Computation of crowded geodesics on the universal Teichmüller spaceMS75

The problems of geometric shape classification, identification, and cliquing are important considerations in pattern theory and computer vision. A popular mathematical approach is to consider the space of shapes as a metrized Riemannian manifold and to subsequently compute distances between points (i.e., shapes) on the manifold. We consider a particular metrization of planar shapes that has the attractive properties of scale and translation invariance: the Weil-Peterson metric on the universal Teichmueller space. This metrization reduces the Riemannian structure to anaylsis of univariate functions. However, sufficiently complicated shapes induce "crowding" in the univariate functional representations: extremely fine structure that must be resolved in order to correctly model the topological shape space. We present a numerical technique that is effective at resolving shapes with reasonable amounts of crowding, and provide associated computational results. Finally, we identify current shortcomings and discuss in-development solutions to these problems.

This presentation is part of Minisymposium “MS75 - Geometric methods for shape analysis with applications to biomedical imaging and computational anatomy, Part II (2 parts)
organized by: Joan Alexis Glaunès (MAP5, Université Paris Descartes) , Sergey Kushnarev (Singapore University of Technology and Design) , Mario Micheli (Harvey Mudd College) .

Akil Narayan (Scientific Computing and Imaging (SCI) Institute, the University of Utah)
Sergey Kushnarev (Singapore University of Technology and Design)
image registration, shape analysis, shape space