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Variational Methods in Shape Analysis

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Handbook of Mathematical Methods in Imaging

Abstract

The concept of a shape space is linked both to concepts from geometry and from physics. On one hand, a path-based viscous flow approach leads to Riemannian distances between shapes, where shapes are boundaries of objects that mainly behave like fluids. On the other hand, a state-based elasticity approach induces a (by construction) non-Riemannian dissimilarity measure between shapes, which is given by the stored elastic energy of deformations matching the corresponding objects. The two approaches are both based on variational principles. They are analyzed with regard to different applications, and a detailed comparison is given.

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Acknowledgements

The model proposed in section “Viscous Fluid-Based Shape Space” has been developed in cooperation with Leah Bar and Guillermo Sapiro from the University of Minnesota. Benedikt Wirth has been funded by the Bonn International Graduate School in Mathematics. Furthermore, the work was supported by the Deutsche Forschungsgemeinschaft, SPP 1253 “Optimization with Partial Differential Equations.” Part of Figs. 3–4 and 19–23 have been taken from [83], the results from Figs. 6, 8, and 10–18 stem from [67, 69].

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  1. Institute for Numerical Simulation, Bonn University, Bonn, Germany

    Martin Rumpf & Benedikt Wirth

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  1. Martin Rumpf
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Correspondence to Martin Rumpf .

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  1. Computational Science Center, University of Vienna, Vienna, Austria

    Otmar Scherzer

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Rumpf, M., Wirth, B. (2015). Variational Methods in Shape Analysis. In: Scherzer, O. (eds) Handbook of Mathematical Methods in Imaging. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0790-8_56

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