<s>
Statistical	O
shape	O
analysis	O
and	O
statistical	O
shape	O
theory	O
in	O
computational	B-Algorithm
anatomy	I-Algorithm
(	O
CA	O
)	O
is	O
performed	O
relative	O
to	O
templates	O
,	O
therefore	O
it	O
is	O
a	O
local	O
theory	O
of	O
statistics	O
on	O
shape	O
.	O
</s>
<s>
Template	O
estimation	O
in	O
computational	B-Algorithm
anatomy	I-Algorithm
from	O
populations	O
of	O
observations	O
is	O
a	O
fundamental	O
operation	O
ubiquitous	O
to	O
the	O
discipline	O
.	O
</s>
<s>
Linear	B-Language
algebra	I-Language
is	O
one	O
of	O
the	O
central	O
tools	O
to	O
modern	O
engineering	O
.	O
</s>
<s>
Central	O
to	O
linear	B-Language
algebra	I-Language
is	O
the	O
notion	O
of	O
an	O
orbit	O
of	O
vectors	O
,	O
with	O
the	O
matrices	O
forming	O
groups	B-Algorithm
(	O
matrices	O
with	O
inverses	O
and	O
identity	O
)	O
which	O
act	O
on	O
the	O
vectors	O
.	O
</s>
<s>
In	O
linear	B-Language
algebra	I-Language
the	O
equations	O
describing	O
the	O
orbit	O
elements	O
the	O
vectors	O
are	O
linear	O
in	O
the	O
vectors	O
being	O
acted	O
upon	O
by	O
the	O
matrices	O
.	O
</s>
<s>
In	O
computational	B-Algorithm
anatomy	I-Algorithm
the	O
space	O
of	O
all	O
shapes	O
and	O
forms	O
is	O
modeled	O
as	O
an	O
orbit	O
similar	O
to	O
the	O
vectors	O
in	O
linear-algebra	O
,	O
however	O
the	O
groups	B-Algorithm
do	O
not	O
act	O
linear	O
as	O
the	O
matrices	O
do	O
,	O
and	O
the	O
shapes	O
and	O
forms	O
are	O
not	O
additive	O
.	O
</s>
<s>
In	O
computational	B-Algorithm
anatomy	I-Algorithm
addition	O
is	O
essentially	O
replaced	O
by	O
the	O
law	O
of	O
composition	O
.	O
</s>
<s>
Several	O
group	B-Algorithm
actions	I-Algorithm
in	I-Algorithm
computational	I-Algorithm
anatomy	I-Algorithm
have	O
been	O
defined	O
.	O
</s>
<s>
The	O
high-dimensional	O
diffeomorphism	O
groups	B-Algorithm
used	O
in	O
computational	B-Algorithm
anatomy	I-Algorithm
are	O
generated	O
via	O
smooth	O
flows	O
which	O
satisfy	O
the	O
Lagrangian	O
and	O
Eulerian	O
specification	O
of	O
the	O
flow	O
fields	O
satisfying	O
the	O
ordinary	O
differential	O
equation	O
:	O
</s>
<s>
The	O
central	O
statistical	O
model	O
of	O
computational	B-Algorithm
anatomy	I-Algorithm
in	O
the	O
context	O
of	O
medical	B-Application
imaging	I-Application
is	O
the	O
source-channel	O
model	O
of	O
Shannon	O
theory	O
;	O
the	O
source	O
is	O
the	O
deformable	O
template	O
of	O
images	O
,	O
the	O
channel	O
outputs	O
are	O
the	O
imaging	O
sensors	O
with	O
observables	O
.	O
</s>
<s>
The	O
variation	O
in	O
the	O
anatomical	O
configurations	O
are	O
modelled	O
separately	O
from	O
the	O
Medical	B-Application
imaging	I-Application
modalities	O
Computed	O
Axial	O
Tomography	O
machine	O
,	O
MRI	O
machine	O
,	O
PET	B-Application
machine	O
,	O
and	O
others	O
.	O
</s>
<s>
Maximum	B-General_Concept
a	I-General_Concept
posteriori	I-General_Concept
estimation	I-General_Concept
(	O
MAP	O
)	O
estimation	O
is	O
central	O
to	O
modern	O
statistical	O
theory	O
.	O
</s>
<s>
Given	O
the	O
observed	O
image	O
,	O
MAP	B-General_Concept
estimation	I-General_Concept
maximizes	O
the	O
posterior	O
:	O
</s>
<s>
In	O
the	O
Bayesian	O
random	O
orbit	O
model	O
of	O
computational	B-Algorithm
anatomy	I-Algorithm
the	O
observed	O
MRI	O
images	O
are	O
modelled	O
as	O
a	O
conditionally	O
Gaussian	O
random	O
field	O
with	O
mean	O
field	O
,	O
with	O
a	O
random	O
unknown	O
transformation	O
of	O
the	O
template	O
.	O
</s>
<s>
The	O
MAP	B-General_Concept
estimation	I-General_Concept
problem	O
is	O
to	O
estimate	O
the	O
unknown	O
template	O
given	O
the	O
observed	O
MRI	O
images	O
.	O
</s>
<s>
This	O
is	O
accomplished	O
using	O
the	O
expectation-maximization	B-Algorithm
(	O
EM	O
)	O
algorithm	O
.	O
</s>
<s>
The	O
orbit-model	O
is	O
exploited	O
by	O
associating	O
the	O
unknown	O
to	O
be	O
estimated	O
flows	O
to	O
their	O
log-coordinates	O
via	O
the	O
Riemannian	O
geodesic	O
log	O
and	O
exponential	O
for	O
computational	B-Algorithm
anatomy	I-Algorithm
the	O
initial	O
vector	O
field	O
in	O
the	O
tangent	O
space	O
at	O
the	O
identity	O
so	O
that	O
,	O
with	O
the	O
mapping	O
of	O
the	O
hyper-template	O
.	O
</s>
