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Journal of Mathematical Imaging and Vision

, Volume 58, Issue 1, pp 147–161 | Cite as

A Discontinuous Galerkin Method for the Subjective Surfaces Problem

  • Leon Bungert
  • Vadym Aizinger
  • Michael Fried
Article

Abstract

The work formulates and evaluates the local discontinuous Galerkin method for the subjective surfaces problem based on the curvature driven level set equation. A new mixed formulation simplifying the treatment of nonlinearities is proposed. The numerical algorithm is evaluated using several artificial and realistic test cases.

Keywords

Local discontinuous Galerkin method Image segmentation Subjective surfaces Slope limiting Anisotropic diffusion Mixed formulation Divergence form Edge detection 

References

  1. 1.
    Aizinger, V.: A geometry independent slope limiter for the discontinuous Galerkin method. In: Krause, E., Shokin, Y., Resch, M., Kröner, D., Shokina, N. (eds.) Computational Science and High Performance Computing IV. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 115, pp. 207–217. Springer, Berlin (2011)CrossRefGoogle Scholar
  2. 2.
    Aizinger, V., Bungert, L., Fried M.: Comparison of two mixed discontinuous Galerkin formulations for the subjective surfaces problem (in preparation)Google Scholar
  3. 3.
    Aizinger, V., Dawson, C.: A discontinuous Galerkin method for two-dimensional flow and transport in shallow water. Adv. Water Resour. 25(1), 67–84 (2002)CrossRefzbMATHGoogle Scholar
  4. 4.
    Aizinger, V., Dawson, C., Cockburn, B., Castillo, P.: The local discontinuous Galerkin method for contaminant transport. Adv. Water Resour. 24(1), 73–87 (2000)CrossRefzbMATHGoogle Scholar
  5. 5.
    Albrecht, T., Dedner, A., Lüthi, M., Vetter, T.: Finite element surface registration incorporating curvature, volume preservation, and statistical model information. Comput. Math. Methods Med. 2013, 674273 (2013). doi: 10.1155/2013/674273
  6. 6.
    Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bänsch, E., Mikula, K.: A coarsening finite element strategy in image selective smoothing. Comput. Vis. Sci. 1(1), 53–61 (1997)CrossRefzbMATHGoogle Scholar
  8. 8.
    Bänsch, E., Mikula, K.: Adaptivity in 3D image processing. Comput. Vis. Sci. 4(1), 21–30 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Barth, T.J., Jespersen, D.C.: The design and application of upwind schemes on unstructured meshes. In: Proceedings of AIAA 27th Aerospace Sciences Meeting, Reno (1989)Google Scholar
  10. 10.
    Baswaraj, D., Govardhan, A., Premchand, P.: Active contours and image segmentation: The current state of the art. Global J. Comput. Sci. Technol. (2012). http://computerresearch.org/index.php/computer/article/view/568
  11. 11.
    Becker, J., Preußer, T., Rumpf, M.: PDE methods in flow simulation post processing. Comput. Vis. Sci. 3(3), 159–167 (2000)CrossRefzbMATHGoogle 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.
    Chan, T.F., Moelich, M., Sandberg, B.: Some recent developments in variational image segmentation. In: Image Processing Based on Partial Differential Equations, pp. 175–210. Springer (2007)Google Scholar
  14. 14.
    Chan, T.F., Sandberg, B., Vese, L.A.: Active contours without edges for vector-valued images. J. Vis. Commun. Image Represent. 11(2), 130–141 (2000)CrossRefGoogle Scholar
  15. 15.
    Chan, T.F., Vese, L.A.: Active contours without edges. IEEE Trans. Image Process. 10(2), 266–277 (2001)CrossRefzbMATHGoogle Scholar
  16. 16.
    Chung, E., Lee, C.S.: A staggered discontinuous Galerkin method for the convection–diffusion equation. J. Numer. Math. 20(1), 1–31 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Cockburn, B., Kanschat, G., Schötzau, D.: A locally conservative LDG method for the incompressible Navier–Stokes equations. Math. Comput. 74(251), 1067–1095 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Cockburn, B., Kanschat, G., Schötzau, D., Schwab, C.: Local discontinuous Galerkin methods for the Stokes system. SIAM J. Numer. Anal. 40(1), 319–343 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Cockburn, B., Shu, C.: The local discontinuous Galerkin method for time-dependent convection–diffusion systems. SIAM J. Numer. Anal. 35(6), 2440–2463 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Cohen, L.D., Cohen, I.: Finite-element methods for active contour models and balloons for 2-D and 3-D images. IEEE Trans. Pattern Anal. Mach. Intell. 15(11), 1131–1147 (1993)CrossRefGoogle Scholar
  21. 21.
    Corsaro, S., Mikula, K., Sarti, A., Sgallari, F.: Semi-implicit covolume method in 3D image segmentation. SIAM J. Sci. Comput. 28, 2248–2265 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Dawson, C., Aizinger, V., Cockburn, B.: The local discontinuous Galerkin method for contaminant transport problems. In: Cockburn, B., Karniadakis, G.E., Shu, C.-W. (eds.) Discontinuous Galerkin Methods. Lecture Notes in Computational Science and Engineering, pp. 309–314. Springer, Berlin (2000)CrossRefGoogle Scholar
  23. 23.
    Evans, L.C., Spruck, J.: Motion of level sets by mean curvature. i. J. Differ. Geom. 33(3), 635–681 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Frank, F., Reuter, B., Aizinger V.: FESTUNG—The Finite Element Simulation Toolbox for UNstructured Grids. http://www.math.fau.de/FESTUNG (2015)
  25. 25.
    Frank, F., Reuter, B., Aizinger, V., Knabner, P.: FESTUNG: A MATLAB / GNU Octave toolbox for the discontinuous Galerkin method, part I: diffusion operator. Comput. Math. Appl. 70(1), 11–46 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Fried, M.: Berechnung des Krm̈mungsflusses von Niveauflächen (in German). Master’s thesis, University of Freiburg (1993)Google Scholar
  27. 27.
    Fried, M.: Multichannel image segmentation using adaptive finite elements. Comput. Vis. Sci. 12(3), 125–135 (2009)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Fried, M., Mikula, K.: Efficient subjective surfaces segmentation by adaptive finite elements. Lecture presented at the IMI International Workshop on Computational Photography and Aesthetics, 12 (2009)Google Scholar
  29. 29.
    Frolkovič, P., Mikula, K.: Flux-based level set method: a finite volume method for evolving interfaces. Appl. Numer. Math. 57(4), 436–454 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Frolkovič, P., Mikula, K.: High-resolution flux-based level set method. SIAM J. Sci. Comput. 29(2), 579–597 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Handlovičova, A., Mikula, K., Sgallari, F.: Semi-implicit complementary volume scheme for solving level set like equations in image processing and curve evolution. Numer. Math. 93(4), 675–695 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Iacono, F., May, G., Mller, S., Schfer, R.: An adaptive multiwavelet-based DG discretization for compressible fluid flow. In: Computational Fluid Dynamics 2010—Proceedings of the 6th International Conference on Computational Fluid Dynamics, ICCFD 2010, pp. 813–820 (2011)Google Scholar
  33. 33.
    Kanizsa, G.: Subjective contours. Sci. Am. 234(4), 48–52 (1976)Google Scholar
  34. 34.
    Kass, Michael, Witkin, Andrew, Terzopoulos, Demetri: Snakes: active contour models. Int. J. Comput. Vis. 1(4), 321–331 (1988)Google Scholar
  35. 35.
    Kačur, J., Mikula, K.: Solution of nonlinear diffusion appearing in image smoothing and edge detection. Appl. Numer. Math. 17(1), 47–59 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Kuzmin, D.: A vertex-based hierarchical slope limiter for adaptive discontinuous Galerkin methods. J. Comput. Appl. Math. 233(12), 3077–3085 (2010). Finite Element Methods in Engineering and Science (FEMTEC 2009)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Liu, Y., Shu, C., Tadmor, E., Zhang, M.: Central discontinuous Galerkin methods on overlapping cells with a nonoscillatory hierarchical reconstruction. SIAM J. Numer. Anal. 45(6), 2442–2467 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Menet, S., Saint-Marc, P., Medioni, G.: Active contour models: overview, implementation and applications. In: Systems, Man and Cybernetics, 1990. Conference Proceedings, IEEE International Conference on, pp. 194–199 (1990)Google Scholar
  39. 39.
    Mikula, K., Ohlberger, M., Urban, J.: Inflow-implicit/outflow-explicit finite volume methods for solving advection equations. Appl. Numer. Math. 85, 16–37 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Mikula, K., Peyriéras, N., Remešíková, M., Sarti., A.: 3D embryogenesis image segmentation by the generalized subjective surface method using the finite volume technique. In: Eymard, R., Hérard, J.-M. (eds.) Finite Volumes for Complex Applications V, pp. 585–592. Wiley, London (2008)Google Scholar
  41. 41.
    Mikula, K., Sarti, A.: Parallel co-volume subjective surface method for 3D medical image segmentation. In: Deformable Models, Topics in Biomedical Engineering. International Book Series, pp. 123–160. Springer, New York (2007)Google Scholar
  42. 42.
    Mikula, K., Sarti, A., Sgallari, F.: Co-volume level set method in subjective surface based medical image segmentation. In: Handbook of Biomedical Image Analysis, pp. 583–626. Springer (2005)Google Scholar
  43. 43.
    Mikula, K., Sarti, A., Sgallari, F.: Semi-implicit co-volume level set method in medical image segmentation. In: Deformable Models, Handbook of biomedical image analysis: Segmentation and Registration Models, pp. 583–626. Springer, New York (2007)Google Scholar
  44. 44.
    Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42(5), 577–685 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces (Applied Mathematical Sciences). Springer, New York, 2003 edition (2002)Google Scholar
  46. 46.
    Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79, 12–49 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Papari, G., Petkov, N.: Edge and line oriented contour detection: state of the art. Image Vis. Comput. 29(2), 79–103 (2011)CrossRefGoogle Scholar
  48. 48.
    Peraire, J., Persson, P.-O.: The compact discontinuous Galerkin (CDG) method for elliptic problems. SIAM J. Sci. Comput. 30(4), 1806–1824 (2007)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12(7), 629–639 (1990)CrossRefGoogle Scholar
  50. 50.
    Randrianarivony, M.: Adaptive discontinuous Galerkin B-splines on Parametric Geometries. Lecture Notes in Computer Science (Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 6785 LNCS(PART 4):59–74 (2011)Google Scholar
  51. 51.
    Reed, H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, NM (1973)Google Scholar
  52. 52.
    Reuter, B., Aizinger, V., Wieland, M., Frank, F., Knabner, P.: FESTUNG: A MATLAB / GNU Octave toolbox for the discontinuous Galerkin method. Part II: advection operator and slope limiting. Comput. Math. Appl. 72(7), 1896–1925 (2016)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Rivière, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. Society for Industrial and Applied Mathematics, Philadelphia (2008)Google Scholar
  54. 54.
    Sapiro, G.: Geometric Partial Differential Equations and Image Analysis, 1st edn. Cambridge University Press, Cambridge (2001)CrossRefzbMATHGoogle Scholar
  55. 55.
    Sarti, A., Citti, G.: Subjective surfaces and riemannian mean curvature flow graphs. Acta Math. Univ. Comen. 70, 85–104 (2001)MathSciNetzbMATHGoogle Scholar
  56. 56.
    Sarti, A., Malladi, R., Sethian, J.A.: Subjective surfaces: a method for completing missing boundaries. Proc. Natl. Acad. Sci. 97(12), 6258–6263 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Sarti, A., Malladi, R., Sethian, J.A.: Subjective surfaces: a geometric model for boundary completition. Int. J. Comput. Vis. 46(3), 201–221 (2002)CrossRefzbMATHGoogle Scholar
  58. 58.
    Sethian, J.A.: Numerical algorithms for propagating interfaces: Hamilton–Jacobi equations and conservation laws. J. Differ. Geom. 31(1), 131–161 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Sethian, J.A.: Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, 2nd edn. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  60. 60.
    Stamm, B., Wihler, T.P.: A total variation discontinuous Galerkin approach for image restoration. Int. J. Numer. Anal. Model. 12(1), 81–93 (2015)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Wang, Z., Qi, F., Zhou, F.: A discontinuous finite element method for image denoising. Lecture Notes in Computer Science (Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 4141 LNCS:116–125 (2006)Google Scholar

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Applied Mathematics 1Friedrich-Alexander Universität Erlangen-NürnbergErlangenGermany
  2. 2.Applied Mathematics 3Friedrich-Alexander Universität Erlangen-NürnbergErlangenGermany

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