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Thin-Plate Splines Between Riemannian Manifolds

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http://pubman.mpdl.mpg.de/cone/persons/resource/persons84235

Steinke,  F
Department Empirical Inference, Max Planck Institute for Biological Cybernetics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons83958

Hein,  M
Department Empirical Inference, Max Planck Institute for Biological Cybernetics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons84193

Schölkopf,  B
Department Empirical Inference, Max Planck Institute for Biological Cybernetics, Max Planck Society;

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Citation

Steinke, F., Hein, M., & Schölkopf, B. (2008). Thin-Plate Splines Between Riemannian Manifolds. Talk presented at Workshop on Geometry and Statistics of Shapes 2008. Bonn, Germany.


Cite as: http://hdl.handle.net/11858/00-001M-0000-0013-C94F-D
Abstract
With the help of differential geometry we describe a framework to define a thin-plate spline like energy for maps between arbitrary Riemannian manifolds. The so-called Eells energy only depends on the intrinsic geometry of the input and output manifold, but not on their respective representation. The energy can then be used for regression between manifolds, we present results for cases where the outputs are rotations, sets of angles, or points on 3D surfaces. In the future we plan to also target regression where the output is an element of "shape space", understood as a Riemannian manifold. One could also further explore the meaning of the Eells energy when applied to diffeomorphisms between shapes, especially with regard to its potential use as a distance measure between shapes that does not depend on the embedding or the parametrisation of the shapes.