Advertisement

Abstract

In this paper, we propose a two-stage approach for color image segmentation, which is inspired by minimal surface smoothing. Indeed, the first stage is to find a smooth solution to a convex variational model related to minimal surface smoothing. The classical primal-dual algorithm can be applied to efficiently solve the minimization problem. Once the smoothed image u is obtained, in the second stage, the segmentation is done by thresholding. Here, instead of using the classical K-means to find the thresholds, we propose a hill-climbing procedure to find the peaks on the histogram of u, which can be used to determine the required thresholds. The benefit of such approach is that it is more stable and can find the number of segments automatically. Finally, the experiment results illustrate that the proposed algorithm is very robust to noise and exhibits superior performance for color image segmentation.

Keywords

image segmentation minimal surface primal-dual method total variation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Acar, R., Vogel, C.R.: Analysis of bounded variation penalty methods for ill-posed problems. Inverse Problems 10(6), 1217–1229 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Achanta, R., Estrada, F.J., Wils, P., Süsstrunk, S.: Salient region detection and segmentation. In: Gasteratos, A., Vincze, M., Tsotsos, J.K. (eds.) ICVS 2008. LNCS, vol. 5008, pp. 66–75. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  3. 3.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems, vol. 254. Clarendon Press Oxford (2000)Google Scholar
  4. 4.
    Bae, E., Lellmann, J., Tai, X.-C.: Convex relaxations for a generalized chan-vese model. In: Heyden, A., Kahl, F., Olsson, C., Oskarsson, M., Tai, X.-C. (eds.) EMMCVPR 2013. LNCS, vol. 8081, pp. 223–236. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  5. 5.
    Bae, E., Yuan, J., Tai, X.C.: Global minimization for continuous multiphase partitioning problems using a dual approach. International Journal of Computer Vision 92(1), 112–129 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Boykov, Y., Kolmogorov, V.: Computing geodesics and minimal surfaces via graph cuts. In: Proceedings of the Ninth IEEE International Conference on Computer Vision, pp. 26–33. IEEE (2003)Google Scholar
  7. 7.
    Bresson, X., Chan, T.F.: Fast dual minimization of the vectorial total variation norm and applications to color image processing. Inverse Problems and Imaging 2(4), 455–484 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Brown, E.S., Chan, T.F., Bresson, X.: Completely convex formulation of the chan-vese image segmentation model. International journal of computer vision 98(1), 103–121 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Cai, X., Chan, R., Zeng, T.: A two-stage image segmentation method using a convex variant of the mumford–shah model and thresholding. SIAM Journal on Imaging Sciences 6(1), 368–390 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. International Journal of Computer Vision 22(1), 61–79 (1997)CrossRefzbMATHGoogle Scholar
  11. 11.
    Caselles, V., Kimmel, R., Sapiro, G., Sbert, C.: Minimal surfaces: A geometric three dimensional segmentation approach. Numerische Mathematik 77(4), 423–451 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. Journal of Mathematical Imaging and Vision 40(1), 120–145 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Chan, R., Yang, H., Zeng, T.: A two-stage image segmentation method for blurry images with poisson or multiplicative gamma noise. SIAM Journal on Imaging Sciences 7(1), 98–127 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Chan, T.F., Esedoglu, S., Nikolova, M.: Algorithms for finding global minimizers of image segmentation and denoising models. SIAM Journal on Applied Mathematics 66(5), 1632–1648 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Chan, T.F., Vese, L.A.: Active contours without edges. IEEE Transactions on Image Processing 10(2), 266–277 (2001)CrossRefzbMATHGoogle Scholar
  16. 16.
    Deimling, K.: Nonlinear functional analysis. Courier Dover Publications (2013)Google Scholar
  17. 17.
    Ding, Z., Jia, J., Li, D.: Fast clustering segmentation method combining hill-climbing for color image (2011)Google Scholar
  18. 18.
    Goldstein, T., Osher, S.: The split bregman method for l1-regularized problems. SIAM Journal on Imaging Sciences 2(2), 323–343 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
  20. 20.
    Jin, R., Kou, C., Liu, R., Li, Y.: A color image segmentation method based on improved k-means clustering algorithm. In: Zhong, Z. (ed.) Proceedings of the International Conference on Information Engineering and Applications (IEA) 2012. LNEE, vol. 217, pp. 499–505. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  21. 21.
    Kimmel, R.: Numerical geometry of images: Theory, algorithms, and applications. Springer (2004)Google Scholar
  22. 22.
    Kluzner, V., Wolansky, G., Zeevi, Y.Y.: Minimal surfaces, measure-based metric and image segmentation. Technion-IIT, Department of Electrical Engineering (2006)Google Scholar
  23. 23.
    Kluzner, V., Wolansky, G., Zeevi, Y.Y.: Geometric approach to measure-based metric in image segmentation. Journal of Mathematical Imaging and Vision 33(3), 360–378 (2009)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Li, C., Xu, C., Gui, C., Fox, M.D.: Level set evolution without re-initialization: a new variational formulation. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition, CVPR 2005, vol. 1, pp. 430–436. IEEE (2005)Google Scholar
  25. 25.
    Li, F., Ng, M.K., Zeng, T.Y., Shen, C.: A multiphase image segmentation method based on fuzzy region competition. SIAM Journal on Imaging Sciences 3(3), 277–299 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Martin, D., Fowlkes, C., Tal, D., Malik, J.: A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics. In: Proc. 8th Int’l Conf. Computer Vision, vol. 2, pp. 416–423 (2001)Google 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.
    Nimbarte, N.M., Mushrif, M.M.: Multi-level thresholding algorithm for color image segmentation. In: 2010 Second International Conference on Computer Engineering and Applications (ICCEA), pp. 231–233. IEEE (2010)Google Scholar
  29. 29.
    Ohashi, T., Aghbari, Z., Makinouchi, A.: Hill-climbing algorithm for efficient color-based image segmentation. In: IASTED International Conference on Signal Processing, Pattern Recognition, and Applications, pp. 17–22 (2003)Google Scholar
  30. 30.
    Otsu, N.: A threshold selection method from gray-level histograms. Automatica 11(285-296), 23–27 (1975)Google Scholar
  31. 31.
    Singh, M., Patel, P., Khosla, D., Kim, T.: Segmentation of functional mri by k-means clustering. IEEE Transactions on Nuclear Science 43(3), 2030–2036 (1996)CrossRefGoogle Scholar
  32. 32.
    Sochen, N., Kimmel, R., Malladi, R.: A general framework for low level vision. IEEE Transactions on Image Processing 7(3), 310–318 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Storath, M., Weinmann, A.: Fast partitioning of vector-valued images. SIAM Journal on Imaging Sciences 7(3), 1826–1852 (2014)CrossRefMathSciNetGoogle Scholar
  34. 34.
    Vese, L.A., Chan, T.F.: A multiphase level set framework for image segmentation using the mumford and shah model. International Journal of Computer Vision 50(3), 271–293 (2002)CrossRefzbMATHGoogle Scholar
  35. 35.
    Weber, A.: The usc-sipi image database. Signal and Image Processing Institute of the University of Southern California (1997), http://sipi.usc.edu/services/database
  36. 36.
    Yuan, J., Bae, E., Tai, X.C.: A study on continuous max-flow and min-cut approaches. In: 2010 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 2217–2224. IEEE (2010)Google Scholar
  37. 37.
    Yuan, J., Bae, E., Tai, X.-C., Boykov, Y.: A continuous max-flow approach to potts model. In: Daniilidis, K., Maragos, P., Paragios, N. (eds.) ECCV 2010, Part VI. LNCS, vol. 6316, pp. 379–392. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  38. 38.
    Zhang, R., Bresson, X., Chan, T.F., Tai, X.C.: Four color theorem and convex relaxation for image segmentation with any number of regions. Inverse Problems and Imaging 7(3), 1099–1113 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Zhu, W., Tai, X.C., Chan, T.: Augmented lagrangian method for a mean curvature based image denoising model. Inverse Problems and Imaging 7(4), 1409–1432 (2013)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Zhi Li
    • 1
  • Tieyong Zeng
    • 1
  1. 1.Department of MathematicsHong Kong Baptist UniversityHong KongChina

Personalised recommendations