Second order elastic metrics on the shape space of curves

Martin Bauer, Martins Bruveris, Philipp Harms and Jakob Møller-Andersen

Abstract

Second order Sobolev metrics on the space of regular unparametrized planar curves have several desirable completeness properties not present in lower order metrics, but numerics are still largely missing. In this paper, we present algorithms to numerically solve the initial and boundary value problems for geodesics. The combination of these algorithms allows to compute Karcher means in a Riemannian gradient-based optimization scheme. Our framework has the advantage that the constants determining the weights of the zero, first, and second order terms of the metric can be chosen freely. Moreover, due to its generality, it could be applied to more general spaces of mapping. We demonstrate the effectiveness of our approach by analyzing a collection of shapes representing physical objects.

Session

Workshop: 1st International Workshop on DIFFerential Geometry in Computer Vision for Analysis of Shapes, Images and Trajectories (DIFF-CV)

Files

Paper (PDF, 820K)

DOI

10.5244/C.29.DIFFCV.9
https://dx.doi.org/10.5244/C.29.DIFFCV.9

Citation

Bibtex

@inproceedings{DIFFCV2015_9,
	title={Second order elastic metrics on the shape space of curves},
	author={Martin Bauer and Martins Bruveris and Philipp Harms and Jakob Møller-Andersen},
	year={2015},
	month={September},
	pages={9.1-9.11},
	articleno={9},
	numpages={11},
	booktitle={Proceedings of the 1st International Workshop on DIFFerential Geometry in Computer Vision for Analysis of Shapes, Images and Trajectories (DIFF-CV 2015)},
	publisher={BMVA Press},
	editor={H. Drira, S. Kurtek, and P. Turaga},
	doi={10.5244/C.29.DIFFCV.9},
	isbn={1-901725-56-1},
	url={https://dx.doi.org/10.5244/C.29.DIFFCV.9}
}