In this chapter we consider the choice of shape distance, which is required to fully define the non-Euclidean shape metric space. We consider the partial Procrustes, full Procrustes and Riemannian shape distances. In the case of 2D point configurations the shape space is the complex projective space, and in the special case of triangles in two dimensions the shape space is a sphere. Finally, we consider different choices of coordinates in the tangent space to shape space, which are particularly useful for practical statistical inference.
We shall initially describe the partial and full Procrustes distances. Consider two configuration matrices from k points in m dimensions X1 and X2 with pre-shapes Z1 and Z2. First we minimize over rotations to find the closest Euclidean distance between Z1 and Z2.
Definition 4.1 The partial Procrustes distance dP is obtained by matching the pre-shapes Z1 and Z2 of X1 and X2 as closely as possible over rotations. So,
where Zj = HXj/||HXj||, j = 1, 2.
Here ‘inf’ denotes the infimum, and we will use ‘sup’ for the supremum.
Result 4.1 The partial Procrustes distance is given by:
where λ1 ≥ λ2 ≥ … ≥ λm − 1 ≥ |λm| are the square roots of the eigenvalues of ZT1Z2ZT2Z1, ...