A Continuum Mechanical Approach to Geodesics in Shape Space
In this paper concepts from continuum mechanics are used to define geodesic paths in the space of shapes, where shapes are implicitly described as boundary contours of objects. The proposed shape metric is derived from a continuum mechanical notion of viscous dissipation. A geodesic path is defined as the family of shapes such that the total amount of viscous dissipation caused by an optimal material transport along the path is minimized. The approach can easily be generalized to shapes given as segment contours of multi-labeled images and to geodesic paths between partially occluded objects. The proposed computational framework for finding such a minimizer is based on the time discretization of a geodesic path as a sequence of pairwise matching problems, which is strictly invariant with respect to rigid body motions and ensures a 1–1 correspondence along the induced flow in shape space. When decreasing the time step size, the proposed model leads to the minimization of the actual geodesic length, where the Hessian of the pairwise matching energy reflects the chosen Riemannian metric on the underlying shape space. If the constraint of pairwise shape correspondence is replaced by the volume of the shape mismatch as a penalty functional, one obtains for decreasing time step size an optical flow term controlling the transport of the shape by the underlying motion field. The method is implemented via a level set representation of shapes, and a finite element approximation is employed as spatial discretization both for the pairwise matching deformations and for the level set representations. The numerical relaxation of the energy is performed via an efficient multi-scale procedure in space and time. Various examples for 2D and 3D shapes underline the effectiveness and robustness of the proposed approach.
KeywordsRiemannian shape space Geodesics Variational time discretization Viscous fluids Rigid body motion invariance
Unable to display preview. Download preview PDF.
- Black, M. J., & Anandan, P. (1993). A framework for the robust estimation of optical flow. In Fourth international conference on computer vision, ICCV-93 (pp. 231–236). Google Scholar
- Eckstein, I., Pons, J., Tong, Y., Kuo, C., & Desbrun, M. (2007). Generalized surface flows for mesh processing. In Eurographics symposium on geometry processing. Google Scholar
- Fletcher, P., & Whitaker, R. (2006). Riemannian metrics on the space of solid shapes. In MICCAI 2006: Med Image Comput Assist Interv. Google Scholar
- Fuchs, M., Jüttler, B., Scherzer, O., & Yang, H. Shape metrics based on elastic deformations. Journal of Mathematical Imaging and Vision, to appear, 2009. Google Scholar
- Kapur, T., Yezzi, L., & Zöllei, L. (2001). A variational framework for joint segmentation and registration. In IEEE CVPR—MMBIA (pp. 44–51). Google Scholar
- Liu, X., Shi, Y., Dinov, I., & Mio, W. (2010). A computational model of multidimensional shape. doi:10.1007/s11263-010-0323-0
- Truesdell, C., & Noll, W. (2004). The non-linear field theories of mechanics. Berlin: Springer. Google Scholar
- Yezzi, A. J., & Mennucci, A. (2005). Conformal metrics and true “gradient flows” for curves. In ICCV 2005: Proceedings of the 10th IEEE international conference on computer vision (pp. 913–919). Google Scholar
- Zolésio, J.-P. (2004). Shape topology by tube geodesic. In IFIP conference on system modeling and optimization No 21 (pp. 185–204). Google Scholar