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 )
.