Tangent functional canonical correlation analysis for densities and shapes, with applications to multimodal imaging data

TL;DR

This paper introduces a novel tangent space-based canonical correlation analysis method for nonlinear functional data like densities and shapes, aiding multimodal imaging data analysis in biomedical research.

Contribution

It develops a new approach combining tangent space linearization with standard CCA for analyzing complex manifold-valued functional data.

Findings

Effective in capturing associations between densities and shapes.

Demonstrates utility on MRI data of brain tumors.

Facilitates visualization of canonical variate directions.

Abstract

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.