A metric on shape space with explicit geodesics

作者：Younes Laurent^*; Michor Peter W; Shah Jayant; Mumford David

来源：Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni, 2008, 19(1): 25-57.

DOI：10.4171/rlm/506

• shape space
• diffeomorphism group
• Riemannian metric
• Stiefel manifold

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 explicitly describe the geodesics, first in the parametric case, then by modding out the parametrization 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.

• 出版日期2008
• 单位东北大学