Simulation-based analysis of rigid alignment bias in PCA of weight-bearing foot shapes.

Abstract

Generalized Procrustes Analysis (GPA) is used for rigid alignment in statistical shape modeling to remove positional and size variation prior to Principal Component Analysis (PCA). GPA assumes isotropic variation, which is violated in boundary-constrained structures such as weight-bearing feet and may distort PCA-based estimates of shape variability. This study presents a simulation-based framework to quantify alignment-induced bias in PCA of 3D foot point clouds with predefined deformation modes derived from real foot scans, for which the deformation direction and magnitude serve as ground truth. Eight simulated datasets with increasing foot-length heterogeneity (±0-50 mm) were aligned using standard GPA and a constrained variant that preserved sole-ground contact. Bias was quantified using directional agreement with ground truth and differences in explained variance. Standard GPA introduced systematic bias even without foot-length variation. While the constrained approach reduced bias at low heterogeneity levels, both methods showed increasing distortion with greater variation. Alignment-induced bias was accompanied by a redistribution of variance across components, with increasing concentration in the dominant mode. Increasing sample size reduced variability in the estimated principal component directions but did not affect bias magnitude. These findings demonstrate that alignment procedures can systematically distort the covariance structure in PCA-based shape models in the presence of anatomical or mechanical constraints. The proposed framework provides a basis for quantifying such effects and evaluating strategies to mitigate alignment-induced bias.