Abstract
We present a novel framework to exert topology control over a level set evolution. Level set methods offer several advantages over parametric active contours, in particular automated topological changes. In some applications, where some a priori knowledge of the target topology is available, topological changes may not be desirable. This is typically the case in biomedical image segmentation, where the topology of the target shape is prescribed by anatomical knowledge. However, topologically constrained evolutions often generate topological barriers that lead to large geometric inconsistencies. We introduce a topologically controlled level set framework that greatly alleviates this problem. Unlike existing work, our method allows connected components to merge, split or vanish under some specific conditions that ensure that the genus of the initial active contour (i.e. its number of handles) is preserved. We demonstrate the strength of our method on a wide range of numerical experiments and illustrate its performance on the segmentation of cortical surfaces and blood vessels.
Similar content being viewed by others
References
Adalsteinsson, D., & Sethian, J. A. (1995). A fast level set method for propagating interfaces. Journal of Computational Physics, 118(2), 269–277.
Bardinet, E., Cohen, L. D., & Ayache, N. (1998). A parametric deformable model to fit unstructured 3D data. Computer Vision and Image Understanding, 71(1), 39–54.
Bertrand, G. (1994). Simple points, topological numbers and geodesic neighborhoods in cubic grids. Pattern Recognition Letters, 15(10), 1003–1011.
Bertrand, G. (1996). A boolean characterization of three-dimensional simple points. Pattern Recognition Letters, 17, 115–124.
Caselles, V., Kimmel, R., & Sapiro, G. (1997). Geodesic active contours. The International Journal of Computer Vision, 22(1), 61–79.
Dale, A. D., Fischl, B., & Sereno, M. I. (1999). Cortical surface-based analysis, I: segmentation and surface reconstruction. NeuroImage, 9, 179–194.
Davatzikos, C., & Bryan, R. N. (1996). Using a deformable surface model to obtain a shape representation of the cortex. IEEE Transactions on Medical Imaging, 15, 758–795.
DoCarmo, M. P. (1976). Differential geometry of curves and surfaces. New York: Prentice-Hall.
Duan, Y., Yang, L., Qin, H., & Samaras, D. (2004). Shape reconstruction from 3D and 2D data using PDE-based deformable surfaces. In European conference on computer vision (Vol. 3, pp. 238–251).
Faugeras, O., & Keriven, R. (1998). Variational principles, surface evolution, PDE’s, level set methods and the stereo problem. IEEE Transactions on Image Processing, 7(3), 336–344.
Fischl, B., Liu, A., & Dale, A. M. (2001). Automated manifold surgery: Constructing geometrically accurate and topologically correct models of the human cerebral cortex. IEEE Transactions on Medical Imaging, 20, 70–80.
Fua, P., & Leclerc, Y. G. (1995). Object-centered surface reconstruction: Combining multi-image stereo and shading. The International Journal of Computer Vision, 16(1), 35–56.
Goldenberg, R., Kimmel, R., Rivlin, E., & Rudzsky, M. (2002). Cortex segmentation: a fast variational geometric approach. IEEE Transactions on Medical Imaging, 21(2), 1544–1551.
Goldlücke, B., & Magnor, M. (2004). Space-time isosurface evolution for temporally coherent 3D reconstruction. In International conference on computer vision and pattern recognition (Vol. 1, pp. 350–355).
Guskov, I., & Wood, Z. (2001). Topological noise removal. In Graphics proceedings (Vol. I, pp. 19–26).
Han, X., Xu, C., Braga-Neto, U., & Prince, J. L. (2002). Topology correction in brain cortex segmentation using a multiscale, graph-based approach. IEEE Transactions on Medical Imaging, 21(2), 109–121.
Han, X., Xu, C., & Prince, J. L. (2003). A topology preserving level set method for geometric deformable models. IEEE Transactions on Pattern Analysis and Machine Intelligence, 25(6), 755–768.
Hatcher, A. (2002). Algebraic topology. Cambridge: Cambridge University Press.
Jin, H., Soatto, S., & Yezzi, A. J. (2003). Multi-view stereo beyond Lambert. In International conference on computer vision and pattern recognition (Vol. 1, pp. 171–178).
Kass, M., Witkin, A., & Terzopoulos, D. (1987). Snakes: active contour models. The International Journal of Computer Vision, 1(4), 321–331.
Kriegeskorte, N., & Goeble, R. (2001). An efficient algorithm for topologically segmentation of the cortical sheet in anatomical mr volumes. NeuroImage, 14, 329–346.
Lorensen, W. E., & Cline, H. E. (1987). Marching cubes: a high-resolution 3D surface reconstruction algorithm. ACM Computer Graphics, 21(4), 163–170.
MacDonald, D., Kabani, N., Avis, D., & Evens, A. C. (2000). Automated 3D extraction of inner and outer surfaces of cerebral cortex from mri. NeuroImage, 12, 340–356.
Mangin, J.-F., Frouin, V., Bloch, I., Regis, J., & Lopez-Krahe, J. (1995). From 3D magnetic resonance images to structural representations of the cortex topography using topology preserving deformations. Journal of Mathematical Imaging and Vision, 5, 297–318.
Metaxas, D. N., & Terzopoulos, D. (1993). Shape and nonrigid motion estimation through physics-based synthesis. IEEE Transactions on Pattern Analysis and Machine Intelligence, 15(6), 580–591.
Osher, S., & Sethian, J. A. (1988). Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. Journal of Computational Physics, 79(1), 12–49.
Paragios, N., & Deriche, R. (2005). Geodesic active regions and level set methods for motion estimation and tracking. Computer Vision and Image Understanding, 97(3), 259–282.
Pons, J.-P. (2005). Methodological and applied contributions to the deformable models framework. PhD dissertation, Ecole Nationale des Ponts et Chaussées, 18 November 2005.
Pons, J.-P., Keriven, R., & Faugeras, O. (2007). Multi-view stereo reconstruction and scene flow estimation with a global image-based matching score. The International Journal of Computer Vision, 72(2), 179–193.
Ségonne, F. (2005). Segmentation of medical images under topological constraints. PhD dissertation, Massachusetts Institute of Technology, December 12 2005.
Ségonne, F., Pons, J.-P., Grimson, E., & Fischl, B. (2005). A novel level set framework for the segmentation of medical images under topology control. In Workshop on computer vision for biomedical image applications: current techniques and future trends (pp. 135–145).
Ségonne, F., Pacheco, J., & Fischl, B. (2007). Geometrically-accurate topology simplification of triangulated cortical surfaces using non-separating loops. IEEE Transactions on Medical Imaging, 26(4), 518–529.
Shattuck, D. W., & Leahy, R. M. (2001). Automated graph based analysis and correction of cortical volume topology. IEEE Transactions on Medical Imaging, 20(11), 1167–1177.
Taubin, G., Cukierman, F., Sullivan, S., Ponce, J., & Kriegman, D. J. (1994). Parameterized families of polynomials for bounded algebraic curve and surface fitting. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16(3), 287–303.
Xu, C., Pham, D. L., Rettmann, M. E., Yu, D. N., & Prince, J. L. (1999). Reconstruction of the human cerebral cortex from magnetic resonance images. IEEE Transactions on Medical Imaging, 18, 467–480.
Yezzi, A. J., & Soatto, S. (2003). Deformotion: Deforming motion, shape average and the joint registration and approximation of structures in images. The International Journal of Computer Vision, 53(2), 153–167.
Zhao, H., Osher, S., Merriman, B., & Kang, M. (2000). Implicit and non-parametric shape reconstruction from unorganized points using a variational level set method. Computer Vision and Image Understanding, 80(3), 295–314.
Author information
Authors and Affiliations
Corresponding author
Electronic Supplementary Material
Rights and permissions
About this article
Cite this article
Ségonne, F. Active Contours Under Topology Control—Genus Preserving Level Sets. Int J Comput Vis 79, 107–117 (2008). https://doi.org/10.1007/s11263-007-0102-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11263-007-0102-8