Skip to main content
Log in

A variational model and its numerical solution for local, selective and automatic segmentation

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

Variational region-based segmentation models can serve as effective tools for identifying all features and their boundaries in an image. To adapt such models to identify a local feature defined by geometric constraints, re-initializing iterations towards the feature offers a solution in some simple cases but does not in general lead to a reliable solution. This paper presents a dual level set model that is capable of automatically capturing a local feature of some interested region in three dimensions. An additive operator spitting method is developed for accelerating the solution process. Numerical tests show that the proposed model is robust in locally segmenting complex image structures.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams, R., Bischof, L.: Seeded region growing. IEEE Trans. Pattern Anal. Mach. Intell. 16(6), 641–647 (1994)

    Article  Google Scholar 

  2. Aubert, G., Kornprobst, P.: Mathematical problems in image processing: partial differential equations and the calculus of variations. Springer (2001)

  3. Badshah, N., Chen, K.: Image selective segmentation under geometrical constraints using an active contour approach. Commun. Comput. Phys. 7(4), 759–778 (2009)

    MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  5. Chambolle, A.: Image segmentation by variational methods: Mumford and Shah functional and the discrete approximations. SIAM J. Appl. Math. 55(3), 827–863 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chan, T.F., Esedoglu, S., Nikolova, M.: Algorithms for finding global minimizers of image segmentation and denoising models. SIAM J. Appl. Math. 66(5), 16321648 (2004)

    MathSciNet  Google Scholar 

  7. Chan, T.F., Shen, J.H.: Image processing and analysis—variational, PDE, wavelet, and stochastic methods. SIAM Publications (2005)

  8. Chan, T.F., Vese, L.A.: Active contours without edges. UCLA CAM Report, 98–53 (1998)

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

    Article  MATH  Google Scholar 

  10. Chan, T.F., Vese, L.A.: Active contour and segmentation models using geometric PDE’s for medical imaging. In: Malladi, R. (ed.) Geometric Methods in Bio-Medical Image Processing",Series:Mathematics and Visualization. Springer (2002)

  11. Cheng, Y.: Mean shift, mode seeking, and clustering. IEEE Trans. Pattern Anal. Mach. Intell. 17(8), 790–799 (1995)

    Article  Google Scholar 

  12. Comaniciu, D., Meer, P., Member, S.: Mean shift: A robust approach toward feature space analysis. IEEE Trans. Pattern. Anal.Mach. Intell. 24, 603–619 (2002)

    Article  Google Scholar 

  13. Gout, C., Guyader, C.L., Vese, L.A.: Segmentation under geometrical consitions with geodesic active contour and interpolation using level set methods. Num. Algoritm. 39, 155–173 (2005)

    Article  MATH  Google Scholar 

  14. Guyader, C.L., Gout, C.: Geodesic active contour under geometrical conditions theory and 3D applications. Num. Algoritm. 48, 105–133 (2008)

    Article  MATH  Google Scholar 

  15. Kass, M., Witkin, A., Terzopoulos, D.: Snakes: Active contour models. Int. J. Comp. Vision 1(4), 321–331 (1988)

    Article  Google Scholar 

  16. Koepfler, G., Lopez, C., Morel, J.M.: A multiscale algorithm for image segmentation by variational method. SIAM J. Numer. Anal. 31, 282–299 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  17. Li, C., Kao, C., Gore, J., Ding, Z.: Implicit active contours driven by local binary fitting energyIn: Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 1–7. IEEE Computer Society, Washington, DC (2007)

    Google Scholar 

  18. Li, C., Kao, C., Gore, J.C., Ding, Z.: Minimization of region-scalable fitting energy for image segmentation. IEEE Trans. Image Process. 17(10), 1940–1949 (2008)

    Article  MathSciNet  Google Scholar 

  19. Li, C., Xu, C., Gui, C., Fox, M.D.: Level set evolution without re-initialization: A new variational formulationProceedings of IEEE Conference on Computer Vision and Pattern Recognition (CVPR), vol. 1 pp. 430–436. IEEE Computer Society, Washington, DC (2005)

    Google Scholar 

  20. Lie, J., Lysaker, M., Tai, X.C.: A variant of the level set method and applications to image segmentation. Math. Comput. 75(255), 1155–1174 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  21. Lu, T., Neittaanmaki, P., Tai, X.C.: A parallel splitting-up method for partial differential equations and its application to navier-stokes equations. RAIRO Math. Model. Numer. Anal. 26(6), 673–708 (1992)

    MATH  MathSciNet  Google Scholar 

  22. Malik, J., Leung, T., Shi, J.: Contour and texture analysis for image segmentation. Int. J. Comput. Vis. 43, 7–27 (2001). http://dl.acm.org/citation.cfm?id=543015.543016

    Article  MATH  Google Scholar 

  23. Malladi, R., Sethian, J.A.: A real-time algorithm for medical shape recoveryIn: Proceedings of International Conference on Computer Vision, pp. 304–310. Mumbai (1998)

  24. Mitiche, A., Ben-Ayed, I.: Variational and Level Set Methods in Image Segmentation. Springer (2010)

  25. Mumford, D., Shah, J.: Optimal approximation by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42, 577–685 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  26. Ning, J., Zhang, L., Zhang, D., Wub, C.: Interactive image segmentation by maximal similarity based region merging. Pattern Recognit. 43(2), 445–456 (2010)

    Article  MATH  Google Scholar 

  27. Samson, C., Blanc-Féraud, L., Aubert, G., Zerubia, J.: A level set model for image classification. Int. J. Comput. Vision 40, 187–197 (2000)

    Article  MATH  Google Scholar 

  28. Sapiro, G., Kimmel, R., Caselles, V.: Object detection and measurements in medical images via geodesic deformable contours. In: Melter, R.A., Wu, A.Y., Bookstein, F.L., Green, W.D. (eds.) Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series of Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series vol. 2573, pp. 366–378 (1995)

  29. Sen, D., Pal, S.K.: Histogram thresholding using fuzzy and rough measures of association error. IEEE Trans. Image Process. 18(4), 879 –888 (2009)

    Article  MathSciNet  Google Scholar 

  30. Sethian, J.A.: Curvature and the evolution of fronts. Commun. Math. Phys. 101, 487–499 (1985). doi:10.1007/BF01210742

    Article  MATH  MathSciNet  Google Scholar 

  31. Sethian, J.A.: Level set methods and fast marching methods: Evolving interfaces in computational geometry, fluid Mechanics, computer vision, and materials science. Cambridge University Press (1999). http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/0521642043

  32. Székely, A., Kelemen, C., Brechbühler, G.: Gerig, Segmentation of 3D objects from MRI volume data using constrained elastic deformations of flexible Fourier surface models. In: Ayache, N. (ed.) Computer Vision, Virtual Reality and Robotics in Medicine of Lecture Notes in Computer Science. vol. 905 pp 493–505. Springer Berlin, Heidelberg (1995)

    Chapter  Google Scholar 

  33. Trottenberg, U., Schuller, A.: Multigrid. Academic Press, Inc., Orlando (2001)

    MATH  Google Scholar 

  34. Vese, L.A., Chan, T.F.: A multiphase level set framework for image segmentation using the mumford and shah model. Int. J. Comput. Vis. 50(3), 271–293 (2002)

    Article  MATH  Google Scholar 

  35. Vincent, L., Soille, P.: Watersheds in digital spaces: An efficient algorithm based on immersion simulations. IEEE Trans. Pattern Anal. Mach. Intell. 13(6), 583–598 (1991)

    Article  Google Scholar 

  36. Weickert, J., Kühne, G.: Geometric Level Set Methods in Imaging, Vision, and Graphics, pp. 43–57. (1995). doi:10.1007/0-387-21810-6_3

  37. Weickert, J., Romeny, B., Viergever, M.: Efficient and reliable schemes for nonlinear diffusion filtering. IEEE Trans. Image Process. 7(3), 398 –410. (1998)

    Article  Google Scholar 

  38. Xie, X., Mirmehdi, M.: Radial basis function based level set interpolation and evolution for deformable modelling. Image Vision Comput. 29(3), 167–177 (2011)

    Article  Google Scholar 

  39. Zhang, J., Chen, K., Yu, B.: A multigrid algorithm for the 3D Chan-Vese model of variational image segmentation. Int. J. Comput. Math. 89(2), 160–189 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  40. Zhang, Y., Matuszewski, B.J., Histace, A., Precioso, F., Kilgallon, J., Moore, C.: Boundary delineation in prostate imaging using active contour segmentation method with interactively defined object regions. Proc. Prostate Cancer Imaging’2010 LNCS 6367, 131–142 (2010)

    Google Scholar 

  41. Zucker, S.W.: Region growing: childhood and adolescence. Comput. Graph. Image Process. 5, 382–399 (1976)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ke Chen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rada, L., Chen, K. A variational model and its numerical solution for local, selective and automatic segmentation. Numer Algor 66, 399–430 (2014). https://doi.org/10.1007/s11075-013-9741-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-013-9741-8

Keywords

Mathematics Subject Classifications

Navigation