Dynamic Shapes Average∗

A framework for computing shape statistics in general, and average in particular, for dynamic shapes is introduced in this paper. Given a metric d(·, ·) on the set of static shapes, the empirical mean of N static shapes,C1, . . . , CN , is defined by argminC 1 N ∑N i=1 d(C,Ci) . The purpose of this paper is to extend this shape average work to the case of N dynamic shapes and to give an efficient algorithm to compute it. The key concept is to combine the static shape statistics approach with a time-alignment step. To align the time scale while performing the shape average we use dynamic time warping, adapted to deal with dynamic shapes. The proposed technique is independent of the particular choice of the shape metric d(·, ·). We present the underlying concepts, a number of examples, and conclude with a variational formulation to address the dynamic shape average problem. We also demonstrate how to use these results for comparing different types of dynamics. Although only average is addressed in this paper, other shape statistics can be similarly obtained following the framework here proposed.