Distances and algorithms to compare sets of shapes for automated biological morphometrics

In this thesis we present the Generalized Dataset Procrustes Distance, the basis for an automated framework to compare datasets of rigid biological shapes. It is based on a pairwise shape comparison algorithm that generalizes Procrustes Analysis in three dimensions and is closely related to the Iterative Closest Point (ICP) algorithm. It is not restricted by the topology of the shapes and is completely automatic, with only one parameter, namely the number of points to consider in each shape. The framework is based on an optimization problem on the whole dataset, which is assumed to consist of multiple similar shapes. Its backbone is the computation of the Minimum Spanning Tree of a complete graph where each vertex represents a shape in the dataset and the weights represent new distances between the shapes. Each of these distances is (relatively) expensive to compute, but we can reduce the number of shape comparisons required to compute the MST by exploiting that the distances satisfy the triangle inequality. This new framework provides morphologists with a tool to compare 3D scans of bones in a way that requires no human interaction and thus both reduces the time necessary for sample preparation and is free of human bias. Furthermore, the output of the algorithm can be interpreted by the well established procedures of the Morphometrics community, which facilitates its adoption and use. We solve the Generalized Dataset Procrustes Problem for three datasets of biological importance: the calcanei, astragali and grooming claws of primates.