Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sasaki Metric for Spline Models of Manifold-Valued Trajectories

Esfandiar Nava-Yazdani, Felix Ambellan, Martin Hanik, Christoph von Tycowicz

TL;DR

The paper addresses the problem of analyzing manifold-valued trajectories by replacing the intractable infinite-dimensional space with a finite-dimensional cubic Bézier spline model on a manifold. It introduces a Sasaki-metric-based pullback, yielding a natural Riemannian structure on the Bézierfold that supports intrinsic regression and principal geodesic analysis, enabling group-level mean trajectories. The key contributions are a bijective F-map from Bézier splines to (TU)^{L+1}, a pulled-back Sasaki metric for efficient geodesic computations, and an application to 2010–2021 Atlantic hurricane tracks demonstrating improved intensity classification over conventional Euclidean and elastic metrics. Practically, the geometry-aware framework improves discriminability of hurricane categories and offers scalable, intrinsic analysis for manifold-valued spatiotemporal data.

Abstract

We propose a generic spatiotemporal framework to analyze manifold-valued measurements, which allows for employing an intrinsic and computationally efficient Riemannian hierarchical model. Particularly, utilizing regression, we represent discrete trajectories in a Riemannian manifold by composite B\' ezier splines, propose a natural metric induced by the Sasaki metric to compare the trajectories, and estimate average trajectories as group-wise trends. We evaluate our framework in comparison to state-of-the-art methods within qualitative and quantitative experiments on hurricane tracks. Notably, our results demonstrate the superiority of spline-based approaches for an intensity classification of the tracks.

Sasaki Metric for Spline Models of Manifold-Valued Trajectories

TL;DR

The paper addresses the problem of analyzing manifold-valued trajectories by replacing the intractable infinite-dimensional space with a finite-dimensional cubic Bézier spline model on a manifold. It introduces a Sasaki-metric-based pullback, yielding a natural Riemannian structure on the Bézierfold that supports intrinsic regression and principal geodesic analysis, enabling group-level mean trajectories. The key contributions are a bijective F-map from Bézier splines to (TU)^{L+1}, a pulled-back Sasaki metric for efficient geodesic computations, and an application to 2010–2021 Atlantic hurricane tracks demonstrating improved intensity classification over conventional Euclidean and elastic metrics. Practically, the geometry-aware framework improves discriminability of hurricane categories and offers scalable, intrinsic analysis for manifold-valued spatiotemporal data.

Abstract

We propose a generic spatiotemporal framework to analyze manifold-valued measurements, which allows for employing an intrinsic and computationally efficient Riemannian hierarchical model. Particularly, utilizing regression, we represent discrete trajectories in a Riemannian manifold by composite B\' ezier splines, propose a natural metric induced by the Sasaki metric to compare the trajectories, and estimate average trajectories as group-wise trends. We evaluate our framework in comparison to state-of-the-art methods within qualitative and quantitative experiments on hurricane tracks. Notably, our results demonstrate the superiority of spline-based approaches for an intensity classification of the tracks.
Paper Structure (14 sections, 1 theorem, 22 equations, 6 figures)

This paper contains 14 sections, 1 theorem, 22 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Mathematical Framework and Method
  3. Background
  4. Bézier Splines
  5. The Bézierfold
  6. Sasaki Metric for the Bézierfold
  7. Application: Hurricane Tracks
  8. Dataset
  9. Spline Model
  10. Group-level Analysis
  11. Intensity Classification
  12. Computational Performance
  13. Conclusion and Future Work
  14. Acknowledgments

Key Result

Theorem 1

The Bézierfold $\mathcal{B}$ can be given the structure of a smooth $(2L+2)n$-dimensional manifold.

Figures (6)

  • Figure 1: Cubic Bézier spline with two segments on the 2-dimensional sphere $S^2$. Footpoints are shown in green while vectors are in orange. Here, $u_0^{(i)}:=\textnormal{log}_{p_0^{(0)}}(p_1^{(0)})$, $u_0^{(1)}:=\textnormal{log}_{p_0^{(1)}}(p_1^{(1)})$, and $u_3^{(1)}:=-\textnormal{log}_{p_3^{(1)}}(p_2^{(1)})$. Control points that are not used in the representation are gray.
  • Figure 2: Maximum sustained wind (in knots): Color-coded for 2010 hurricane tracks (right) and histogram of maxima for all hurricanes (left).
  • Figure 3: Left: Histogram of $R^2$ values for the regressed cubic Bézier curves. Right: Two exemplary hurricane tracks (white) together with regressed one- and two-segment curves (gray/black) with $R^2$ values of $0.995$/$0.998$ and $0.916$/$0.993$, respectively.
  • Figure 4: Two-segment spline Fréchet mean (black) and first two dominant modes of variation (gray; left: first mode, right: second mode) for hurricane tracks.
  • Figure 5: PGA loading plot of the first two principal geodesic modes for proposed (right) and $L^2$ (left) metric. Group-wise means are highlighted with black boundaries.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Definition 2.1: De Casteljau's Algorithm on Manifolds
  • Definition 2.2: Bézierfold of cubic splines
  • Theorem 1
  • proof