<s>
Diffeomorphometry	B-Algorithm
is	O
the	O
metric	O
study	O
of	O
imagery	O
,	O
shape	O
and	O
form	O
in	O
the	O
discipline	O
of	O
computational	B-Algorithm
anatomy	I-Algorithm
(	O
CA	O
)	O
in	O
medical	B-Application
imaging	I-Application
.	O
</s>
<s>
The	O
study	O
of	O
images	O
in	O
computational	B-Algorithm
anatomy	I-Algorithm
rely	O
on	O
high-dimensional	O
diffeomorphism	O
groups	O
which	O
generate	O
orbits	O
of	O
the	O
form	O
,	O
in	O
which	O
images	O
can	O
be	O
dense	O
scalar	O
magnetic	O
resonance	O
or	O
computed	O
axial	O
tomography	O
images	O
.	O
</s>
<s>
For	O
deformable	O
shapes	O
these	O
are	O
the	O
collection	O
of	O
manifolds	B-Architecture
,	O
points	O
,	O
curves	O
and	O
surfaces	O
.	O
</s>
<s>
The	O
diffeomorphisms	O
move	O
the	O
images	O
and	O
shapes	O
through	O
the	O
orbit	O
according	O
to	O
which	O
are	O
defined	O
as	O
the	O
group	B-Algorithm
actions	I-Algorithm
of	I-Algorithm
computational	I-Algorithm
anatomy	I-Algorithm
.	O
</s>
<s>
The	O
study	O
of	O
metrics	O
on	O
groups	O
of	O
diffeomorphisms	O
and	O
the	O
study	O
of	O
metrics	O
between	O
manifolds	B-Architecture
and	O
surfaces	O
has	O
been	O
an	O
area	O
of	O
significant	O
investigation	O
.	O
</s>
<s>
In	O
Computational	B-Algorithm
anatomy	I-Algorithm
,	O
the	O
diffeomorphometry	B-Algorithm
metric	O
measures	O
how	O
close	O
and	O
far	O
two	O
shapes	O
or	O
images	O
are	O
from	O
each	O
other	O
.	O
</s>
<s>
The	O
group	O
is	O
thusly	O
made	O
into	O
a	O
smooth	O
Riemannian	B-Architecture
manifold	I-Architecture
with	O
Riemannian	O
metric	O
associated	O
to	O
the	O
tangent	O
spaces	O
at	O
all	O
.	O
</s>
<s>
The	O
Riemannian	O
metric	O
satisfies	O
at	O
every	O
point	O
of	O
the	O
manifold	B-Architecture
there	O
is	O
an	O
inner	O
product	O
inducing	O
the	O
norm	O
on	O
the	O
tangent	O
space	O
that	O
varies	O
smoothly	O
across	O
.	O
</s>
<s>
In	O
the	O
Riemannian	O
orbit	O
model	O
of	O
Computational	B-Algorithm
anatomy	I-Algorithm
,	O
diffeomorphisms	O
acting	O
on	O
the	O
forms	O
do	O
n't	O
act	O
linearly	O
.	O
</s>
<s>
Measuring	O
the	O
lengths	O
of	O
the	O
geodesic	O
flow	O
between	O
coordinates	O
systems	O
in	O
the	O
orbit	O
of	O
shapes	O
is	O
called	O
diffeomorphometry	B-Algorithm
.	O
</s>
<s>
Shapes	O
in	O
Computational	B-Algorithm
Anatomy	I-Algorithm
(	O
CA	O
)	O
are	O
studied	O
via	O
the	O
use	O
of	O
diffeomorphic	O
mapping	O
for	O
establishing	O
correspondences	O
between	O
anatomical	O
coordinate	O
systems	O
.	O
</s>
<s>
In	O
this	O
setting	O
,	O
3-dimensional	O
medical	B-Application
images	I-Application
are	O
modelled	O
as	O
diffemorphic	O
transformations	O
of	O
some	O
exemplar	O
,	O
termed	O
the	O
template	O
,	O
resulting	O
in	O
the	O
observed	O
images	O
to	O
be	O
elements	O
of	O
the	O
random	O
orbit	O
model	O
of	O
CA	O
.	O
</s>
<s>
For	O
images	O
these	O
are	O
defined	O
as	O
,	O
with	O
for	O
charts	O
representing	O
sub-manifolds	O
denoted	O
as	O
.	O
</s>
<s>
The	O
orbit	O
of	O
shapes	O
and	O
forms	O
in	O
Computational	B-Algorithm
Anatomy	I-Algorithm
are	O
generated	O
by	O
the	O
group	O
action	O
,	O
.	O
</s>
<s>
This	O
distance	O
provides	O
a	O
right-invariant	O
metric	O
of	O
diffeomorphometry	B-Algorithm
,	O
invariant	O
to	O
reparameterization	O
of	O
space	O
since	O
for	O
all	O
,	O
</s>
<s>
Software	B-Application
suites	I-Application
containing	O
a	O
variety	O
of	O
diffeomorphic	O
mapping	O
algorithms	O
include	O
the	O
following	O
:	O
</s>
