Diffeomorphometry and geodesic positioning systems for human anatomy.

The Computational Anatomy project has largely been a study of large deformations within a Riemannian framework as an efficient point of view for generating metrics between anatomical configurations. This approach turns D'Arcy Thompson's comparative morphology of human biological shape and form into a metrizable space. Since the metric is constructed based on the geodesic length of the flows of diffeomorphisms connecting the forms, we call it diffeomorphometry. Just as importantly, since the flows describe algebraic group action on anatomical submanifolds and associated functional measurements, they become the basis for positioning information, which we term geodesic positioning. As well the geodesic connections provide Riemannian coordinates for locating forms in the anatomical orbit, which we call geodesic coordinates. These three components taken together - the metric, geodesic positioning of information, and geodesic coordinates - we term the geodesic positioning system. We illustrate via several examples in human and biological coordinate systems and machine learning of the statistical representation of shape and form.