Geodesics in the TPS Space

Valerio Varano, Stefano Gabriele, Franco Milicchio, Stefan Shlager, Ian Dryden and Paolo Piras
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Valerio Varano: Department of Architecture, Roma Tre University, 00184 Rome, Italy
Stefano Gabriele: Department of Architecture, Roma Tre University, 00184 Rome, Italy
Franco Milicchio: Department of Engineering, Roma Tre University, 00146 Rome, Italy
Stefan Shlager: Department of Biological Anthropology, University of Freiburg, 79106 Freiburg, Germany
Ian Dryden: School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK
Paolo Piras: CPIA3 Roma, 00186 Roma, Italy

Mathematics, 2022, vol. 10, issue 9, 1-20

Abstract: 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.

Keywords: shape analysis; geodesics; thin plate spline (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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