On Some Novel Uses of Strain Tensors Beyond Visualization in Modern Shape Analysis

Modern Shape Analysis integrates mathematical and statistical methods to study morphology using geometric data from 2D and 3D digitization. Beyond visualization, local deformation analysis offers deeper insight into longitudinal changes like ontogenetic growth. This study explores strain tensors as analytical tools for shape transformations, evaluating Thin Plate Spline, Quadratic Trend, and Cubic Trend interpolants in estimating deformation tensors at landmarks. Simulated 2D data and 3D hominid skulls confirm that tensor-based multivariate analyses, such as Principal Component Analysis, yield results comparable to Parallel Transport methods. Thin Plate Spline more accurately recovers assigned tensors than least square interpolants, making it preferable for local deformation analysis. This method integrates traditional shape comparisons, proving particularly useful for studying ontogenetic trajectories and biomechanical deformations.