Nonlinear Regression on Manifolds for Shape Analysis using Intrinsic B\'ezier Splines

TL;DR

This paper introduces a nonlinear regression framework on manifolds using Riemannian Bézier splines, enabling flexible modeling of complex data evolutions beyond geodesic assumptions.

Contribution

It presents a constructive approach for Riemannian spline regression with efficient evaluation, extending manifold regression to nonlinear trajectories using Bézier curves.

Findings

Effective reconstruction of periodic mitral valve motion

Analysis of femoral shape changes during osteoarthritis

Bézier spline regression outperforms linear models in flexibility

Abstract

Intrinsic and parametric regression models are of high interest for the statistical analysis of manifold-valued data such as images and shapes. The standard linear ansatz has been generalized to geodesic regression on manifolds making it possible to analyze dependencies of random variables that spread along generalized straight lines. Nevertheless, in some scenarios, the evolution of the data cannot be modeled adequately by a geodesic. We present a framework for nonlinear regression on manifolds by considering Riemannian splines, whose segments are B\'ezier curves, as trajectories. Unlike variational formulations that require time-discretization, we take a constructive approach that provides efficient and exact evaluation by virtue of the generalized de Casteljau algorithm. We validate our method in experiments on the reconstruction of periodic motion of the mitral valve as well as the analysis of femoral shape changes during the course of osteoarthritis, endorsing B\'ezier spline regression as an effective and flexible tool for manifold-valued regression.