Diffeomorphometry and geodesic positioning systems for human anatomy

Abstract

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 <i>diffeomorphometry</i>. 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 <i>geodesic positioning</i>. As well the geodesic connections provide Riemannian coordinates for locating forms in the anatomical orbit, which we call <i>geodesic coordinates</i>. These three components taken together - the metric, geodesic positioning of information, and geodesic coordinates - we term the <i>geodesic positioning system</i>. We illustrate via several examples in human and biological coordinate systems and machine learning of the statistical representation of shape and form.

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