  The classification of shapes is of great interest in diverse areas ranging
from medical imaging to computer vision and beyond. While many statistical
frameworks have been developed for the classification problem, most are
strongly tied to early formulations of the problem - with an object to be
classified described as a vector in a relatively low-dimensional Euclidean
space. Statistical shape data have two main properties that suggest a need for
a novel approach: (i) shapes are inherently infinite dimensional with strong
dependence among the positions of nearby points, and (ii) shape space is not
Euclidean, but is fundamentally curved. To accommodate these features of the
data, we work with the square-root velocity function of the curves to provide a
useful formal description of the shape, pass to tangent spaces of the manifold
of shapes at different projection points which effectively separate shapes for
pairwise classification in the training data, and use principal components
within these tangent spaces to reduce dimensionality. We illustrate the impact
of the projection point and choice of subspace on the misclassification rate
with a novel method of combining pairwise classifiers.
