Spline-Based Smoothing of Noisy Discrete Curves in the Frenet-Serret Framework: Sensitivity Analysis of Curvature and Torsion Estimation via CSI and TSI Indices for Analytically Defined Space Curves

Abstract

This study investigates the robustness of Frenet-Serret curvature (kappa) and torsion (tau) estimates derived from noisy discretely-sampled three-dimensional space curves, with emphasis on the comparative performance of cubic spline and cubic Hermite interpolation methods. Accurate estimation of these geometric invariants is essential for reliable analysis of curves arising in signal processing and shape reconstruction; yet, the higher-order derivatives required for their computation exhibit pronounced sensitivity to measurement noise. We examine curves constructed through a Hilbert transform-based parameterization of the form r(t)=X(t),A(t)sin phi(t),g(t), where discrete samples are contaminated with additive white Gaussian noise at varying signal-to-noise ratios. Reconstruction is performed using cubic spline interpolation, which ensures global C2 continuity, as well as cubic Hermite spline interpolation, which provides C1 continuity with local tangent control. Frenet frame computations are then applied via regularized finite difference schemes. To characterize noise amplification theoretically, we derive the Curvature Stability Index (CSI) and Torsion Stability Index (TSI) as first-order variance bounds under the delta method. While these indices formalize the derivative-order dependence of noise sensitivity, Monte Carlo simulations reveal that empirical variance exceeds theoretical predictions by factors of 104 to 106, indicating dominance of nonlinear error propagation. Nevertheless, the indices establish that torsion instability arises fundamentally from third-order derivative structure rather than ground-truth magnitude. Numerical experiments across three geometric regimes constant-invariant helices, variable-curvature helices, and planar curves with identically zero torsion demonstrate that the ratio of the torsion root mean square error to curvature root mean square error consistently ranges from 6.5 to 9.8. This disparity persists even in the degenerate planar case, where tau equivalent to 0 analytically, confirming that torsion sensitivity is an intrinsic property of the Frenet-Serret formulation. Across all configurations, cubic spline reconstruction yields lower Monte Carlo mean RMSE and reduced empirical variance compared to Hermite spline, providing superior stability for derivative-based invariant estimation.