Discrete Geodesic Regression in Shape Space
A new approach for the effective computation of geodesic regression curves in shape spaces is presented. Here, one asks for a geodesic curve on the shape manifold that minimizes a sum of dissimilarity measures between given two- or three-dimensional input shapes and corresponding shapes along the regression curve. The proposed method is based on a variational time discretization of geodesics. Curves in shape space are represented as deformations of suitable reference shapes, which renders the computation of a discrete geodesic as a PDE constrained optimization for a family of deformations. The PDE constraint is deduced from the discretization of the covariant derivative of the velocity in the tangential direction along a geodesic. Finite elements are used for the spatial discretization, and a hierarchical minimization strategy together with a Lagrangian multiplier type gradient descent scheme is implemented. The method is applied to the analysis of root growth in botany and the morphological changes of brain structures due to aging.
Unable to display preview. Download preview PDF.
- 2.Kilian, M., Mitra, N.J., Pottmann, H.: Geometric modeling in shape space. ACM Transactions on Graphics 26, #64, 1–8 (2007)Google Scholar
- 3.Beg, M.F., Miller, M., Trouvé, A., Younes, L.: Computational anatomy: Computing metrics on anatomical shapes. In: Proceedings of 2002 IEEE ISBI, pp. 341–344 (2002)Google Scholar
- 4.Miller, M.I., Trouvé, A., Younes, L.: The metric spaces, Euler equations, and normal geodesic image motions of computational anatomy. In: Proceedings of the 2003 International Conference on Image Processing, vol. 2, pp. 635–638. IEEE (2003)Google Scholar
- 7.Davis, B., Fletcher, P.T., Bullitt, E., Joshi, S.: Population shape regression from random design data. In: Proceedings of IEEE International Conference on Computer Vision (2007)Google Scholar
- 9.Fletcher, P.T.: Geodesic regression on Riemannian manifolds. In: MICCAI Workshop on Mathematical Foundations of Computational Anatomy (MFCA), pp. 75–86 (2011)Google Scholar
- 22.Rumpf, M., Wirth, B.: Variational time discretization of geodesic calculus (2012), http://de.arxiv.org/abs/1210.2097
- 23.Fletcher, R.: Practical Methods of Optimization, 2nd edn. John Wiley & Sons (1987)Google Scholar