We consider the task of computing shape statistics and classification
of 3D anatomical structures (as continuous, parameterized surfaces) under a Riemannian
framework. This task requires a Riemannian metric that allows: (1) reparameterizations
of surfaces by isometries, and (2) efficient computations of
geodesic paths between surfaces. These tools allow for computing Karcher means
and covariances (using tangent PCA) for shape classes, and a probabilistic classification
of surfaces into disease and control classes. In a separate paper [13],
we introduced a mathematical representation of surfaces, called q-maps, and we
used the L2 metric on the space of q-maps to induce a Riemannian metric on
the space of parameterized surfaces. We also developed a path-straightening algorithm
for computing geodesic paths [14]. This process requires optimal reparameterizations
(deformations of grids) of surfaces and achieves a superior
alignment of geometric features across surfaces. The resulting means and covariances
are better representatives of the original data and lead to parsimonious
shape models. These two moments specify a normal probability model on shape
classes, which are then used for classifying test shapes. Through improved random
sampling and a higher classification performance, we demonstrate the success
of this model over some past methods. In addition to toy objects, we use the
Detroit Fetal Alcohol and Drug Exposure Cohort data to study brain structures
and present classification results for the Attention Deficit Hyperactivity Disorder
cases and controls in this study.We find that using the mean and covariance structure
of the given data, we are able to attain a 88% classification rate, which is an
improvement over a previously reported result of 82% on the same data