Tensor biometrics for changes in cranial shape

D'Arcy Thompson's method of Cartesian transformations can be adapted to the biometric study of group differences and trends in shape. The key to this extension is the symmetric tensor, a mathematical representation of the linear deformation taking one triangle onto another. These tensors may easily be computed from data that come in the form of Cartesian coordinates of homologous landmarks, such as are customary in roentgenographic cephalometrics. This article shows how the tensors make possible the rigorous statistical analysis of populations of shapes without requiring the specification of particular shape measures in advance. Instead, the findings of a study include the extraction of particular shape measures most clearly manifesting the group difference or treatment effect under study. Only conventional multivariate statistical techniques, such as Hotelling's T2, are involved. Configurations of many landmarks can be analysed via the display of their mean difference as a biorthogonal grid pair, perusal of which leads to the selection of triangles of landmarks for more rigorous analysis. The techniques are exemplified in two cephalometric studies: the description and prediction of normal craniofacial growth, and a comparison of the craniofacial deformities associated with Apert's and Crouzon's syndromes.