Landmark methods for forms without landmarks: morphometrics of group differences in outline shape

Morphometrics, a new branch of statistics, combines tools from geometry, computer graphics and biometrics in techniques for the multivariate analysis of biological shape variation. Although medical image analysts typically prefer to represent scenes by way of curving outlines or surfaces, the most recent developments in this associated statistical methodology have emphasized the domain of landmark data: size and shape of configurations of discrete, named points in two or three dimensions. This paper introduces a combination of Procrustes analysis and thin-plate splines, the two most powerful tools of landmark-based morphometrics, for multivariate analysis of curving outlines in samples of biomedical images. The thin-plate spline is used to assign point-to-point correspondences, called semi-landmarks, between curves of similar but variable shape, while the standard algorithm for Procrustes shape averages and shape coordinates is altered to accord with the ways in which semi-landmarks formally differ from more traditional landmark loci. Subsequent multivariate statistics and visualization proceed mainly as in the landmark-based methods. The combination provides a range of complementary filters, from high pass to low pass, for effects on outline shape in grouped studies. The low-pass version is based on the spectrum of the spline, the high pass, on a familiar special case of Procrustes analysis. This hybrid method is demonstrated in a comparison of the shape of the corpus callosum from mid-sagittal sections of MRI of 25 human brains, 12 normal and 13 with schizophrenia.