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International Conference on Scale-Space Theories in Computer Vision

Scale-Space 1999: Scale-Space Theories in Computer Vision pp 58–69Cite as

Unfolding the Cerebral Cortex Using Level Set Methods

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1682))

Abstract

Level set methods provide a robust way to implement ge- ometric flows, but they suffer from two problems which are relevant when using smoothing flows to unfold the cortex: the lack of point- correspondence between scales and the inability to implement tangential velocities. In this paper, we suggest to solve these problems by driving the nodes of a mesh with an ordinary differential equation. We state that this approach does not suffer from the known problems of Lagrangian methods since all geometrical properties are computed on the fixed (Eu- lerian) grid. Additionally, tangential velocities can be given to the nodes, allowing the mesh to follow general evolution equations, which could be crucial to achieving the final goal of minimizing local metric distortions. To experiment with this approach, we derive area and volume preserv- ing mean curvature flows and use them to unfold surfaces extracted from MRI data of the human brain.

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References

  1. M. Bertalmio, G. Sapiro, and G. Randall. Region tracking on level-sets methods. IEEE Transactions On Medical Imaging, to appear.

    Google Scholar 

  2. Lia Bronsard and Barbara Stoth. Volume-preserving mean curvature flow as a limit of a nonlocal ginzburg-landau equation. SIAM Journal on Mathematical Analysis, 28(4):769–807, July 1997.

    Article  MATH  MathSciNet  Google Scholar 

  3. V. Caselles, R. Kimmel, and G. Sapiro. Geodesic active contours. The International Journal of Computer Vision, 22(1):61–79, 1997.

    Article  MATH  Google Scholar 

  4. V. Caselles, R. Kimmel, G. Sapiro, and C. Sbert. 3d active contours. In M-O. Berger, R. Deriche, I. Herlin, J. Jare, and J-M. Morel, editors, Images,Wavelets and PDEs, volume 219 of Lecture Notes in Control and Information Sciences, pages 43–49. Springer, June 1996.

    Google Scholar 

  5. Y.G. Chen, Y. Giga, and S. Goto. Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Differential Geometry, 33:749–786, 1991.

    MATH  MathSciNet  Google Scholar 

  6. David L. Chopp. Computing minimal surfaces via level set curvature flow. Journal of Computational Physics, 106:77–91, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  7. D.L. Chopp and J.A. Sethian. Flow under curvature: singularity formation, minimal surfaces, and geodesics. Experimental Mathematics, 2(4):235–255, 1993.

    MATH  MathSciNet  Google Scholar 

  8. Anders M. Dale and Martin I. Sereno. Improved localization of cortical activity by combining eeg and meg with mri cortical surface reconstruction: A linear approach. Journal of Cognitive Neuroscience, 5(2):162–176, 1993.

    Article  Google Scholar 

  9. Rachid Deriche, Stephane Bouvin, and Olivier. Faugeras. Front propagation and level-set approach for geodesic active stereovision. In Third Asian Conference On Computer Vision, Bombay, India, January 1998.

    Google Scholar 

  10. M. P. DoCarmo. Differential Geometry of Curves and Surfaces. Prentice-Hall, 1976.

    Google Scholar 

  11. Joachim Escher and Gieri Simonett. The volume preserving mean curvature flow near spheres. Proceedings of the american Mathematical Society, 126(9):2789–2796, September 1998.

    Article  MATH  MathSciNet  Google Scholar 

  12. D. C. Van Essen, H.A. Drury, S. Joshi, and M.I. Miller. Functional and structural mapping of human cerebral cortex: Solutions are in the surfaces. In Proceedings of the National Academy of Science, 1998.

    Google Scholar 

  13. L.C. Evans and J. Spruck. Motion of level sets by mean curvature: I. Journal of Differential Geometry, 33:635–681, 1991.

    MATH  MathSciNet  Google Scholar 

  14. Olivier Faugeras and Renaud Keriven. Variational principles, surface evolution, pde’s, level set methods and the stereo problem. IEEE Trans. on Image Processing, 7(3):336–344, March 1998.

    Article  MATH  MathSciNet  Google Scholar 

  15. M. Gage and R.S. Hamilton. The heat equation shrinking convex plane curves. J. of Differential Geometry, 23:69–96, 1986.

    MATH  MathSciNet  Google Scholar 

  16. M. Grayson. The heat equation shrinks embedded plane curves to round points. J. of Differential Geometry, 26:285–314, 1987.

    MATH  MathSciNet  Google Scholar 

  17. Eduard Harabetian and Stanley Osher. Regularization of ill-posed problems via the level set approach. SIAM J. APPL. MATH, 58(6):1689–1706, December 1998.

    Article  MATH  MathSciNet  Google Scholar 

  18. L. Lorigo, O. Faugeras, W.E.L. Grimson, R. Keriven, R. Kikinis, and C-F. Westin. Co-dimension 2 geodesic active contours for mra segmentation. In International Conference on Information Processing in Medical Imaging, June 1999.

    Google Scholar 

  19. R. Malladi, J. A. Sethian, and B.C. Vemuri. Shape modeling with front propagation: A level set approach. PAMI, 17(2):158–175, February 1995.

    Google Scholar 

  20. R. Malladi and J.A. Sethian. Image processing: Flows under min/max curvature and mean curvature. Graphical Models and Image Processing, 58(2):127–141, March 1996.

    Article  MathSciNet  Google Scholar 

  21. Peter J. Olver, Guillermo Sapiro, and Allen Tannenbaum. Invariant geometric evolutions of surfaces and volumetric smoothing. SIAM J. APPL. MATH, 57(1):176–194, February 1997.

    Article  MATH  MathSciNet  Google Scholar 

  22. G. Orban. Cerebral Cortex, chapter 9, pages 359–434. Plenum Press, New York, 1997.

    Google Scholar 

  23. S. Osher and J. Sethian. Fronts propagating with curvature dependent speed: algorithms based on the Hamilton-Jacobi formulation. Journal of Computational Physics, 79:12–49, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  24. N. Paragios and R. Deriche. A PDE-based Level Set Approach for Detection and Tracking of Moving Objects. In Proceedings of the 6th International Conference on Computer Vision, Bombay,India, January 1998. IEEE Computer Society Press.

    Google Scholar 

  25. A. H. Salden. Dynamic Scale-Space Paradigms. PhD thesis, Utrecht University, The Netherlands, 1996.

    Google Scholar 

  26. G. Sapiro and A. Tannenbaum. Area and length preserving geometric invariant scale-spaces. PAMI, 17(1):67–72, January 1995.

    Google Scholar 

  27. J. A. Sethian. Level Set Methods. Cambridge University Press, 1996.

    Google Scholar 

  28. R. B. H. Tootell, J. D. Mendola, N. K. Hadjikhani, P. J. Leden, A. K. Liu, J. B. Reppas, M. I. Sereno, and A. M. Dale. Functional analysis of v3a and related areas in human visual cortex. The Journal of Neuroscience, 17(18):7060–7078, September 1997.

    Google Scholar 

  29. K. Zilles, E. Armstrong, A. Schleicher, and H.-J. Kretschmann. The Human Pattern of Gyrification in the Cerebral Cortex, pages 173–179. 1988.

    Google Scholar 

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Author information

Authors and Affiliations

  1. I.N.R.I.A, BP. 93, 2004 Route des Lucioles, 06902 Sophia Antipolis Cedex, France

    Gerardo Hermosillo, Olivier Faugeras & José Gomes

Authors
  1. Gerardo Hermosillo
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  2. Olivier Faugeras
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  3. José Gomes
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Editor information

Editors and Affiliations

  1. The IT University in Copenhagen, Glentevej 67, DK-2400, Copenhagen NV, Denmark

    Mads Nielsen

  2. Department of Computer Science, University of Copenhagen, Universitetsparken 1, DK-2100, Copenhagen OE, Denmark

    Peter Johansen  & Ole Fogh Olsen  & 

  3. Department of Mathematics and Computer Science, University of Mannheim, D-68131, Mannheim, Germany

    Joachim Weickert

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© 1999 Springer-Verlag Berlin Heidelberg

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Hermosillo, G., Faugeras, O., Gomes, J. (1999). Unfolding the Cerebral Cortex Using Level Set Methods. In: Nielsen, M., Johansen, P., Olsen, O.F., Weickert, J. (eds) Scale-Space Theories in Computer Vision. Scale-Space 1999. Lecture Notes in Computer Science, vol 1682. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48236-9_6

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  • DOI: https://doi.org/10.1007/3-540-48236-9_6

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-66498-7

  • Online ISBN: 978-3-540-48236-9

  • eBook Packages: Springer Book Archive

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