Computational anatomy: an emerging discipline

This paper studies mathematical methods in the emerging new discipline of Computational Anatomy. Herein we formalize the Brown/Washington University model of anatomy following the global pattern theory introduced in [1, 2], in which anatomies are represented as deformable templates, collections of 0, 1, 2, 3-dimensional manifolds. Typical structure is carried by the template with the variabilities accommodated via the application of random transformations to the background manifolds. The anatomical model is a quadruple $\left ( \Omega , H, I, P \right )$, the background space $\Omega \dot = {U_\alpha }{M_\alpha }$ of 0, 1, 2, 3-dimensional manifolds, the set of diffeomorphic transformations on the background space ${H} : \Omega \leftrightarrow \Omega$, the space of idealized medical imagery $I$, and $P$ the family of probability measures on $H$. The group of diffeomorphic transformations $H$ is chosen to be rich enough so that a large family of shapes may be generated with the topologies of the template maintained. For normal anatomy one deformable template is studied, with $\left ( \Omega , H, I \right )$ corresponding to a homogeneous space [3], in that it can be completely generated from one of its elements, $I = {HI_{temp}}, {I_{temp}} \in I$. For disease, a family of templates ${U_\alpha }I_{temp}^\alpha$ are introduced of perhaps varying dimensional transformation classes. The complete anatomy is a collection of homogeneous spaces ${I_{total}} = {U_\alpha }\left ( {I^\alpha }, {H^\alpha } \right )$.