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Computing and Visualization in Science

, Volume 12, Issue 4, pp 171–188 | Cite as

Composite finite elements for 3D image based computing

  • Florian Liehr
  • Tobias PreusserEmail author
  • Martin Rumpf
  • Stefan Sauter
  • Lars Ole Schwen
Regular article

Abstract

We present an algorithmical concept for modeling and simulation with partial differential equations (PDEs) in image based computing where the computational geometry is defined through previously segmented image data. Such problems occur in applications from biology and medicine where the underlying image data has been acquired through, e.g. computed tomography (CT), magnetic resonance imaging (MRI) or electron microscopy (EM). Based on a level-set description of the computational domain, our approach is capable of automatically providing suitable composite finite element functions that resolve the complicated shapes in the medical/biological data set. It is efficient in the sense that the traversal of the grid (and thus assembling matrices for finite element computations) inherits the efficiency of uniform grids away from complicated structures. The method’s efficiency heavily depends on precomputed lookup tables in the vicinity of the domain boundary or interface. A suitable multigrid method is used for an efficient solution of the systems of equations resulting from the composite finite element discretization. The paper focuses on both algorithmical and implementational details. Scalar and vector valued model problems as well as real applications underline the usability of our approach.

Keywords

Coarse Grid Lookup Table Multigrid Method Virtual Node Regular Node 
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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References

  1. 1.
    Adams, L., Li, Z.: The immersed interface/multigrid methods for interface problems. SIAM J. Sci. Comput. 24(2), 463–479 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Ambrosio, L., Tortorelli, V.M.: On the approximation of free discontinuity problems. Boll. dell’Unione Matematica Ital. B 6(7), 105–123 (1992)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Babuška, I., Melenk, J.: The partition of unity method. Int. J. Numer. Methods Eng. 40, 727–758 (1997)zbMATHCrossRefGoogle Scholar
  4. 4.
    Belytschko, T., Moës, N., Usui, S., Parimi, C.: Arbitrary discontinuities in finite elements. Int. J. Numer. Methods Eng. 50(4), 993–1013 (2001)zbMATHCrossRefGoogle Scholar
  5. 5.
    Beyer, R.P., LeVeque, R.J.: Analysis of a one-dimensional model for the immersed boundary method. SIAM J. Numer. Anal. 29(2), 332–364 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bornemann, F., Rasch, C.: Finite-element discretization of static Hamilton–Jacobi equations based on a local variation principle. Comput. Vis. Sci. 9(2), 57–69 (2004) arXiv:math.NA/0403517Google Scholar
  7. 7.
    Calhoun, D.: A Cartesian grid method for solving the two-dimensional streamfunction-vorticity equations in irregular regions. J. Comput. Phys. 176, 231–275 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Calhoun, D., LeVeque, R.J.: A Cartesian grid finite-volume method for the advection-diffusion equation in irregular geometries. J. Comput. Phys. 157, 143–180 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Caselles, V., Catté, F., Coll, T., Dibos, F.: A geometric model for active contours in image processing. Numerische Mathematik 66, 1–31 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. Int. J. Comput. Vis. 22(1), 61–79 (1997)zbMATHCrossRefGoogle Scholar
  11. 11.
    Chan, T., Vese, L.: A level set algorithm for minimizing the Mumford–Shah functional in image processing. In: Proceedings of the 1st IEEE Workshop on Variational and Level Set Methods in Computer Vision, pp. 161–168 (2001)Google Scholar
  12. 12.
    Cheng, S.-W., Dey, T.K., Edelsbrunner, H., Facello, M.A., Teng, S.-H.: Sliver exudation. J. ACM 47(5), 883–904 (2000)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Ciarlet, P.G.: The finite element method for elliptic problems. Number 40 in Classics in applied mathematics. SIAM (2002)Google Scholar
  14. 14.
    Cremers, D., Schnörr, C., Weickert, J.: Diffusion-snakes: combining statistical shape knowledge and image information in a variational framework. IEEE Workshop on Variational and Levelset Methods, pp. 137–144 (2001)Google Scholar
  15. 15.
    Deuflhard, P., Weiser, M., Seebaß, M.: A new nonlinear elliptic multilevel FEM applied to regional hyperthermia. Comput. Vis. Sci. 3(3), 115–120 (2000)zbMATHCrossRefGoogle Scholar
  16. 16.
    Droske, M., Preusser, T., Rumpf, M.: A multilevel segmentation method. In: Proceedings of Vision, Modeling and Visualization (VMV), pp. 327–336. Saarbrücken, Germany (2000)Google Scholar
  17. 17.
    Frauböse, N., Sauter, S.: Composite finite elements and multi-grid. Part I: Convergence theory in 1-d. In: Proceedings of the 17th GAMM-Seminar Leipzig on Construction of Grid Generation Algorithms, pp. 69–86 (2001)Google Scholar
  18. 18.
    Goméz-Benito, M., García-Aznar, J., Doblaré, M.: Finite element prediction of proximal femoral fracture patterns under different loads. J. Biomech. Eng. 127(1), 9–14 (2005)CrossRefGoogle Scholar
  19. 19.
    Hackbusch, W.: Multi-Grid Methods and Applications. Springer Series in Computational Mathematics, vol. 4. Springer, Heidelberg (1985)Google Scholar
  20. 20.
    Hackbusch, W., Sauter, S.: Composite finite elements for the approximation of PDEs on domains with complicated micro-structures. Numerische Mathematik 75, 447–472 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Hahn, H.K., Lentschig, M.G., Deimling, M., Terwey, B., Peitgen, H.-O.: MRI-based volumetry of intra- and extracerebral liquor spaces. In: CARS, pp. 401–407 (2001)Google Scholar
  22. 22.
    Höllig, K., Reif, U., Wipper, J.: Weighted extended B-spline approximation of Dirichlet problems. SIAM J. Numer. Anal. 39(2), 442–462 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Kass, M., Witkin, A., Terzopoulos, D.: Snakes: active contour models. Int. J. Comput. Vis. 1, 321–331 (1988)CrossRefGoogle Scholar
  24. 24.
    Kornhuber, R., Krause, R., Sander, O., Deuflhard, P., Ertel, S.: A monotone multigrid solver for two body contact problems in biomechanics. Comput. Vis. Sci. 11(1), 3–15 (2008)Google Scholar
  25. 25.
    Kröger, T., Altrogge, I., Preusser, T., et al.: Numerical simulation of radio frequency ablation with state dependent material parameters in three space dimensions. In: MICCAI 2006. Lecture Notes in Computer Science, vol. 4191, pp. 380–388. Springer, Heidelberg (2006)Google Scholar
  26. 26.
    Kuhnigk, J.M., Dicken, V., Bornemann, L., Bakai, A., Wormanns, D., Krass, S., Peitgen, H.O.: Morphological segmentation and partial volume analysis for volumetry of solid pulmonary lesions in thoracic CT scans. IEEE Trans. Med. Imaging 25(4), 417–434 (2006)CrossRefGoogle Scholar
  27. 27.
    LeVeque, R.J., Li, Z.L.: The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 31(4), 1019–1044 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Li, Z.: A fast iterative algorithm for elliptic interface problems. SIAM J. Numer. Anal. 35(1), 230–254 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Li, Z.: The immersed interface method using a finite element formulation. Appl. Numer. Math. 27, 253–267 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Li, Z., Lin, T., Wu, X.: New Cartesian grid methods for interface problems using the finite element formulation. Numerische Mathematik 1996(1), 61–98 (2003)CrossRefMathSciNetGoogle Scholar
  31. 31.
    Lorensen, W.E., Cline, H.E.: Marching cube: a high resolution 3D surface construction algorithm. Comput. Graph. 21(4), 163–169 (1987)Google Scholar
  32. 32.
    Melenk, J.M., Babuška, I.: The partition of unity finite element method. Research Report 96-01, Eidgenössische Technische Hochschule Zürich, Seminar für angewandte Mathematik (1996)Google Scholar
  33. 33.
    Müller, H., Wehle, M.: Visualization of implicit surfaces using adaptive tetrahedrizations. In: Proceedings of Scientific Visualization Conference (Dagstuhl ’97) (1997)Google Scholar
  34. 34.
    Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79, 12–49 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Parvizian, J., Düster, A., Rank, E.: Finite cell method. Comput. Mech. 41(1), 121–133 (2007)CrossRefMathSciNetGoogle Scholar
  36. 36.
    Preusser, T., Rumpf, M.: An adaptive finite element method for large scale image processing. J. Vis Commun. Image Representation 11, 183–195 (2000)CrossRefGoogle Scholar
  37. 37.
    Preusser, T., Rumpf, M., Sauter, S., Schwen, L.O., et al.: Three-dimensional composite finite elements for scalar problems with jumping coefficients. In preparation (2007)Google Scholar
  38. 38.
    Preusser, T., Rumpf, M., Schwen, L.O.: Finite element simulation of bone microstructures. In: Proceedings of the 14th Finite Element Workshop. University of Ulm (2007, to appear)Google Scholar
  39. 39.
    Rech, M., Sauter, S., Smolianski, A.: Two-scale composite finite element method for the Dirichlet problem on complicated domains. Numerische Mathematik 102, 681–708 (2006)Google Scholar
  40. 40.
    Russo, G., Smereka, P.: A remark on computing distance functions. J. Comput. Phys. 163(1), 51–67 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Sauter, S.A., Warnke, R.: Composite finite elements for elliptic boundary value problems with discontinuous coefficients. Computing 77, 29–55 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Schenk, A., Prause, G., Peitgen, H.-O.: Efficient semiautomatic segmentation of 3D objects in medical images. In: Proceedings of MICCAI. Lecture Notes in Computer Science, vol. 1935, pp. 186–195. Springer, Heidelberg (2000)Google Scholar
  43. 43.
    Schwen, L.O., et al.: Computing elastic deformation of vertebrae using composite finite elements. In preparation (2007)Google Scholar
  44. 44.
    Segonne, F., Dale, A.M., Busa, E., Glessner, M., Salat, D., Hahn, H.K., Fischl, B.: A hybrid approach to the skull stripping problem in MRI. Neuroimage 22(3), 1060–1075 (2004)CrossRefGoogle Scholar
  45. 45.
    Sethian, J.A., Wiegmann, A.: Structural boundary design via level set and immersed interface methods. J. Comput. Phys. 163(2), 489–528 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Stazi, F.L., Budyn, E., Chessa, J., Belytschko, T.: An extended finite element method with higher-order elements for curved cracks. Comput. Mech. 31, 38–48 (2003)zbMATHCrossRefGoogle Scholar
  47. 47.
    Strang, G., Fix, G.J.: An Analysis of the Finite Element Method. Wellesley-Cambridge Press, Wellesley (1973)zbMATHGoogle Scholar
  48. 48.
    Strouboulis, T., Copps, K., Babuška, I.: The generalized finite element method. Comput. Methods Appl. Mech. Eng. 190, 4081–4193 (2001)zbMATHCrossRefGoogle Scholar
  49. 49.
    Teng, S.-H., Wong, C.W.: Unstructured mesh generation: theory, practice and applications. Int. J. Comput. Geometry Appl. 10(3), 227–266 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  50. 50.
    Wiegmann, A., Bube, K.P.: The immersed interface method for nonlinear differential equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 35(1), 177–200 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    Wittek, A., Kikinis, R., Warfield, S.K., Brain, M.K.: Shift computation using a fully nonlinear biomechanical model. In: Proceedings of MICCAI, LNCS, pp. 583–590. Springer, Heidelberg (2005)Google Scholar
  52. 52.
    Xu, C., Prince, J.L.: Snakes, shapes, and gradient vector flow. IEEE Trans. Image Proces. 7(3), 359–369 (1998)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Florian Liehr
    • 1
  • Tobias Preusser
    • 2
    Email author
  • Martin Rumpf
    • 1
  • Stefan Sauter
    • 3
  • Lars Ole Schwen
    • 1
  1. 1.INS, Institute for Numerical SimulationUniversity of BonnBonnGermany
  2. 2.CeVis, Center of Complex Systems and VisualizationUniversity of BremenBremenGermany
  3. 3.Institute for MathematicsUniversity of ZurichZurichSwitzerland

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