[Paper Review] Computing distances and geodesics between manifold-valued curves in the SRV framework

Alice Le Brigant|arXiv (Cornell University)|Jan 11, 2016

Morphological variations and asymmetry42 references19 citations

TL;DR

This paper introduces a reparametrization-invariant Riemannian metric on the space of manifold-valued curves using the Square Root Velocity Function (SRVF) framework, enabling computation of geodesics and distances that account for both the manifold's geometry and curve reparameterizations. The method pulls back a natural metric on the tangent bundle TM to define a first-order Sobolev metric, yielding explicit geodesic equations and exponential/Jacobi field computations, validated on hyperbolic half-plane curves for radar signal analysis applications.

ABSTRACT

This paper focuses on the study of open curves in a Riemannian manifold M, and proposes a reparametrization invariant metric on the space of such paths. We use the square root velocity function (SRVF) introduced by Srivastava et al. to define a Riemannian metric on the space of immersions M'=Imm([0,1],M) by pullback of a natural metric on the tangent bundle TM'. This induces a first-order Sobolev metric on M' and leads to a distance which takes into account the distance between the origins in M and the L2-distance between the SRV representations of the curves. The geodesic equations for this metric are given and exploited to define an exponential map on M'. The optimal deformation of one curve into another can then be constructed using geodesic shooting, which requires to characterize the Jacobi fields of M'. The particular case of curves lying in the hyperbolic half-plane is considered as an example, in the setting of radar signal processing.

Motivation & Objective

• To develop a reparametrization-invariant Riemannian metric on the space of open, oriented curves in a Riemannian manifold M.
• To extend the SRVF framework beyond Euclidean spaces to general Riemannian manifolds, preserving geometric fidelity along the entire curve.
• To derive explicit geodesic equations and compute the exponential and Jacobi fields for optimal curve deformation via geodesic shooting.
• To apply the framework to the hyperbolic half-plane, motivated by its equivalence to the statistical manifold of Gaussian densities in radar signal processing.
• To compute the Fréchet mean of curves in this space for statistical analysis of locally stationary radar signals.

Proposed method

• Define a Riemannian metric G on the space of immersions M = Imm([0,1], M) as the pullback of a natural L2 metric on the tangent bundle TM via the SRVF transformation.
• Ensure reparametrization invariance by constructing G such that Gc◦φ(h◦φ, k◦φ) = Gc(h,k) for all φ ∈ Diff+([0,1]).
• Derive the geodesic equations for G using the pullback structure, enabling numerical computation of geodesics via geodesic shooting.
• Construct the exponential map on M using the geodesic equations to compute optimal deformations between curves.
• Characterize Jacobi fields along geodesics to support the geodesic shooting algorithm and shape space analysis.
• Apply the framework to the hyperbolic half-plane H, leveraging its known metric and geodesic structure to validate algorithms on simulated radar data.

Experimental results

Research questions

• RQ1 How can a reparametrization-invariant Riemannian metric be defined on the space of manifold-valued curves that respects the intrinsic geometry of the base manifold?
• RQ2 What is the structure of the geodesics and exponential map in this new metric, and how can they be computed efficiently?
• RQ3 How does this SRVF-based approach compare to prior methods that rely on parallel transport to a single tangent space?
• RQ4 Can this framework be effectively applied to non-Euclidean spaces such as the hyperbolic half-plane, particularly in signal processing contexts?
• RQ5 What is the behavior of the Fréchet mean under this metric, and how well does it represent a set of curves in a non-flat space?

Key findings

• The proposed metric is reparametrization invariant and induces a Riemannian structure on the space of curves that accounts for both the distance between curve origins in M and the L2-distance between their SRV representations.
• The geodesic equations are derived explicitly using the pullback structure, enabling numerical computation of geodesics via geodesic shooting with the exponential and Jacobi field maps.
• The method avoids the need to parallel transport all curve data to a single tangent space, thus preserving more geometric information from the manifold's curvature along the entire curve.
• In the hyperbolic half-plane, the framework successfully computes the Fréchet mean of 11 radar signal curves, with the mean curve (in black) accurately capturing the average evolution of a reflection coefficient under small rotor speed variations.
• The algorithm is validated on simulated helicopter radar data, showing that the mean curve serves as a robust reference signature for target recognition, with red and blue ends indicating highest and lowest rotor speeds respectively.
• Theoretical proofs confirm the correctness of the logarithm, exponential, and parallel transport maps in the hyperbolic half-plane, with explicit formulas derived for geodesics and vector transport using Möbius transformations and matrix exponentials.