Statistical Modeling of Curves Using Shapes and Related Features

Abstract

Motivated by the problems of analyzing protein backbones, diffusion tensor magnetic resonance imaging (DT-MRI) fiber tracts in the human brain, and other problems involving curves, in this study we present some statistical models of parameterized curves, in , in terms of combinations of features such as shape, location, scale, and orientation. For each combination of interest, we identify a representation manifold, endow it with a Riemannian metric, and outline tools for computing sample statistics on these manifolds. An important characteristic of the chosen representations is that the ensuing comparison and modeling of curves is invariant to how the curves are parameterized. The nuisance variables, including parameterization, are removed by forming quotient spaces under appropriate group actions. In the case of shape analysis, the resulting spaces are quotient spaces of Hilbert spheres, and we derive certain wrapped truncated normal densities for capturing variability in observed curves. We demonstrate these models using both artificial data and real data involving DT-MRI fiber tracts from multiple subjects and protein backbones from the Shape Retrieval Contest of Non-rigid 3D Models (SHREC) 2010 database.

KEY WORDS:

• DT-MRI fibers
• Path-straightening
• Protein backbones
• Shape analysis
• Square-root function
• Square-root velocity function

Acknowledgments

The authors thank Dr Adam W. Anderson for providing the DT-MRI data used in this work [National Institutes of Health (NIH) grant: R01NS058639]. This research was also supported by the grant ONR N00014-09-1-0664 and the National Science Foundation (NSF) grant DMS-0915003(AS).