Journal of Mathematical Imaging and Vision

, Volume 55, Issue 1, pp 105–124 | Cite as

Segmentation and Restoration of Images on Surfaces by Parametric Active Contours with Topology Changes

  • Heike BenninghoffEmail author
  • Harald Garcke


In this article, a new method for segmentation and restoration of images on two-dimensional surfaces is given. Active contour models for image segmentation are extended to images on surfaces. The evolving curves on the surfaces are mathematically described using a parametric approach. For image restoration, a diffusion equation with Neumann boundary conditions is solved in a postprocessing step in the individual regions. Numerical schemes are presented which allow to efficiently compute segmentations and denoised versions of images on surfaces. Also topology changes of the evolving curves are detected and performed using a fast sub-routine. Finally, several experiments are presented where the developed methods are applied on different artificial and real images defined on different surfaces.


Image segmentation Images on surfaces Evolving curves on surfaces Active contours Parametric method Mumford–Shah Chan–Vese Topology changes Triple junctions Image restoration Finite element approximation 


  1. 1.
    Balažovjech, M., Mikula, K., Petrášová, M., Urbán, J.: Lagrangean method with topological changes for numerical modelling of forest fire propagation. In: Proceedings of ALGORITMY 2012, 19th Conference on Scientific Computing, pp. 42–52. Vysoké Tatry, Podbansk’v, Slovakia (2012)Google Scholar
  2. 2.
    Barrett, J.W., Garcke, H., Nürnberg, R.: A parametric finite element method for fourth order geometric evolution equations. J. Comput. Phys. 222(1), 441–467 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barrett, J.W., Garcke, H., Nürnberg, R.: On the variational approximation of combined second and fourth order geometric evolution equations. SIAM J. Sci. Comput. 29(3), 1006–1041 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Barrett, J.W., Garcke, H., Nürnberg, R.: Numerical approximation of gradient flows for closed curves in \({\mathbb{R}}^d\). IMA J. Numer. Anal. 30(1), 4–60 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Benninghoff, H., Garcke, H.: Efficient image segmentation and restoration using parametric curve evolution with junctions and topology changes. SIAM J. Imaging Sci. 7(3), 1451–1483 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bertalmio, M., Cheng, L.T., Osher, S., Sapiro, G.: Variational problems and partial differential equations on implicit surfaces: the framework and examples in image processing and pattern formation. J. Comput. Phys. 174(2), 759–780 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. Int. J. Comput. Vision 22(1), 61–79 (1997)CrossRefzbMATHGoogle Scholar
  8. 8.
    Chan, T.F., Esedoglu, S., Nikolova, M.: Algorithms for finding global minimizers of image segmentation and denoising models. SIAM J. Appl. Math. 66, 1632–1648 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chan, T.F., Sandberg, B.Y., Vese, L.A.: Active contours without edges for vector-valued images. J. Vis. Commun. Image R. 11(2), 130–141 (2000)CrossRefGoogle Scholar
  10. 10.
    Chan, T.F., Vese, L.A.: Active contours without edges. IEEE Trans. Image Process. 10(2), 266–277 (2001)CrossRefzbMATHGoogle Scholar
  11. 11.
    Cheng, L.T., Burchard, P., Merriman, B., Osher, S.: Motion of curves constrained on surfaces using a level-set approach. J. Comput. Phys. 175(2), 604–644 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Čunderlík, R., Mikula, K., Tunega, M.: Nonlinear diffusion filtering of data on the Earth’s surface. J. Geod. 87(2), 143–160 (2013)CrossRefGoogle Scholar
  13. 13.
    Davis, T.: Algorithm 832: UMFPACK, an unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw. 30(2), 196–199 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dziuk, G., Elliott, C.M.: Finite element methods for surface PDEs. Acta Numer. 22, 289–396 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Garcke, H., Wieland, S.: Surfactant spreading on thin viscous films: nonnegative solutions of a coupled degenerate system. SIAM J. Math. Anal. 37(6), 2025–2048 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kass, M., Witkin, A., Terzopoulos, D.: Snakes: active contour models. Int. J. Comput. Vision 1(4), 321–331 (1988)CrossRefzbMATHGoogle Scholar
  17. 17.
    Kimmel, R.: Intrinsic scale space for images on surfaces: the geodesic curvature flow. Graph. Model. Im. Proc. 59(5), 365–372 (1997)CrossRefGoogle Scholar
  18. 18.
    Krüger, M., Delmas, P., Gimelfarb, G.: Active contour based segmentation of 3D surfaces. In: Proceedings of the European Conference on Computer Vision, pp. 350–363. Marseille, France (2008)Google Scholar
  19. 19.
    Lai, R., Chan, T.F.: A framework for intrinsic image processing on surfaces. Comput. Vis. Image Und. 115(12), 1647–1661 (2011)CrossRefGoogle Scholar
  20. 20.
    Lang, S.: Introduction to differentiable manifolds, 2nd edn. Universitext. Springer, Berlin (2002)zbMATHGoogle Scholar
  21. 21.
    Lee, J.M.: Introduction to smooth manifolds, graduate texts in mathematics, vol. 218. Springer, Heidelberg (2002)Google Scholar
  22. 22.
    Mikula, K., Urbán, J.: New fast and stable Lagrangean method for image segmentation. In: Proceedings of the 5th International Congress on Image and Signal Processing (CISP 2012), pp. 834–842. Chongquing (2012)Google Scholar
  23. 23.
    Mikula, K., Ševčovič, D.: Evolution of curves on a surface driven by the geodesic curvature and external force. Appl. Anal. 85(4), 345–362 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Mumford, D., Shah, J.: Optimal approximation by piecewise smooth functions and associated variational problems. Commun. Pure. Appl. Math. 42, 577–685 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    NASA: NASA Earth Observations (2014).
  26. 26.
    Osher, S., Sethian, J.A.: Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Paysan, P., Knothe, R., Amberg, B., Romdhani, S., Vetter, T.: A 3D face model for pose and illumination invariant face recognition. In: Proceedings of the 6th IEEE International Conference on Advanced Video and Signal Based Surveillance (AVSS) for Security. Safety and Monitoring in Smart Environments, pp. 296–301. Genova (2009)Google Scholar
  28. 28.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60(1–4), 259–268 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Spira, A., Kimmel, R.: Geometric curve flows on parametric manifolds. J. Comput. Phys. 223, 235–249 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Tian, L., Macdonald, C.B., Ruuth, S.J.: Segmentation on surfaces with the closest point method. In: Proceedings of the 16th IEEE International Conference on Image Processing, pp. 3009–3012. Cairo (2009)Google Scholar
  31. 31.
    Turk, G., Levoy, M.: Zippered polygon meshes from range images. In: Proceedings of the 21st annual conference on Computer graphics and interactive techniques (SIGGRAPH ’94), pp. 311–318. ACM, New York (1994)Google Scholar
  32. 32.
    Wu, C., Tai, X.C.: Augmented Lagrangian method dual methods, and split bregman iteration for ROF, vectorial TV, and high order models. SIAM J. Imaging Sci. 3(3), 300–339 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Wu, C., Zhang, J., Duan, Y., Tai, X.C.: Augmented Lagrangian method for total variation based image restoration and segmentation over triangulated surfaces. J. Sci. Comput. 50(1), 145–166 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Zhou, L., Li, J.: Image segmentation on implicit surface based on Chan-Vese model. J. Theor. Appl. Inf. Technol. 48(1), 206–209 (2013)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Deutsches Zentrum für Luft- und Raumfahrt (DLR)WeßlingGermany
  2. 2.Fakultät für MathematikUniversität RegensburgRegensburgGermany

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