  Motivated by the problem of nonparametric inference in high level digital
image analysis, we introduce a general extrinsic approach for data analysis on
Hilbert manifolds with a focus on means of probability distributions on such
sample spaces. To perform inference on these means, we appeal to the concept of
neighborhood hypotheses from functional data analysis and derive a one-sample
test. We then consider analysis of shapes of contours lying in the plane. By
embedding the corresponding sample space of such shapes, which is a Hilbert
manifold, into a space of Hilbert-Schmidt operators, we can define extrinsic
mean shapes of planar contours and their sample analogues. We apply the general
methods to this problem while considering the computational restrictions faced
when utilizing digital imaging data. Comparisons of computational cost are
provided to another method for analyzing shapes of contours.
