Regression in quotient metric spaces with a focus on elastic curves

TL;DR

This paper introduces regression models for curve-valued data in quotient metric spaces, focusing on re-parametrization invariance, with applications to medical imaging and shape analysis.

Contribution

It generalizes linear regression to quotient metric spaces for curve data, addressing re-parametrization invariance and irregular sampling issues.

Findings

Successfully modeled hippocampal shape variations with respect to age, disease, and sex.

Demonstrated the effectiveness of spline-based regression for irregularly sampled curves.

Validated the approach through simulations and real MRI data analysis.

Abstract

We propose regression models for curve-valued responses in two or more dimensions, where only the image but not the parametrization of the curves is of interest. Examples of such data are handwritten letters, movement paths or outlines of objects. In the square-root-velocity framework, a parametrization invariant distance for curves is obtained as the quotient space metric with respect to the action of re-parametrization, which is by isometries. With this special case in mind, we discuss the generalization of 'linear' regression to quotient metric spaces more generally, before illustrating the usefulness of our approach for curves modulo re-parametrization. We address the issue of sparsely or irregularly sampled curves by using splines for modeling smooth conditional mean curves. We test this model in simulations and apply it to human hippocampal outlines, obtained from Magnetic Resonance Imaging scans. Here we model how the shape of the irregularly sampled hippocampus is related to age, Alzheimer's disease and sex.