Biometrics, biomathematics and the morphometric synthesis
 Fred L. Bookstein
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Abstract
At the core of contemporarymorphometrics—the quantitative study of biological shape variation—is a synthesis of two originally divergent methodological styles. One contributory tradition is the multivariate analysis of covariance matrices originally developed as biometrics and now dominant across a broad expanse of applied statistics. This approach, couched solely in the linear geometry of covariance structures, ignores biomathematical aspects of the original measurements. The other tributary emphasizes the direct visualization of changes in biological form. However, making objective the biological meaning of the features seen in those diagrams was always problematical; also, the representation of variation, as distinct from pairwise difference, proved infeasible.
To combine these two variants of biomathematical modeling into a valid praxis for quantitative studies of biological shape was a goal earnestly sought though most of this century. That goal was finally achieved in the 1980s when techniques from mathematical statistics, multivariate biometrics, nonEuclidean geometry and computer graphics were combined in a coherent new system of tools for the complete regionalized quantitative analysis oflandmark points together with the biomedical images in which they are seen.
In this morphometric synthesis, correspondence of landmarks (biologically labeled geometric points, like “bridge of the nose”) across specimens is taken as a biomathematical primitive. The shapes of configurations of landmarks are defined as equivalence classes with respect to the Euclidean similarity group and then represented as single points in David Kendall'sshape space, a Riemannian manifold with Procrustes distance as metric. All conventional multivariate strategies carry over to the study of shape variation and covariation when shapes are interpreted in the tangent space to the shape manifold at an average shape. For biomathematical interpretation of such analyses, one needs a basis for the tangent space compatible with the reality of local biotheoretical processes and explanations at many different geometric scales, and one needs graphics for visualizing average shape differences and other statistical contrasts there. Both of these needs are managed by thethinplate spline, a deformation function that has an unusually helpful linear algebra. The spline also links the biometrics of landmarks to deformation analysis of the images from which the landmarks originally arose.
This article reviews the history and principal tools of this synthesis in their biomathematical and biometrical context and demonstrates their usefulness in a study of focal neuroanatomical anomalies in schizophrenia.
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 Title
 Biometrics, biomathematics and the morphometric synthesis
 Journal

Bulletin of Mathematical Biology
Volume 58, Issue 2 , pp 313365
 Cover Date
 19960301
 DOI
 10.1007/BF02458311
 Print ISSN
 00928240
 Online ISSN
 15229602
 Publisher
 Kluwer Academic Publishers
 Additional Links
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 Authors

 Fred L. Bookstein ^{(1)}
 Author Affiliations

 1. University of Michigan, 300 North Ingalls Building, 481092007, Ann Arbor, MI, USA