HUANG, YOU, ABSIL, GALLIVAN: KARCHER MEAN IN ELASTIC SHAPE ANALYSIS 1 Karcher Mean in Elastic Shape Analysis*

In the framework of elastic shape analysis, a shape is invariant to scaling, translation, rotation and reparameterization. Since this framework does not yield a closed form of geodesic between two shapes, iterative methods have been proposed. In particular, path straightening methods have been proposed and used for computing a geodesic that is invariant to curve scaling and translation. Path straightening can then be exploited within a coordinate-descent algorithm that computes the best rotation and reparameterization of the end point curves [13]. A Riemannian quasi-Newton method to compute a geodesic invariant to scaling, translation, rotation and reparameterization has been given in [15] and shown to be more efficient than the coordinate-descent/path-straightening approach. This paper extends [15] by showing that using the new approach to the geodesic when computing the Karcher mean yields a faster algorithm.