A Novel Infimum Dimension Nash Embedding for 3D Projective Shape Data

Music City Center 

An isometric embedding of a Riemannian manifold in Euclidean space is an embedding such that the Riemannian structure on each tangent space is induced by the scalar product in the ambient space; such an embedding preserves path lengths, angles, areas and volumes. Such a mapping preserves much of the geometric information contained in the original Riemannian manifold where the data is valued allowing for some computationally advantages. In addition if the Riemmanian manifold is homogeneous then we would also want the such an embedding, to be equivariant. The Veronese-Whitney (VW) embedding is an isometric equivariant embedding for 3D projective shape data. It is in fact the only embedding used in the analysis of 3D projective shape data, at the present time. In this paper we consider a novel equivariant isometric embedding, the Nash embedding into a Euclidean space of a lower dimension than the Euclidean space where the Veronese-Whitney embedding is valued . We compare the performance of the novel Nash embedding-based statistical techniques with those  based on the VW embedding in Monte Carlo studies and a real data example.

Extrinsic shape analysis

Nash embedding

Nonparametric test

Three-dimensional projective shape analysis 

Section on Statistics in Imaging