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 largedisplacement diffeomorphic 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.