<s>
Computational	B-Algorithm
anatomy	I-Algorithm
(	O
CA	O
)	O
is	O
a	O
discipline	O
within	O
medical	B-Application
imaging	I-Application
focusing	O
on	O
the	O
study	O
of	O
anatomical	O
shape	O
and	O
form	O
at	O
the	O
visible	O
or	O
gross	O
anatomical	O
scale	O
of	O
morphology	O
.	O
</s>
<s>
The	O
field	O
is	O
broadly	O
defined	O
and	O
includes	O
foundations	O
in	O
anatomy	O
,	O
applied	O
mathematics	O
and	O
pure	O
mathematics	O
,	O
including	O
medical	B-Application
imaging	I-Application
,	O
neuroscience	O
,	O
physics	O
,	O
probability	O
,	O
and	O
statistics	O
.	O
</s>
<s>
It	O
focuses	O
on	O
the	O
anatomical	O
structures	O
being	O
imaged	O
,	O
rather	O
than	O
the	O
medical	B-Application
imaging	I-Application
devices	O
.	O
</s>
<s>
The	O
central	O
focus	O
of	O
the	O
sub-field	O
of	O
computational	B-Algorithm
anatomy	I-Algorithm
within	O
medical	B-Application
imaging	I-Application
is	O
mapping	O
information	O
across	O
anatomical	O
coordinate	O
systems	O
most	O
often	O
dense	O
information	O
measured	O
within	O
a	O
magnetic	O
resonance	O
image	O
(	O
MRI	O
)	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
has	O
been	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
see	O
Figure	O
)	O
.	O
</s>
<s>
The	O
importance	O
of	O
the	O
source-channel	O
model	O
is	O
that	O
the	O
variation	O
in	O
the	O
anatomical	O
configuration	O
are	O
modelled	O
separated	O
from	O
the	O
sensor	O
variations	O
of	O
the	O
Medical	B-Application
imagery	I-Application
.	O
</s>
<s>
The	O
random	O
orbit	O
model	O
of	O
Computational	B-Algorithm
Anatomy	I-Algorithm
first	O
appeared	O
in	O
modelling	O
the	O
change	O
in	O
coordinates	O
associated	O
to	O
the	O
randomness	O
of	O
the	O
group	O
acting	O
on	O
the	O
templates	O
,	O
which	O
induces	O
the	O
randomness	O
on	O
the	O
source	O
of	O
images	O
in	O
the	O
anatomical	O
orbit	O
of	O
shapes	O
and	O
forms	O
and	O
resulting	O
observations	O
through	O
the	O
medical	B-Application
imaging	I-Application
devices	O
.	O
</s>
<s>
For	O
the	O
study	O
of	O
deformable	O
shape	O
in	O
CA	O
,	O
the	O
high-dimensional	O
diffeomorphism	O
groups	O
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>
In	O
the	O
random	B-Algorithm
orbit	I-Algorithm
model	I-Algorithm
of	I-Algorithm
computational	I-Algorithm
anatomy	I-Algorithm
,	O
the	O
entire	O
flow	O
is	O
reduced	O
to	O
the	O
initial	O
condition	O
which	O
forms	O
the	O
coordinates	O
encoding	O
the	O
diffeomorphism	O
.	O
</s>
<s>
From	O
the	O
initial	O
condition	O
then	O
geodesic	O
positioning	O
with	O
respect	O
to	O
the	O
Riemannian	O
metric	O
of	O
Computational	B-Algorithm
anatomy	I-Algorithm
solves	O
for	O
the	O
flow	O
of	O
the	O
Euler-Lagrange	O
equation	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>
The	O
MAP	B-General_Concept
estimator	I-General_Concept
of	O
segmentation	O
is	O
the	O
maximizer	O
given	O
,	O
which	O
involves	O
the	O
mixture	O
over	O
all	O
atlases	O
.	O
</s>
<s>
In	O
the	O
Bayesian	O
random	B-Algorithm
orbit	I-Algorithm
model	I-Algorithm
of	I-Algorithm
computational	I-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
algorithm	I-Algorithm
.	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>
