Eigenshapes

Eigenshapes are a concept arising from the field of geometric morphometrics, which uses statistical methods to analyze shape variation. They represent the principal modes of shape variation within a collection of objects. Imagine a set of related biological specimens, such as a group of bird skulls. If these skulls vary in size and shape, eigenshapes capture the dominant patterns of this variation.

Mathematically, eigenshapes are derived from a covariance matrix of landmark positions, typically after a Procrustes analysis to remove differences due to size, orientation, and position. The process involves performing a principal component analysis (PCA) on the superimposed landmark coordinates. The resulting principal components are then transformed back into shape variables, and these are the eigenshapes. Each eigenshape is associated with an eigenvalue, which indicates the amount of shape variation explained by that particular mode. The first eigenshape accounts for the largest portion of the variation, the second for the next largest, and so on.

Visualizing eigenshapes involves plotting the mean shape and then deforming it along the direction of an eigenshape. This allows researchers to see how the shape changes as it moves along a particular axis of variation. For instance, an eigenshape for bird skulls might show variation in the length of the beak or the height of the cranial vault. Eigenshapes are valuable tools for understanding evolutionary trends, developmental processes, and functional adaptations by revealing the fundamental ways in which shapes differ within a population or across species.