K-GMRF: Kinetic Gauss-Markov Random Field for First-Principles Covariance Tracking on Lie Groups

• URL: http://arxiv.org/abs/2603.19601v1
• Date: Fri, 20 Mar 2026 03:16:36 GMT
• Title: K-GMRF: Kinetic Gauss-Markov Random Field for First-Principles Covariance Tracking on Lie Groups
• Abstract summary: We propose K-GMRF, an online, training-free framework for covariance tracking.<n>Our method interprets observations as torques driving a latent angular velocity, propagated via a structure-preserving symplectic integrator.<n>We theoretically prove that this second-order dynamics achieves zero steady-state error under constant rotation.
• Score: 8.489406212619164
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Tracking non-stationary covariance matrices is fundamental to vision yet hindered by existing estimators that either neglect manifold constraints or rely on first-order updates, incurring inevitable phase lag during rapid evolution. We propose K-GMRF, an online, training-free framework for covariance tracking that reformulates the problem as forced rigid-body motion on Lie groups. Derived from the Euler-Poincaré equations, our method interprets observations as torques driving a latent angular velocity, propagated via a structure-preserving symplectic integrator. We theoretically prove that this second-order dynamics achieves zero steady-state error under constant rotation, strictly superior to the proportional lag of first-order baselines. Validation across three domains demonstrates robust tracking fidelity: (i) on synthetic ellipses, K-GMRF reduces angular error by 30x compared to Riemannian EMA while maintaining stability at high speeds; (ii) on SO(3) stabilization with 20% dropout, it decreases geodesic error from 29.4° to 9.9°; and (iii) on OTB motion-blur sequences, it improves loU from 0.55 to 0.74 on BlurCar2 with a 96% success rate. As a fully differentiable symplectic module, K-GMRF provides a plug-and-play geometric prior for data-constrained scenarios and an interpretable layer within modern deep architectures.

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