Riemannian metrics for statistics on shapes: parallel transport and scale invariance


To be able to statistically compare evolutions of image timeseries data requires a method to express these evolutions in a common coordinate system. This requires a mechanism to transport evolutions between coordinate systems: e.g., parallel transport has been used for large displacement di ffeomorphic metric mapping (LDDMM) approaches. A common purpose to study evolutions is to assess local tissue growth or decay as observed in the context of neurodevelopment or neurodegeneration. Hence, preserving this information under transport is important to allow for faithful statistical analysis in the common coordinate system. Most basically, we require scale invariance. Here, we show that a scale invariant metric does not exist in the LDDMM setting. We illustrate the impact of this non-invariance on parallel transport. We also propose a new class of Riemannian metrics on shapes which preserves the variation of a global indicator such as volume under parallel transport.

Proceedings of the 4th MICCAI workshop on Mathematical Foundations of Computational Anatomy (MFCA)
Marc Niethammer
Marc Niethammer
Professor of Computer Science

My research interests include image registration, image segmentation, shape analysis, machine learning, and biomedical applications.