Fast Global Minimization of the Active Contour/Snake Model

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.

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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)

  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)

    Article  Google 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)

    Article  Google Scholar 

  5. 5.

    Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. Int. J. Comput. Vis. 22(1), 61–79 (1997)

    MATH  Article  Google 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)

    MATH  Article  Google 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)

    MATH  Google 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)

    MATH  Article  Google 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)

    Google 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)

    Google 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)

  14. 14.

    Chan, T., Vese, L.: Active contours without edges. IEEE Trans. Image Process. 10(2), 266–277 (2001)

    MATH  Article  Google 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)

    MATH  Article  Google 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)

    MATH  Article  Google 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)

    MATH  Article  Google Scholar 

  18. 18.

    Carter, J.: Dual methods for total variation-based image restoration. Ph.D. thesis, UCLA (2001)

  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)

    Article  Google 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)

    Article  Google Scholar 

  22. 22.

    Alliney, S.: Digital filters as absolute norm regularizers. IEEE Trans. Signal Process. 40(6), 1548–1562 (1992)

    MATH  Article  Google Scholar 

  23. 23.

    Alliney, S.: Recursive median filters of increasing order: a variational approach. IEEE Trans. Signal Process. 44(6), 1346–1354 (1996)

    Article  Google 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)

    Article  Google Scholar 

  25. 25.

    Cheon, E., Paranjpye, A.: Noise removal by total variation minimization. UCLA MATH 199 project report, adviser. L. Vese (2002)

  26. 26.

    Nikolova, M.: Minimizers of cost-functions involving nonsmooth data-fidelity terms. SIAM J. Numer. Anal. 40(3), 965–994 (2002)

    MATH  Article  Google 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)

    Article  Google Scholar 

  29. 29.

    Chan, T., Esedoḡlu, S.: Aspects of total varation regularized L 1 function approximation. UCLA CAM Report 04-07 (2004)

  30. 30.

    Meyer, Y.: Oscillating patterns in image processing and in some nonlinear evolution equations. The Fifteenth Dean Jacquelines B. Lewis Memorial Lectures (2001)

  31. 31.

    Vese, L., Osher, S.: Modeling textures with total variation minimization and oscillating patterns. J. Sci. Comput. 19, 553–572 (2003)

    MATH  Article  Google 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)

  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)

    MATH  Article  Google 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)

  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)

    Article  Google Scholar 

  36. 36.

    Appleton, B., Talbot, H.: Globally optimal geodesic active contours. J. Math. Imaging Vis. 23(1), 67–86 (2005)

    Article  Google 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)

    MATH  Article  Google Scholar 

  39. 39.

    Gomes, J., Faugeras, O.: Reconciling distance functions and level sets. J. Vis. Commun. Image Represent. 11, 209–223 (2000)

    Article  Google 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)

    Google 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)

    MATH  Article  Google Scholar 

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Correspondence to Xavier Bresson.

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Research supported by NIH U54RR021813, NSF DMS-0312222, NSF ACI-0321917 and NSF DMI-0327077.

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Bresson, X., Esedoḡlu, S., Vandergheynst, P. et al. Fast Global Minimization of the Active Contour/Snake Model. J Math Imaging Vis 28, 151–167 (2007). https://doi.org/10.1007/s10851-007-0002-0

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Keywords

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