Principal geodesic analysis in the space of discrete shells

Martin Rumpf
Rheinische Friedrich-Wilhelms-Universität Bonn

Important sources of shape variability, such as articulated motion of body models or soft tissue dynamics, are highly nonlinear and are usually superposed on top of rigid body motion which must be factored out. We propose a novel, nonlinear, rigid body motion invariant Principal Geodesic Analysis (PGA) that allows us to analyse this variability, compress large variations based on statistical shape analysis and fit a model to measurements. For given input shape data sets we show how to compute a low dimensional approximating submanifold on the space of discrete shells, making our approach a hybrid between a physical and statistical model. General discrete shells can be projected onto the submanifold and sparsely represented by a small set of coefficients.
We propose two different approaches: one based on a rigid body motion invariant, discrete metric and the other on a representation of triangular surfaces via dihedral angles and edge length and a discrete Gauss Codazzi equation.

We demonstrate two specific applications: model-constrained mesh editing and reconstruction of a dense animated mesh from sparse motion capture markers using the statistical knowledge as a prior.

This is joint work with B. Heeren, C. Zhang, J. Sassen, and W. Smith

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