Comparative evaluation of spline-based and Laplace-based interpolation for joint torque field reconstruction: a simulation study

Accurate interpolation of joint torque fields across joint angle combinations is essential in experimental and clinical biomechanics. Although thin-plate spline (TPS) interpolation is widely used, its exact interpolation property renders it sensitive to outliers. The present study compared TPS, thin-plate smoothing spline (TPSS) with two smoothing levels, and discrete Laplace interpolation under controlled simulation conditions. Two ground truth surfaces were defined: a physiologically motivated single-peak surface representing plantarflexion torque as a function of ankle and knee angles and a two-peak surface representing a more complex spatial structure. Noisy measurement conditions were simulated by sampling each surface on a jittered 5 × 5 grid, adding proportional Gaussian noise (5% or 10%), and introducing outliers (0-5 points, ±10% or ±20%) or missing data (0-5 points). Each condition was simulated 1000 times, and reconstruction accuracy was quantified by root mean square error (RMSE) and absolute deviation from the median, with method comparisons performed using Wilcoxon signed-rank tests. For the single-peak surface, TPSS with greater smoothing consistently yielded the lowest RMSE across outlier conditions, whereas TPS was most adversely affected by outliers. For the two-peak surface, TPS outperformed other methods, and Laplace interpolation consistently produced the largest RMSE. Under missing-data conditions, spline-based methods showed comparable or superior accuracy to Laplace interpolation, despite its theoretical suitability for handling missing data. These findings empirically demonstrate condition-dependent differences in interpolation accuracy across methods and suggest that method selection should be guided by prior knowledge or preliminary assessment of the spatial complexity of the target torque field.