuous multiphase partitioning problems using a dual approach. International Journal of Computer Vision, 92(1), 112–129. MathSciNetCrossRefGoogle Scholar
  6. Bae, E., Yuan, J., Tai, X.-C., & Boykov, Y. (2010). A study on continuous max-flow and min-cut approaches. Part II: multiple linearly ordered labels (UCLA CAM Report 10-62). Google Scholar
  7. Bertsekas, D. P. (1982). Constrained optimization and Lagrange multiplier methods. New York: Academic Press. zbMATHGoogle Scholar
  8. Bresson, X., Esedoglu, S., Vandergheynst, P., Thiran, J., & Osher, S. (2007). Fast global minimization of the active contour/snake models. Journal of Mathematical Imaging and Vision, 28(2), 151–167. MathSciNetCrossRefGoogle Scholar
  9. Brown, E. S., Chan, T. F., & Bresson, X. (2009). Convex formulations for piecewise constant Mumford-Shah image segmentation (UCLA CAM Report 09-66). Google Scholar
  10. Brown, E. S., Chan, T. F., & Bresson, X. (2010). A convex relaxation method for a class of vector-valued minimization problems with applications to Mumford-Shah segmentation (UCLA CAM Report 10-43). Google Scholar
  11. Chambolle, A., Cremers, D., & Pock, T. (2008). A convex approach for computing minimal partitions (Technical report TR-2008-05). Bonn: Dept. of Computer Science, University of Bonn. Google Scholar
  12. Chambolle, A., & Pock, T. (2010). A first-order primal-dual algorithm for convex problems with applications to imaging (R.I. 685). CMAP, Ecole Polytechnique. Google Scholar
  13. Chan, T. F., Esedoḡlu, S., & Nikolova, M. (2006). Algorithms for finding global minimizers of image segmentation and denoising models. SIAM Journal on Applied Mathematics, 66(5), 1632–1648. MathSciNetzbMATHCrossRefGoogle Scholar
  14. Chan, T. F., & Vese, L. A. (2001). Active contours without edges. IEEE Transactions on Image Processing, 10(2), 266–277. zbMATHCrossRefGoogle Scholar
  15. El-Zehiry, N., Xu, S., Sahoo, P., & Elmaghraby, A. (2007). Graph cut optimization for the Mumford-Shah model. In IASTED international conference on visualization, imaging and image processing (pp. 182–187). Google Scholar
  16. Esser, E., Zhang, X., & Chan, T. F. (2010). A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science. SIAM Journal on Imaging Sciences, 3(4), 1015–1046. MathSciNetzbMATHCrossRefGoogle Scholar
  17. Evans, L. C., & Gariepy, R. F. (2000). Measure theory and fine properties of functions. Boca Raton: CRC Press. Google Scholar
  18. Federer, H. (1959). Curvature measures. Transactions of the American Mathematical Society, 93(3), 418–491. MathSciNetzbMATHCrossRefGoogle Scholar
  19. Fleming, W., & Rishel, R. (1960). An integral formula for total gradient variation. Archiv der Mathematik, 11(1), 218–222. MathSciNetzbMATHCrossRefGoogle Scholar
  20. Goldluecke, S., & Cremers, D. (2010). Convex relaxation for multilabel problems with product label spaces. In European conference on computer vision (pp. 225–238). Google Scholar
  21. Goldstein, T., Bresson, X., & Osher, S. (2009). Geometric applications of the split Bregman method: segmentation and surface reconstruction. Journal of Scientific Computing, 45(1–3), 272–293. MathSciNetGoogle Scholar
  22. Goldstein, T., Bresson, X., & Osher, S. (2009). Global minimization of Markov random fields with applications to optical flow (UCLA CAM Report 09-77). Google Scholar
  23. Ishikawa, H. (2003). Exact optimization for Markov random fields with convex priors. IEEE Transactions on Pattern Analysis and Machine Intelligence, 25(10), 1333–1336. CrossRefGoogle Scholar
  24. Lellmann, J., Becker, F., & Schnörr, C. (2009). Convex optimization for multi-class image labeling with a novel family of total variation based regularizers. In International conference on computer vision (pp. 646–653). CrossRefGoogle Scholar
  25. Lellmann, J., Kappes, J., Yuan, J., Becker, F., & Schnörr, C. (2009). Convex multi-class image labeling by simplex-constrained total variation. In International conference on scale space and variational methods in computer vision (pp. 150–162). CrossRefGoogle Scholar
  26. Lellmann, J., & Schnörr, C. (2010). Continuous multiclass labeling approaches and algorithms (Tech. Rep.). Heidelberg: University of Heidelberg. Google Scholar
  27. Lie, J., Lysaker, M., & Tai, X.-C. (2006). A binary level set model and some applications to Mumford-Shah image segmentation. IEEE Transactions on Image Processing, 15(5), 1171–1181. CrossRefGoogle Scholar
  28. Lieb, E. H., & Loss, M. (2001). Analysis. Providence: Am. Math. Soc. zbMATHGoogle Scholar
  29. MacQueen, J. (1967). Some methods for classification and analysis of multivariate observations. In Proceedings of the fifth Berkeley symposium on mathematical statistics and probability (pp. 281–297). Google Scholar
  30. Mumford, D., & Shah, J. (1989). Optimal approximations of piecewise smooth functions and associated variational problems. Communications on Pure and Applied Mathematics, 42, 577–685. MathSciNetzbMATHCrossRefGoogle Scholar
  31. Osher, S., & Sethian, J. A. (1988). Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. Journal of Computational Physics, 79(1), 12–49. MathSciNetzbMATHCrossRefGoogle Scholar
  32. Pock, T., Chambolle, A., Bischof, H., & Cremers, D. (2009). An algorithm for minimizing the Mumford-Shah functional. In IEEE conference on computer vision (ICCV). Google Scholar
  33. Pock, T., Cremers, D., Bischof, H., & Chambolle, A. (2010). Global solutions of variational models with convex regularization. SIAM Journal on Imaging Sciences, 2(3), 1122–1145. MathSciNetCrossRefGoogle Scholar
  34. Pock, T., Schoenemann, T., Graber, G., Bischof, H., & Cremers, D. (2008). A convex formulation of continuous multi-label problems. In European conference on computer vision (ECCV) (pp. 792–805). Google Scholar
  35. Popov, L. D. (1980). A modification of the Arrow-Hurwitz method of search for saddle points. Matematičeskie Zametki, 28(5), 777–784, 803. zbMATHGoogle Scholar
  36. Shekhovtsov, A., Kovtun, I., & Hlavác, V. (2008). Efficient MRF deformation model for non-rigid image matching. Computer Vision and Image Understanding, 112(1), 91–99. CrossRefGoogle Scholar
  37. Strandmark, P., Kahl, F., & Overgaard, N. C. (2009). Optimizing parametric total variation models. In International conference on computer vision (pp. 2240–2247). CrossRefGoogle Scholar
  38. Strang, G. (1983). Maximal flow through a domain. Mathematical Programming, 26(2), 123–143. MathSciNetzbMATHCrossRefGoogle Scholar
  39. Yuan, J., Bae, E., Tai, X.-C., & Boykov, Y. (2010). A study on continuous max-flow and min-cut approaches. Part I: binary labeling (UCLA CAM Report 10-61). Google Scholar
  40. Zach, C., Gallup, D., Frahm, J. M., & Niethammer, M. (2008). Fast global labeling for real-time stereo using multiple plane sweeps. In Vision, modeling, and visualization (pp. 243–252). Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Ethan S. Brown
    • 1
  • Tony F. Chan
    • 2
  • Xavier Bresson
    • 3
    Email author
  1. 1.Department of MathematicsUniversity of California Los AngelesLos AngelesUSA
  2. 2.Hong Kong University of Science and TechnologyHong KongHong Kong
  3. 3.Department of Computer ScienceCity University of Hong KongHong KongHong Kong

Personalised recommendations