Elastic Full Procrustes Analysis of Plane Curves via Hermitian Covariance Smoothing

For shapes of plane curves, the coordinate systems and parametrizations used are often arbitrary and not of interest. In statistical shape analysis, curves are thus frequently considered as equivalence classes of parameterized curves with respect to the shape invariances translation, rotation and scale, as well as re-parameterization (warping), based on the square-root-velocity (SRV) framework. We propose a novel elastic full Procrustes mean for samples of plane curve shapes. Identifying the real plane with the complex numbers, we establish a connection to covariance estimation in irregular/sparse functional data analysis. We introduce Hermitian covariance smoothing and employ it for mean estimation, thereby newly covering the sparse case and improving robustness to outlier contamination compared to existing methods. Necessary for our approach but also of independent interest, we characterize the covariance structure of rotation-invariant bivariate stochastic processes via complex representations, and identify sampling schemes that allow for observing derivatives/SRVtransforms of sparsely sampled curves. In addition, we develop one- and two-way ANOVA for sparse curve shape data, where exact distance computation is not feasible. We demonstrate the performance of our approach in different realistic simulation settings and use it for an ANOVA of tongue shapes during speech production. Proposed methods are implemented in the R package elastes.