GPSA2: combining landmark-free and landmark-based methods in geometric morphometrics

Geometric Morphometrics (GM) revolutionized the way that biologists quantify shape variation among individuals, populations, and species. Traditional GM methods are based on homologous landmarks that can be reliably identified across all specimens in a sample. However, landmark-based studies are limited by the intensive labor required of anatomical experts, and regions of interest are often devoid of landmarks. These limitations inspired the development of many "landmark-free" approaches, but unreliable homology estimation and complicated underlying mathematical bases can make biological interpretation challenging. Here we present GPSA2, a novel method for analyzing surface meshes that combines landmark-based and landmark-free methodology within the familiar framework of Generalized Procrustes Analysis. In a major innovation, our method can incorporate user-defined landmarks into otherwise landmark-free analysis by transforming the landmarks into pointwise shape descriptors that are exploited during iterative homology estimation and superimposition (i.e. "alignment" of objects). GPSA2 also addresses a longstanding issue in morphometrics - the impact of variability in the distribution of sampled points over an object - by introducing a surface area-weighted shape distance metric and superimposition cost function. The improved homology approximation, together with the application of Taubin smoothing and an optional resistant-fit superimposition technique, ensure robust analysis even when a dataset exhibits regions of intense shape variation. We apply GPSA2 to two empirical datasets: 15 primate skulls and 369 mouse bacula. Our analyses show that inclusion of landmarks increases biological accuracy, and that GPSA2 produces summaries of shape variation that are easy to visualize and interpret.