Computational anatomy: an emerging discipline

This paper studies mathematical methods in the emerging new discipline ofComputational 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. Fornormal anatomyone deformable template is studied, with$\left ( \Omega , H, I \right )$corresponding to ahomogeneous space[3], in that it can be completely generated from one of its elements,$I = {HI_{temp}}, {I_{temp}} \in I$. Fordisease, 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