Statistical Analysis on Manifolds: A Nonparametric Approach for Inference on Shape Spaces

This article concerns nonparametric statistics on manifolds with special emphasis on landmark based shape spaces in which a k-ad, i.e., a set of k points or landmarks on an object or a scene, is observed in 2D or 3D, for purposes of identification, discrimination, or diagnostics. Two different notions of shape are considered: reflection shape invariant under all translations, scaling and orthogonal transformations, and affine shape invariant under all affine transformations. A computation of the extrinsic mean reflection shape, which has remained unresolved in earlier works, is given in arbitrary dimensions, enabling one to extend nonparametric inference on Kendall type shape manifolds from 2D to higher dimensions. For both reflection and affine shapes, two sample test statistics are constructed based on appropriate choice of orthonormal frames on the tangent bundle and computations of differentials of projection maps with respect to them at the sample extrinsic mean. The samples in consideration can be either independent or be the outcome of a matched pair experiment. Examples are included to illustrate the theory.