On Parameters of Nonlinear Scalar Dynamics from Video: Invariants, Calibration, and Identifiability

2026-09-29

AI Summary

This study addresses the absence of identifiability theory for video-based parameter estimation in nonlinear scalar dynamical systems. Moving beyond the limitations of linear frameworks, we establish a comprehensive identifiability theory for nonlinear second-order ordinary differential equations. Through state-space analysis and computer vision observation modeling, we reveal the nature of nonlinear coordinate ambiguity and affine alignment mechanisms. Furthermore, we propose a physics-grounded normalization method to determine parameter uniqueness or identify necessary calibration conditions. Extensive experiments on synthetic and real-world videos of pendulum and free-fall dynamics validate our theoretical predictions regarding parameter relationships, coverage effects, and calibration requirements. Ultimately, this work lays a rigorous theoretical foundation for visual parameter identification in nonlinear dynamical systems.

Abstract

Physical parameter estimation from video aims to recover the parameters of a known family of governing dynamical equations from pixel observations. Existing identifiability theory for this setting has focused on linear time-invariant (LTI) second-order systems, leaving open what can be identified for nonlinear scalar dynamics. We develop an identifiability theory for nonlinear scalar second-order ODEs, organized by how their velocity dependence interacts with changes of the learned state coordinate. Under a shared non-collapsed state map and explicit same-state velocity-coverage conditions, we show that parameter identifiability depends on the ODE family: some parameters are uniquely identifiable, while in other families only invariant parameter combinations are identifiable or external physical calibration is required. For laws that are at most linear in velocity, compatibility forces affine coordinate alignment, yielding explicit parameter relations, invariants, and calibration conditions. This affine conclusion extends to broader finite velocity-feature families when coordinate curvature can be separated from the declared velocity dependence. For families admitting a squared-velocity term, nonlinear coordinate ambiguity can remain; a law-derived normalization instead enables affine comparison between canonical laws. Experiments on synthetic systems and real pendulum and free-fall videos support the predicted parameter relations, coverage effects, and calibration requirements.

Problem

• parameter identifiability
• nonlinear scalar dynamics
• physical parameter estimation
• video observations

Innovation

• Nonlinear Scalar Dynamics
• Parameter Identifiability
• Video-based Estimation
• Affine Coordinate Alignment
• Physical Calibration