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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
M. Bertalmio, G. Sapiro, and G. Randall. Region tracking on level-sets methods. IEEE Transactions On Medical Imaging, to appear.
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.
V. Caselles, R. Kimmel, and G. Sapiro. Geodesic active contours. The International Journal of Computer Vision, 22(1):61–79, 1997.
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.
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.
David L. Chopp. Computing minimal surfaces via level set curvature flow. Journal of Computational Physics, 106:77–91, 1993.
D.L. Chopp and J.A. Sethian. Flow under curvature: singularity formation, minimal surfaces, and geodesics. Experimental Mathematics, 2(4):235–255, 1993.
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.
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.
M. P. DoCarmo. Differential Geometry of Curves and Surfaces. Prentice-Hall, 1976.
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.
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.
L.C. Evans and J. Spruck. Motion of level sets by mean curvature: I. Journal of Differential Geometry, 33:635–681, 1991.
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.
M. Gage and R.S. Hamilton. The heat equation shrinking convex plane curves. J. of Differential Geometry, 23:69–96, 1986.
M. Grayson. The heat equation shrinks embedded plane curves to round points. J. of Differential Geometry, 26:285–314, 1987.
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.
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.
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.
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.
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.
G. Orban. Cerebral Cortex, chapter 9, pages 359–434. Plenum Press, New York, 1997.
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.
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.
A. H. Salden. Dynamic Scale-Space Paradigms. PhD thesis, Utrecht University, The Netherlands, 1996.
G. Sapiro and A. Tannenbaum. Area and length preserving geometric invariant scale-spaces. PAMI, 17(1):67–72, January 1995.
J. A. Sethian. Level Set Methods. Cambridge University Press, 1996.
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.
K. Zilles, E. Armstrong, A. Schleicher, and H.-J. Kretschmann. The Human Pattern of Gyrification in the Cerebral Cortex, pages 173–179. 1988.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/3-540-48236-9_6
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-66498-7
Online ISBN: 978-3-540-48236-9
eBook Packages: Springer Book Archive