Two shape metrics for biomedical outline data: bending energy, Procrustes distance, and the biometrical modeling of shape phenomena

Abstract

Quantitative studies of shapes in the real world are often expedited by the tools of morphometrics, a branch of multivariate statistics organized to exploit the origin of such data in real Euclidean geometry. This paper reviews one such tool that has recently been designed for the analysis of corresponding biological outlines of similar shape, such as those that arise from organ or tissue boundaries in medical images. The analysis is a hybrid of two morphometric tools, Procrustes analysis and the thin-plate spline, the properties of which are already well-understood separately. The thin-plate spline optimizes one shape metric, the Procrustes methods another. Their combination permits the averaging of groups of outlines, the rigorous display of sample variation around these averages in full detail, and rigorous multivariate statistical detection and testing of hypotheses linking those patterns to their putative causes or effects. The new tool is demonstrated using outlines of the corpus callosum (connection of the cerebral hemispheres) in mid-sagittal images of 12 normal human brains and 13 brains of schizophrenics.