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 (Omega, H, I, P), the background space Omega = boolean ORalpha M-alpha of 0, 1, 2, 3-dimensional manifolds, the set of diffeomorphic transformations on the background space H : Omega <-> 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 (Omega, H, I) corresponding to a homogeneous space [3], in that it can be completely generated from one of its elements, I = HItemp,I-temp is an element of I. For disease, a family of templates boolean ORalphaItempalpha are introduced of perhaps varying dimensional transformation classes. The complete anatomy is a collection of homogeneous spaces I-total = boolean ORalpha(I-alpha,H-alpha). There are three principal components to computational anatomy studied herein. (1) Computation of large deformation maps: Given any two elements I, I' is an element of I in the same homogeneous anatomy (Omega, H, I), compute diffeomorphisms h from one anatomy to the other I (h-1)reversible arrow(h) I'. This is the principal method by which anatomical structures are understood, transferring the emphasis from the images I is an element of I to the structural transformations h is an element of H that generate them. (2) Computation of empirical probability laws: Given populations of anatomical imagery and diffeomorphisms between them I h(n-1)reversible arrow(hn) I-n, n = 1, . . . , N, generate probability laws P is an element of P on H that represent the anatomical variation reflected by the observed population of diffeomorphisms h(n), n = 1,..., N. (3) Inference and disease testing: Within the anatomy (Omega, H, I, P), perform Bayesian classification and testing for disease and anomaly.
