Geodesics in the TPS Space

Mathematics

Paolo Piras¹  Valerio Varano²  Stefano Gabriele²  Stefan Shlager³  Franco Milicchio⁴  Ian Dryden⁵ 

[1] CPIA3 Roma, 00186 Roma, Italy;Department of Architecture, Roma Tre University, 00184 Rome, Italy;Department of Biological Anthropology, University of Freiburg, 79106 Freiburg, Germany;Department of Engineering, Roma Tre University, 00146 Rome, Italy;School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK;

关键词: shape analysis;    geodesics;    thin plate spline;   

DOI  :  10.3390/math10091562

来源: DOAJ

【 摘 要 】

In shape analysis, the interpolation of shapes’ trajectories is often performed by means of geodesics in an appropriate Riemannian Shape Space. Over the past several decades, different metrics and shape spaces have been proposed, including Kendall shape space, LDDMM based approaches, and elastic contour, among others. Once a Riemannian space is chosen, geodesics and parallel transports can be used to build splines or piecewise geodesics paths. In a recent paper, we introduced a new Riemannian shape space named TPS Space based on the Thin Plate Spline interpolant and characterized by an appropriate metric and parallel transport rule. In the present paper, we further explore the geometry of the TPS Space by characterizing the properties of its geodesics. Several applications show the capability of the proposed formulation to conserve important physical properties of deformation, such as local strains and global elastic energy.

【 授权许可】

Unknown