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 ( Ω , H , I , P ) \left ( \Omega , H, I, P \right ) , the background space Ω = ˙ U α M α \Omega \dot = {U_\alpha }{M_\alpha } of 0, 1, 2, 3-dimensional manifolds, the set of diffeomorphic transformations on the background space H : Ω ↔ Ω {H} : \Omega \leftrightarrow \Omega , the space of idealized medical imagery I I , and P P the family of probability measures on H H . The group of diffeomorphic transformations H 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 ( Ω , H , I ) \left ( \Omega , H, I \right ) corresponding to a homogeneous space [3], in that it can be completely generated from one of its elements, I = H I t e m p , I t e m p ∈ I I = {HI_{temp}}, {I_{temp}} \in I . For disease, a family of templates U α I t e m p α {U_\alpha }I_{temp}^\alpha are introduced of perhaps varying dimensional transformation classes. The complete anatomy is a collection of homogeneous spaces I t o t a l = U α ( I α , H α ) {I_{total}} = {U_\alpha }\left ( {I^\alpha }, {H^\alpha } \right ) .

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