  It is quite common for functional data arising from imaging data to assume
values in infinite-dimensional manifolds. Uncovering associations between two
or more such nonlinear functional data extracted from the same object across
medical imaging modalities can assist development of personalized treatment
strategies. We propose a method for canonical correlation analysis between
paired probability densities or shapes of closed planar curves, routinely used
in biomedical studies, which combines a convenient linearization and dimension
reduction of the data using tangent space coordinates. Leveraging the fact that
the corresponding manifolds are submanifolds of unit Hilbert spheres, we
describe how finite-dimensional representations of the functional data objects
can be easily computed, which then facilitates use of standard multivariate
canonical correlation analysis methods. We further construct and visualize
canonical variate directions directly on the space of densities or shapes.
Utility of the method is demonstrated through numerical simulations and
performance on a magnetic resonance imaging dataset of Glioblastoma Multiforme
brain tumors.
