A Metric on Shape Space with Explicit Geodesics

By Peter W. Michor, David Mumford, J. Shah, Laurent Younes. 2007

250 citations recorded by the source corpus.

This paper studies a specific metric on plane curves that has the property of being isometric to classical manifold (sphere, complex projective, Stiefel, Grassmann) modulo change of parametrization, each of these classical manifolds being associated to specific qualifications of the space of curves (closed-open, modulo rotation etc...) Using these isometries, we are able to explicitely describe the geodesics, first in the parametric case, then by modding out the paremetrization and considering horizontal vectors. We also compute the sectional curvature for these spaces, and show, in particular, that the space of closed curves modulo rotation and change of parameter has positive curvature. Experimental results that explicitly compute minimizing geodesics between two closed curves are finally provided

Subfields: Applied Mathematics, Computational Mechanics, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/1b0449070fb2b27ed222a828f27b94e70358c5c0

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In Applied Mathematics

In Computational Mechanics

In Radiology, Nuclear Medicine and Imaging

Citations in the corpus

Matrix-valued kernels for shape deformation analysis (2013, background): “See, for example, the papers [38, 39, 40, 41, 44, 59] and references therein.”
Properties of Sobolev-type metrics in the space of curves (2006, methodology): “This model may be viewed as a Sobolev-type metric in the space of curves up to rotation, translation and scaling and has been studied in depth in a recent paper [8].”
Optimization Methods on Riemannian Manifolds and Their Application to Shape Space (2012, methodology): “The Riemannian metric of the Stiefel manifold can now be imposed on the the space of smooth closed curves with unit length and fixed base point, which was shown to be equivalent to endowing this shape space with a Sobolev-type metric [26].”
Statistical analysis of trajectories on Riemannian manifolds: Bird migration, hurricane tracking and video surveillance (2014, methodology): “In case M = R2, where the problem involves registration and shape analysis of planar curves, the solution is discussed in Michor and Mumford (2007), Younes et al. (2008), Shah (2008) and Sundaramoorthi et al. (2011).”
Geometry of Diffeomorphism Groups, Complete integrability and Geometric statistics (2013, background): “[29] discusses a metric on the space of immersed curves which is also isometric to an infinite-dimensional round sphere and hence has explicit geodesics.”