Shape Variation in Outline Shapes

A general morphometric method for describing shape variation in a sample consisting of landmarks and multiple outline shapes is developed in this article. A distance metric is developed for such data and is used to embed the data in a low-dimensional Euclidean space. The Euclidean space is used to generate summary statistics such as mean and principal shape variation which are implicitly represented in the original space using elements of the sample. A new distance metric for outline shapes is proposed based on Procrustes distance that does not require the extraction of discrete points along the curve. The outline distance metric can be naturally combined with distances between landmarks. A method for aligning outlines and multiple outlines is developed that minimizes the distance metric. The method is compared with semilandmarks on synthetic data and 2 real data sets. Outline methods produce useful and valid results when suitably constrained by landmarks and are useful visualization aids, but questions remain about their suitability for answering biological questions until appropriate distance metrics can be biologically validated.