Journal of Scientific Computing

, Volume 45, Issue 1–3, pp 272–293 | Cite as

Geometric Applications of the Split Bregman Method: Segmentation and Surface Reconstruction

  • Tom Goldstein
  • Xavier Bresson
  • Stanley Osher
Open Access


Variational models for image segmentation have many applications, but can be slow to compute. Recently, globally convex segmentation models have been introduced which are very reliable, but contain TV-regularizers, making them difficult to compute. The previously introduced Split Bregman method is a technique for fast minimization of L1 regularized functionals, and has been applied to denoising and compressed sensing problems. By applying the Split Bregman concept to image segmentation problems, we build fast solvers which can out-perform more conventional schemes, such as duality based methods and graph-cuts. The convex segmentation schemes also substantially outperform conventional level set methods, such as the Chan-Vese level set-based segmentation algorithm. We also consider the related problem of surface reconstruction from unorganized data points, which is used for constructing level set representations in 3 dimensions. The primary purpose of this paper is to examine the effectiveness of “Split Bregman” techniques for solving these problems, and to compare this scheme with more conventional methods.

Image segmentation Split Bregman Bregman iteration Total variation 


  1. 1.
    Adalsteinsson, D., Sethian, J.: A fast level set method for propagating interfaces. J. Comput. Phys. 118, 269–277 (1995) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Almgren, F., Taylor, J.E., Wang, L.: Curvature-driven flows: a variational approach. SIAM J. Control Optim. 31(2), 387–438 (1993) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Amenta, N., Bern, M.: Surface reconstruction by Voronoi filtering. In: SCG’98: Proceedings of the Fourteenth Annual Symposium on Computational Geometry, pp. 39–48. ACM, New York (1998) CrossRefGoogle Scholar
  4. 4.
    Amenta, N., Bern, M., Kamvysselis, M.: A new Voronoi-based surface reconstruction algorithm. In: SIGGRAPH’98: Proceedings of the 25th Annual Conference on Computer Graphics and Interactive Techniques, pp. 415–421. ACM, New York (1998) CrossRefGoogle Scholar
  5. 5.
    Aujol, J.-F., Chambolle, A.: Dual norms and image decomposition models. Int. J. Comput. Vision 63(1), 85–104 (2005) CrossRefMathSciNetGoogle Scholar
  6. 6.
    Bertsekas, D.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, San Diego (1996) Google Scholar
  7. 7.
    Boissonnat, J.-D.: Geometric structures for three-dimensional shape representation. ACM Trans. Graph. 3(4), 266–286 (1984) CrossRefGoogle Scholar
  8. 8.
    Boykov, Y., Veksler, O., Zabih, R.: Fast approximate energy minimization via graph cuts. IEEE Trans. Pattern Anal. Mach. Intell. 23, 1222–1239 (2001) CrossRefGoogle Scholar
  9. 9.
    Bregman, L.: The relaxation method of finding the common points of convex sets and its application to the solution of problems in convex optimization. USSR Comput. Math. Math. Phys. 7, 200–217 (1967) CrossRefGoogle Scholar
  10. 10.
    Bresson, X., Chan, T.: Active contours based on chambolle’s mean curvature motion. In: IEEE International Conference on Image Processing, pp. 33–36 (2007) Google Scholar
  11. 11.
    Bresson, X., Esedoglu, S., Vandergheynst, P., Thiran, J.-P., Osher, S.: Fast global minimization of the active contour/snake model. J. Math. Imaging Vis. 28, 151–167 (2007) CrossRefMathSciNetGoogle Scholar
  12. 12.
    Burger, M., Hintermuller, M.: Projected gradient flows for bv/level set relaxation. UCLA CAM technical report, 05-40 (2005) Google Scholar
  13. 13.
    Carson, C., Belongie, S., Greenspan, H., Malik, J.: Blobworld: Image segmentation using expectation-maximization and its application to image querying. IEEE Trans. Pattern Anal. Mach. Intell. 24, 1026–1038 (1999) CrossRefGoogle Scholar
  14. 14.
    Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. In: IEEE International Conference on Computer Vision, p. 694 (1995) Google Scholar
  15. 15.
    Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20(1–2), 89–97 (2004) MathSciNetGoogle Scholar
  16. 16.
    Chambolle, A.: An algorithm for mean curvature motion. Interfaces Free Bound. 6(2), 195–218 (2004) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Chambolle, A., Darbon, J.: On total variation minimization and surface evolution using parametric maximum flows. UCLA CAM report 08-19 (2008) Google Scholar
  18. 18.
    Chan, T.F., Vese, L.: Active contours without edges. IEEE Trans. Image Process. 10, 266–277 (2001) MATHCrossRefGoogle Scholar
  19. 19.
    Chan, T.F., Golub, G.H., Mulet, P.: A nonlinear primal-dual method for total variation-based image restoration. SIAM J. Sci. Comput. 20, 1964–1977 (1999) CrossRefMathSciNetGoogle Scholar
  20. 20.
    Chan, T.F., Esedoglu, S., Nikolova, M.: Algorithms for finding global minimizers of image segmentation and denoising models. SIAM J. Appl. Math. 66, 1932–1648 (2006) MathSciNetGoogle Scholar
  21. 21.
    Darbon, J., Sigelle, M.: A fast and exact algorithm for total variation minimization. IbPRIA 2005 3522(1), 351–359 (2005) Google Scholar
  22. 22.
    Donoho, D.L., Johnstone, I.M.: Adapting to unknown smoothness via wavelet shrinkage J. Am. Stat. Assoc. 90(432), 1200–1224 (1995) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Esser, E.: Applications of Lagrangian-based alternating direction methods and connections to split Bregman. UCLA CAM technical report, 09-31 (2009) Google Scholar
  24. 24.
    Felzenszwalb, P.F., Huttenlocher, D.P.: Efficient graph-based image segmentation. Int. J. Comput. Vis. 59(2), 167–181 (2004) CrossRefGoogle Scholar
  25. 25.
    Goldfarb, D., Yin, W.: Parametric maximum flow algorithms for fast total variation minimization. CAAM technical report, TR07-09 (2008) Google Scholar
  26. 26.
    Goldstein, T., Osher, S.: The split Bregman method for l1 regularized problems. UCLA CAM report 08-29 (2008) Google Scholar
  27. 27.
    He, L., Chang, T.-C., Osher, S.: Mr image reconstruction from sparse radial samples by using iterative refinement procedures. In: Proceedings of the 13th Annual Meeting of ISMRM, p. 696 (2006) Google Scholar
  28. 28.
    Hoppe, H., Derose, T., Duchamp, T., Mcdonald, J., Stuetzle, W.: Surface reconstruction from unorganized points. Comput. Graph. 26(2), 71–78 (1992) CrossRefGoogle Scholar
  29. 29.
    Jonasson, L., Bresson, X., Hagmann, P., Cuisenaire, O., Meuli, R., Thiran, J.-P.: White matter fiber tract segmentation in dt-mri using geometric flows. Med. Image Anal. 9(9), 223–236 (2005) CrossRefGoogle Scholar
  30. 30.
    Kass, W., Witkin, A., Terzopoulos, D.: Snakes: Active contour models. Int. J. Comput. Vis. 1(4), 312–331 (2004) Google Scholar
  31. 31.
    Kimmel, R., Bruckstein, A.M.: Regularized Laplacian zero crossings as optimal edge integrators. Int. J. Comput. Vis. 53, 225–243 (2001) CrossRefGoogle Scholar
  32. 32.
    Kolmogorov, V., Zabih, R.: What energy functions can be minimized via graph cuts. IEEE Trans. Pattern Anal. Mach. Intell., pp. 147–159 (2004) Google Scholar
  33. 33.
    Malladi, R., Kimmel, R., Adalsteinsson, D., Sapiro, G., Caselles, V., Sethian, J.A.: A geometric approach to segmentation and analysis of 3d medical images. In: MMBIA’96: Proceedings of the 1996 Workshop on Mathematical Methods in Biomedical Image Analysis (MMBIA’96), Washington, DC, USA, p. 244. IEEE Comput. Soc., Los Alamitos (1996) CrossRefGoogle Scholar
  34. 34.
    Mumford, D., Shah, J.: Optimal approximation by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42, 577–685 (1989) MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Springer, Berlin (2003) MATHGoogle Scholar
  36. 36.
    Osher, S., Fedkiw, R.P.: Level set methods. Technical report, in Imaging, Vision and Graphics (2003) Google Scholar
  37. 37.
    Osher, S., Sethian, J.A.: Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79, 12–49 (1988) MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Osher, S., Burger, M., Goldfarb, D., Xu, J., Yin, W.: An iterative regularization method for total variation-based image restoration. MMS 4, 460–489 (2005) MATHMathSciNetGoogle Scholar
  39. 39.
    Rogers, D.F.: An Introduction to NURBS: With Historical Perspective. Morgan Kaufmann, San Mateo (2001) Google Scholar
  40. 40.
    Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992) MATHCrossRefGoogle Scholar
  41. 41.
    Sethian, J.A.: Level set methods and fast marching methods: Evolving. In: Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge University Press, Cambridge (1999) Google Scholar
  42. 42.
    Setzer, S.: Split Bregman algorithm, Douglas-Rachford splitting and frame shrinkage. In: Proceedings of the Second International Conference on Scale Space Methods and Variational Methods in Computer Vision (2009) Google Scholar
  43. 43.
    Sussman, M., Smereka, P., Osher, S.: A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114(1), 146–159 (1994) MATHCrossRefGoogle Scholar
  44. 44.
    Tschirren, J., Hoffman, E.A., McLennan, G., Sonka, M.: Intrathoracic airway trees: segmentation and airway morphology analysis from low-dose ct scans. IEEE Trans. Med. Imag. 24, 1529–1539 (2005) CrossRefGoogle Scholar
  45. 45.
    Wang, Y., Yin, W., Zhang, Y.: A fast algorithm for image deblurring with total variation regularization. CAAM technical reports (2007) Google Scholar
  46. 46.
    Yezzi, A., Kichenassamy, S., Kumar, A., Olver, P., Tannenbaum, A.: A geometric snake model for segmentation of medical imagery. IEEE Trans. Med. Imag. 16(2), 199–209 (1997) CrossRefGoogle Scholar
  47. 47.
    Yin, W.: Analysis and generalizations of the linearized Bregman method. UCLA CAM technical report, 09-42 (2009) Google Scholar
  48. 48.
    Yin, W.: Pgc: A preflow-push based graph-cut solver. Version 2.32 Google Scholar
  49. 49.
    Yin, W., Osher, S., Goldfarb, D., Darbon, J.: Bregman iterative algorithms for l1-minimization with applications to compressed sensing. SIAM J. Imag. Sci. 1, 142–168 (2008) CrossRefMathSciNetGoogle Scholar
  50. 50.
    Wang, Y., Yang, J., Yin, W., Zhang, Y.: A new alternating minimization algorithm for total variation image reconstruction. SIAM J. Imag. Sci. 1(3), 248–272 (2008) MATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    Zhao, H.-K., Osher, S., Merriman, B., Kang, M.: Implicit and nonparametric shape reconstruction from unorganized data using a variational level set method. Comput. Vis. Image Underst. 80(3), 295–314 (2000) MATHCrossRefGoogle Scholar
  52. 52.
    Zhao, H.-K., Osher, S., Fedkiw, R.: Fast surface reconstruction using the level set method. In: VLSM’01: Proceedings of the IEEE Workshop on Variational and Level Set Methods (VLSM’01), Washington, DC, USA, p. 194. IEEE Comput. Soc., Los Alamitos (2001) CrossRefGoogle Scholar

Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.Department of MathematicsUCLALos AngelesUSA

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