Robust Edge Detection Using Mumford-Shah Model and Binary Level Set Method

  • Li-Lian Wang
  • Yuying Shi
  • Xue-Cheng Tai
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6667)


A new approximation of the Mumford-Shah model is proposed for edge detection, which could handle open-ended curves and closed curves as well. The essential idea is to treat the curves by narrow regions, and use a sharp interface technique to solve the approximate Mumford-Shah model. A fast algorithm based on the augmented Lagrangian method is developed. Numerical results show that the proposed model and method are very efficient and have the potential to be used for edge detections for real complicated images.


Image Segmentation Edge Detection Active Contour Augmented Lagrangian Method Canny Edge Detector 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Alvarez, L., Lions, P., Morel, J.: Image selective smoothing and edge detection by nonlinear diffusion ii. SIAM J. Numer. Anal. 29(3), 845–866 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Ambrosio, L., Tortorelli, V.: Approximation of functions depending on jumps by elliptic functions via gamma-convergence. Comm. Pure Appl. Math. 13, 999–1036 (1990)CrossRefzbMATHGoogle Scholar
  3. 3.
    Ambrosio, L., Tortorelli, V.: On the approximation of functionals depending on jumps by quadratic, elliptic functions. Boll. Un. Mat. Ital. 6-B, 105–123 (1992)zbMATHGoogle Scholar
  4. 4.
    Aubert, G., Kornprobst, P.: Mathematical problems in image processing: partial differential equations and the calculus of variations. Springer-Verlag, New York Inc., Secaucus (2006)zbMATHGoogle Scholar
  5. 5.
    Badshah, N., Chen, K.: Image selective segmentation under geometrical constraints using an active contour approach. Commun. Compu. Phys. 7(4), 759–778 (2010)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Bae, E., Tai, X.: Graph cut optimization for the piecewise constant level set method applied to multiphase image segmentation. In: Tai, X.-C., Mørken, K., Lysaker, M., Lie, K.-A. (eds.) SSVM 2009. LNCS, vol. 5567, pp. 1–13. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  7. 7.
    Basu, S., Mukherjee, D., Acton, S.: Implicit evolution of open ended curves. In: IEEE International Conference on Image Processing, vol. 1, pp. 261–264 (2007)Google Scholar
  8. 8.
    Berkels, B., Rätz, A., Rumpf, M., Voigt, A.: Extracting grain boundaries and macroscopic deformations from images on atomic scale. J. Sci. Comput. 35(1), 1–23 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Braides, A.: Approximation of free-discontinuity problems. Springer, Heidelberg (1998)CrossRefzbMATHGoogle Scholar
  10. 10.
    Brook, A., Kimmel, R., Sochen, N.: Variational restoration and edge detection for color images. J. Math. Imaging Vis. 18(3), 247–268 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Canny, J.: A computational approach to edge detection. IEEE Trans. Pattern. Anal. PAMI-8(6), 679–698 (1986)CrossRefGoogle Scholar
  12. 12.
    Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. Int. J. Comput. Vis. 22(1), 61–79 (1997)CrossRefzbMATHGoogle Scholar
  13. 13.
    Catté, F., Lions, P., Morel, J., Coll, T.: Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal. 29(1), 182–193 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20(1-2), 89–97 (2004)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Chan, T., Vese, L.: Active contours without edges. IEEE Trans. Image Process. 10(2), 266–277 (2001)CrossRefzbMATHGoogle Scholar
  16. 16.
    Dal Maso, G.: Introduction to Γ-convergence. Birkhauser, Basel (1993)CrossRefGoogle Scholar
  17. 17.
    Dal Maso, G., Morel, J., Solimini, S.: A variation method in image segmentation-existence and approximation results. Acta Mathematica 168(1-2), 89–151 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Deriche, R.: Using canny’s criteria to derive a recursively implemented optimal edge detector. Int. J. Comput. Vis. 1(2), 167–187 (1987)CrossRefGoogle Scholar
  19. 19.
    Farouki, R., Neff, C.: Analytic properties of plane offset curves. Computer Aided Geometric Design 7(1-4), 83–99 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Kass, M., Witkin, A., Terzopoulos, D.: Snakes: Active contour models. Int. J. Comput. Vis. 1(4), 321–331 (1988)CrossRefzbMATHGoogle Scholar
  21. 21.
    Leung, S., Zhao, H.: A grid based particle method for evolution of open curves and surfaces. J. Comput. Phys. 228(20), 7706–7728 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Lie, J., Lysaker, M., Tai, X.: A binary level set model and some applications to Mumford-Shah image segmentation. IEEE Trans. Image Process. 15(5), 1171–1181 (2006)CrossRefzbMATHGoogle Scholar
  23. 23.
    Llanas, B., Lantaró, S.: Edge detection by adaptive splitting. J. Sci. Comput. 46(3), 486–518 (2011)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Ma, W., Manjunath, B.: Edgeflow: a technique for boundary detection and image segmentation. IEEE Trans. Image Process. 9(8), 1375–1388 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Meinhardt, E., Zacur, E., Frangi, A., Caselles, V.: 3D edge detection by selection of level surface patches. J. Math. Imaging Vis. 34(1), 1–16 (2009)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Merriman, B., Bence, J., Osher, S.: Motion of multiple functions: a level set approach. J. Comput. Phys. 112(2), 334–363 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math 42(5), 577–685 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Osher, S., Sethian, J.: Fronts propagating with curvature dependent speed: Algorithms based on hamilton-jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Paragios, N., Chen, Y., Faugeras, O.: Handbook of mathematical models in computer vision. Springer-Verlag New York Inc., Secaucus (2006)CrossRefzbMATHGoogle Scholar
  30. 30.
    Perona, P., Malik, J.: Scale-space and edge-detection using anisotropic diffusion. IEEE Trans. Pattern. Anal. 12(7), 629–639 (1990)CrossRefGoogle Scholar
  31. 31.
    Pock, T., Cremers, D., Bischof, H., Chambolle, A.: An algorithm for minimizing the Mumford-Shah functional. In: 12th International Conference on Computer Vision, pp. 1133–1140. IEEE, Los Alamitos (2009)Google Scholar
  32. 32.
    Smereka, P.: Spiral crystal growth. Physica D: Nonlinear Phenomena 138(3-4), 282–301 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Smith, S.: Edge thinning used in the susan edge detector. Technical Report, TR95SMS5 (1995)Google Scholar
  34. 34.
    Sun, Y., Wu, P., Wei, G., Wang, G.: Evolution-operator-based single-step method for image processing. Int. J. Biomed. Imaging, 1–28 (2006)Google Scholar
  35. 35.
    Suzuki, Y., Takayama, T., Motoike, I., Asai, T.: A reaction-diffusion model performing stripe-and spot-image restoration and its lsi implementation. Electronics and Communications in Japan (Part III: Fundamental Electronic Science) 90(1), 20–29 (2007)CrossRefGoogle Scholar
  36. 36.
    Tai, X., Christiansen, O., Lin, P., Skjælaaen, I.: Image segmentation using some piecewise constant level set methods with MBO type of projection. International Journal of Computer Vision 73(1), 61–76 (2007)CrossRefGoogle Scholar
  37. 37.
    Tai, X.C., Wu, C.: Augmented lagrangian method, dual methods and split bregman iteration for ROF model. In: Tai, X.-C., Mørken, K., Lysaker, M., Lie, K.-A. (eds.) SSVM 2009. LNCS, vol. 5567, pp. 502–513. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  38. 38.
    Toponogov, V.: Differential geometry of curves and surfaces: a concise guide. Birkhauser, Basel (2006)zbMATHGoogle Scholar
  39. 39.
    Upmanyu, M., Smith, R., Srolovitz, D.: Atomistic simulation of curvature driven grain boundary migration. Interface Sci. 6, 41–58 (1998)CrossRefGoogle Scholar
  40. 40.
    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)CrossRefzbMATHGoogle Scholar
  41. 41.
    Wang, Y., Yang, J., Yin, W., Zhang, Y.: A new alternating minimization algorithm for total variation image reconstruction. SIAM J. Imaging Sci. 1(3), 248–272 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  42. 42.
    Wei, G., Jia, Y.: Synchronization-based image edge detection. EPL (Europhysics Letters) 59(6), 814–819 (2002)CrossRefGoogle Scholar
  43. 43.
    Witkin, A.P.: Scale-space filtering. In: Proc. 8th Int. Joint Conf. Art. Intell., Karlsruhe, Germany, pp. 1019–1022 (1983)Google Scholar
  44. 44.
    Wu, C., Zhang, J., Tai, X.: Augmented lagrangian method for total variation restoration with non-quadratic fidelity. In: UCLA, CAM09-82, pp. 1–26 (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Li-Lian Wang
    • 1
  • Yuying Shi
    • 2
  • Xue-Cheng Tai
    • 1
    • 3
  1. 1.Division of Mathematical Sciences, School of Physical and Mathematical SciencesNanyang Technological UniversitySingapore
  2. 2.Department of Mathematics and PhysicsNorth China Electric Power UniversityBeijingChina
  3. 3.Department of MathematicsUniversity of BergenBergenNorway

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