Journal of Mathematical Imaging and Vision

, Volume 28, Issue 2, pp 151–167 | Cite as

Fast Global Minimization of the Active Contour/Snake Model

  • Xavier Bresson
  • Selim Esedoḡlu
  • Pierre Vandergheynst
  • Jean-Philippe Thiran
  • Stanley Osher
Article

Abstract

The active contour/snake model is one of the most successful variational models in image segmentation. It consists of evolving a contour in images toward the boundaries of objects. Its success is based on strong mathematical properties and efficient numerical schemes based on the level set method. The only drawback of this model is the existence of local minima in the active contour energy, which makes the initial guess critical to get satisfactory results. In this paper, we propose to solve this problem by determining a global minimum of the active contour model. Our approach is based on the unification of image segmentation and image denoising tasks into a global minimization framework. More precisely, we propose to unify three well-known image variational models, namely the snake model, the Rudin–Osher–Fatemi denoising model and the Mumford–Shah segmentation model. We will establish theorems with proofs to determine the existence of a global minimum of the active contour model. From a numerical point of view, we propose a new practical way to solve the active contour propagation problem toward object boundaries through a dual formulation of the minimization problem. The dual formulation, easy to implement, allows us a fast global minimization of the snake energy. It avoids the usual drawback in the level set approach that consists of initializing the active contour in a distance function and re-initializing it periodically during the evolution, which is time-consuming. We apply our segmentation algorithms on synthetic and real-world images, such as texture images and medical images, to emphasize the performances of our model compared with other segmentation models.

Keywords

Active contour Global minimization Weighted total variation norm ROF model Mumford–Shah energy Dual formulation of TV 

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References

  1. 1.
    Kass, M., Witkin, A., Terzopoulos, D.: Snakes: active contour models. Int. J. Comput. Vis. 1(14), 321–331 (1998) Google Scholar
  2. 2.
    Malladi, R., Kimmel, R., Adalsteinsson, D., Sapiro, G., Caselles, V., Sethian, J.: A geometric approach to segmentation and analysis of 3D medical images. In: Mathematical Methods, Biomedical Image Analysis Workshop (1996) Google Scholar
  3. 3.
    Yezzi, A., Kichenassamy, S., Kumar, A., Olver, P., Tannenbaum, A.: A geometric snake model for segmentation of medical imagery. IEEE Trans. Med. Imaging 16(2), 199–209 (1997) CrossRefGoogle Scholar
  4. 4.
    Jonasson, L., Bresson, X., Hagmann, P., Cuisenaire, O., Meuli, R., Thiran, J.: White matter fiber tract segmentation in DT-MRI using geometric flows. Med. Image Anal. 9(3), 223–236 (2005) CrossRefGoogle Scholar
  5. 5.
    Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. Int. J. Comput. Vis. 22(1), 61–79 (1997) MATHCrossRefGoogle Scholar
  6. 6.
    Kichenassamy, S., Kumar, A., Olver, P., Tannenbaum, A., Yezzi, A.: Conformal curvature flows: from phase transitions to active vision. Arch. Ration. Mech. Anal. 134, 275–301 (1996) MATHCrossRefGoogle Scholar
  7. 7.
    Crandall, M., Ishii, H., Lions, P.: Users’ guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27(1), 1–69 (1992) MATHGoogle Scholar
  8. 8.
    Osher, S., Sethian, J.: Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988) MATHCrossRefGoogle Scholar
  9. 9.
    Osher, S.: Level set methods. In: Osher, S., Paragios, N. (eds.) Geometric Level Set Methods in Imaging, Vision and Graphics, pp.3–20. Springer, New York (2003) CrossRefGoogle Scholar
  10. 10.
    Sethian, J.: Level set methods and fast marching methods: evolving interfaces. In: Computational Geometry, Fluid Mechanics, Computer Vision and Material Sciences. Cambridge University Press, Cambridge (1999) Google Scholar
  11. 11.
    Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Springer, New York (2003) MATHGoogle Scholar
  12. 12.
    Jiang, G.-S., Peng, D.: Weighted ENO schemes for Hamilton–Jacobi equations. J. Sci. Comput. 21(6), 2126–2143 (1999) Google Scholar
  13. 13.
    Chan, T., Esedoḡlu, S., Nikolova, M.: Algorithms for finding global minimizers of image segmentation and denoising models, UCLA CAM Report 04-54 (2004) Google Scholar
  14. 14.
    Chan, T., Vese, L.: Active contours without edges. IEEE Trans. Image Process. 10(2), 266–277 (2001) MATHCrossRefGoogle Scholar
  15. 15.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60(1–4), 259–268 (1992) MATHCrossRefGoogle Scholar
  16. 16.
    Mumford, D., Shah, J.: Optimal approximations of piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42, 577–685 (1989) MATHCrossRefGoogle Scholar
  17. 17.
    Chan, T., Golub, G., Mulet, P.: A nonlinear primal-dual method for total variation-based image restoration. SIAM J. Sci. Comput. 20(6), 1964–1977 (1999) MATHCrossRefGoogle Scholar
  18. 18.
    Carter, J.: Dual methods for total variation-based image restoration. Ph.D. thesis, UCLA (2001) Google Scholar
  19. 19.
    Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20(1–2), 89–97 (2004) Google Scholar
  20. 20.
    Aujol, J.-F., Chambolle, A.: Dual norms and image decomposition models. Int. J. Comput. Vis. 63(1), 85–104 (2005) CrossRefGoogle Scholar
  21. 21.
    Aujol, J.-F., Gilboa, G., Chan, T., Osher, S.: Structure-texture image decomposition—modeling, algorithms, and parameter selection. Int. J. Comput. Vis. 67(1), 111–136 (2006) CrossRefGoogle Scholar
  22. 22.
    Alliney, S.: Digital filters as absolute norm regularizers. IEEE Trans. Signal Process. 40(6), 1548–1562 (1992) MATHCrossRefGoogle Scholar
  23. 23.
    Alliney, S.: Recursive median filters of increasing order: a variational approach. IEEE Trans. Signal Process. 44(6), 1346–1354 (1996) CrossRefGoogle Scholar
  24. 24.
    Alliney, S.: A property of the minimum vectors of a regularizing functional defined by means of the absolute norm. IEEE Trans. Signal Process. 45(4), 913–917 (1997) CrossRefGoogle Scholar
  25. 25.
    Cheon, E., Paranjpye, A.: Noise removal by total variation minimization. UCLA MATH 199 project report, adviser. L. Vese (2002) Google Scholar
  26. 26.
    Nikolova, M.: Minimizers of cost-functions involving nonsmooth data-fidelity terms. SIAM J. Numer. Anal. 40(3), 965–994 (2002) MATHCrossRefGoogle Scholar
  27. 27.
    Alliney, S.: A variational approach to remove outliers and impulse noise. J. Math. Imaging Vis. 20(12), 99–120 (2004) Google Scholar
  28. 28.
    Alliney, S.: Weakly constrained minimization. Application to the estimation of images and signals involving constant regions. J. Math. Imaging Vis. 21(2), 155–175 (2004) CrossRefGoogle Scholar
  29. 29.
    Chan, T., Esedoḡlu, S.: Aspects of total varation regularized L 1 function approximation. UCLA CAM Report 04-07 (2004) Google Scholar
  30. 30.
    Meyer, Y.: Oscillating patterns in image processing and in some nonlinear evolution equations. The Fifteenth Dean Jacquelines B. Lewis Memorial Lectures (2001) Google Scholar
  31. 31.
    Vese, L., Osher, S.: Modeling textures with total variation minimization and oscillating patterns. J. Sci. Comput. 19, 553–572 (2003) MATHCrossRefGoogle Scholar
  32. 32.
    Bresson, X., Esedoḡlu, S., Vandergheynst, P., Thiran, J.-P., Osher, S.: Global minimizers of the active Contour/Snake model. UCLA CAM Report 05-04 (2005) Google Scholar
  33. 33.
    Vese, L., Chan, T.: A multiphase level set framework for image segmentation using the Mumford and Shah model. Int. J. Comput. Vis. 50(3), 271–293 (2002) MATHCrossRefGoogle Scholar
  34. 34.
    Cohen, L., Bardinet, E., Ayache, N.: Surface reconstruction using active contour models. In: SPIE International Symposium on Optics, Imaging and Instrumentation (1993) Google Scholar
  35. 35.
    Cohen, L., Kimmel, R.: Global minimum for active contour models: A minimal path approach. Int. J. Comput. Vis. 24(1), 57–78 (1997) CrossRefGoogle Scholar
  36. 36.
    Appleton, B., Talbot, H.: Globally optimal geodesic active contours. J. Math. Imaging Vis. 23(1), 67–86 (2005) CrossRefGoogle Scholar
  37. 37.
    Strang, G.: L 1 and L approximation of vector fields in the plane. In: Nonlinear Partial Differential Equations in Applied Science, pp. 273–288. Kinokuniya, Tokyo (1982) Google Scholar
  38. 38.
    Strang, G.: Maximal flow through a domain. Math. Program. 26(2), 123–143 (1983) MATHCrossRefGoogle Scholar
  39. 39.
    Gomes, J., Faugeras, O.: Reconciling distance functions and level sets. J. Vis. Commun. Image Represent. 11, 209–223 (2000) CrossRefGoogle Scholar
  40. 40.
    Leung, S., Osher, S.: Global minimization of the active contour model with TV-inpainting and two-phase denoising. In: Variational, Geometric, and Level Set Methods in Computer Vision (VLSM). Lecture Notes in Computer Science, vol. 3752, pp. 149–160. Springer, New York (2005) CrossRefGoogle Scholar
  41. 41.
    Chambolle, A., Vore, R.D., Lee, N., Lucier, B.: Nonlinear wavelet image processing: Variational problems, compression, and noise removal through wavelet shrinkage. IEEE Trans. Image Process. 7(3), 319–335 (1998) MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Xavier Bresson
    • 1
  • Selim Esedoḡlu
    • 2
  • Pierre Vandergheynst
    • 1
  • Jean-Philippe Thiran
    • 1
  • Stanley Osher
    • 3
  1. 1.Signal Processing InstituteSwiss Federal Institute of Technology (EPFL)LausanneSwitzerland
  2. 2.Department of MathematicsUniversity of MichiganAnn ArborUSA
  3. 3.Department of MathematicsUniversity of CaliforniaLos AngelesUSA

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