Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Shape splines and stochastic shape evolutions: A second order point of view

Authors: Alain Trouvé and François-Xavier Vialard
Journal: Quart. Appl. Math. 70 (2012), 219-251
MSC (2010): Primary 65D07, 62J02, 37K65, 34F05, 93E14
Published electronically: February 3, 2012
MathSciNet review: 2953101
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Abstract: This article presents a new mathematical framework to perform statistical analysis on time-indexed sequences of 2D or 3D shapes. At the core of this statistical analysis is the task of time interpolation of such data. Current models in use can be compared to linear interpolation for one-dimensional data. We develop a spline interpolation method which is directly related to cubic splines on a Riemannian manifold. Our strategy consists of introducing a control variable on the Hamiltonian equations of the geodesics. Motivated by statistical modeling of spatiotemporal data, we also design a stochastic model to deal with random shape evolutions. This model is closely related to the spline model since the control variable previously introduced is set as a random force perturbing the evolution.

Although we focus on the finite-dimensional case of landmarks, our models can be extended to infinite-dimensional shape spaces, and they provide a first step for a nonparametric growth model for shapes taking advantage of the widely developed framework of large deformations by diffeomorphisms.

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Additional Information

Alain Trouvé
Affiliation: CMLA, Ecole Normale Supérieure de Cachan, CNRS, UniverSud, 61, Avenue du Président Wilson, F-94 235 Cachan Cedex, France
Email: alain.trouve@cmla.ens-cachan.fr

François-Xavier Vialard
Affiliation: Institute for Mathematical Science, Imperial College London, 53 Prince’s Gate, SW7 2PG, London, United Kingdom
Email: francois.xavier.vialard@normalesup.org

DOI: https://doi.org/10.1090/S0033-569X-2012-01250-4
Received by editor(s): March 19, 2010
Published electronically: February 3, 2012
Article copyright: © Copyright 2012 Brown University

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