<s>
Computational	B-Algorithm
anatomy	I-Algorithm
is	O
an	O
interdisciplinary	O
field	O
of	O
biology	O
focused	O
on	O
quantitative	O
investigation	O
and	O
modelling	O
of	O
anatomical	O
shapes	O
variability	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
machine	O
learning	O
,	O
computational	B-Algorithm
mechanics	I-Algorithm
,	O
computational	O
science	O
,	O
biological	O
imaging	O
,	O
neuroscience	O
,	O
physics	O
,	O
probability	O
,	O
and	O
statistics	O
;	O
it	O
also	O
has	O
strong	O
connections	O
with	O
fluid	O
mechanics	O
and	O
geometric	O
mechanics	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>
In	O
computational	B-Algorithm
anatomy	I-Algorithm
,	O
the	O
diffeomorphism	O
group	O
is	O
used	O
to	O
study	O
different	O
coordinate	O
systems	O
via	O
coordinate	O
transformations	O
as	O
generated	O
via	O
the	O
Lagrangian	O
and	O
Eulerian	O
velocities	O
of	O
flow	O
in	O
.	O
</s>
<s>
The	O
flows	O
between	O
coordinates	O
in	O
computational	B-Algorithm
anatomy	I-Algorithm
are	O
constrained	O
to	O
be	O
geodesic	O
flows	O
satisfying	O
the	O
principle	O
of	O
least	O
action	O
for	O
the	O
Kinetic	O
energy	O
of	O
the	O
flow	O
.	O
</s>
<s>
The	O
kinetic	O
energy	O
is	O
defined	O
through	O
a	O
Sobolev	O
smoothness	O
norm	O
with	O
strictly	O
more	O
than	O
two	O
generalized	O
,	O
square-integrable	B-Algorithm
derivatives	O
for	O
each	O
component	O
of	O
the	O
flow	O
velocity	O
,	O
which	O
guarantees	O
that	O
the	O
flows	O
in	O
are	O
diffeomorphisms	O
.	O
</s>
<s>
It	O
also	O
implies	O
that	O
the	O
diffeomorphic	O
shape	O
momentum	B-Algorithm
taken	O
pointwise	O
satisfying	O
the	O
Euler-Lagrange	O
equation	O
for	O
geodesics	O
is	O
determined	O
by	O
its	O
neighbors	O
through	O
spatial	O
derivatives	O
on	O
the	O
velocity	O
field	O
.	O
</s>
<s>
This	O
separates	O
the	O
discipline	O
from	O
the	O
case	O
of	O
incompressible	O
fluids	O
for	O
which	O
momentum	B-Algorithm
is	O
a	O
pointwise	O
function	O
of	O
velocity	O
.	O
</s>
<s>
Computational	B-Algorithm
anatomy	I-Algorithm
intersects	O
the	O
study	O
of	O
Riemannian	B-Architecture
manifolds	I-Architecture
and	O
nonlinear	O
global	O
analysis	O
,	O
where	O
groups	O
of	O
diffeomorphisms	O
are	O
the	O
central	O
focus	O
.	O
</s>
<s>
Emerging	O
high-dimensional	O
theories	O
of	O
shape	O
are	O
central	O
to	O
many	O
studies	O
in	O
computational	B-Algorithm
anatomy	I-Algorithm
,	O
as	O
are	O
questions	O
emerging	O
from	O
the	O
fledgling	O
field	O
of	O
shape	O
statistics	O
.	O
</s>
<s>
The	O
metric	O
structures	O
in	O
computational	B-Algorithm
anatomy	I-Algorithm
are	O
related	O
in	O
spirit	O
to	O
morphometrics	O
,	O
with	O
the	O
distinction	O
that	O
Computational	B-Algorithm
anatomy	I-Algorithm
focuses	O
on	O
an	O
infinite-dimensional	O
space	O
of	O
coordinate	O
systems	O
transformed	O
by	O
a	O
diffeomorphism	O
,	O
hence	O
the	O
central	O
use	O
of	O
the	O
terminology	O
diffeomorphometry	B-Algorithm
,	O
the	O
metric	O
space	O
study	O
of	O
coordinate	O
systems	O
via	O
diffeomorphisms	O
.	O
</s>
<s>
At	O
computational	B-Algorithm
anatomy	I-Algorithm
's	O
heart	O
is	O
the	O
comparison	O
of	O
shape	O
by	O
recognizing	O
in	O
one	O
shape	O
the	O
other	O
.	O
</s>
<s>
Albrecht	O
Durer	O
's	O
Four	O
Books	O
on	O
Human	O
Proportion	O
were	O
arguably	O
the	O
earliest	O
works	O
on	O
computational	B-Algorithm
anatomy	I-Algorithm
.	O
</s>
<s>
The	O
efforts	O
of	O
Noam	O
Chomsky	O
in	O
his	O
pioneering	O
of	O
computational	O
linguistics	O
inspired	O
the	O
original	O
formulation	O
of	O
computational	B-Algorithm
anatomy	I-Algorithm
as	O
a	O
generative	O
model	O
of	O
shape	O
and	O
form	O
from	O
exemplars	O
acted	O
upon	O
via	O
transformations	O
.	O
</s>
<s>
Due	O
to	O
the	O
availability	O
of	O
dense	O
3D	O
measurements	O
via	O
technologies	O
such	O
as	O
magnetic	O
resonance	O
imaging	O
(	O
MRI	O
)	O
,	O
computational	B-Algorithm
anatomy	I-Algorithm
has	O
emerged	O
as	O
a	O
subfield	O
of	O
medical	B-Application
imaging	I-Application
and	O
bioengineering	O
for	O
extracting	O
anatomical	O
coordinate	O
systems	O
at	O
the	O
morphome	O
scale	O
in	O
3D	O
.	O
</s>
<s>
The	O
spirit	O
of	O
this	O
discipline	O
shares	O
strong	O
overlap	O
with	O
areas	O
such	O
as	O
computer	B-Application
vision	I-Application
and	O
kinematics	O
of	O
rigid	O
bodies	O
,	O
where	O
objects	O
are	O
studied	O
by	O
analysing	O
the	O
groups	O
responsible	O
for	O
the	O
movement	O
in	O
question	O
.	O
</s>
<s>
Computational	B-Algorithm
anatomy	I-Algorithm
departs	O
from	O
computer	B-Application
vision	I-Application
with	O
its	O
focus	O
on	O
rigid	O
motions	O
,	O
as	O
the	O
infinite-dimensional	O
diffeomorphism	O
group	O
is	O
central	O
to	O
the	O
analysis	O
of	O
Biological	O
shapes	O
.	O
</s>
<s>
It	O
is	O
a	O
branch	O
of	O
the	O
image	O
analysis	O
and	O
pattern	B-General_Concept
theory	I-General_Concept
school	O
at	O
Brown	O
University	O
pioneered	O
by	O
Ulf	O
Grenander	O
.	O
</s>
<s>
In	O
Grenander	O
's	O
general	O
metric	O
pattern	B-General_Concept
theory	I-General_Concept
,	O
making	O
spaces	O
of	O
patterns	O
into	O
a	O
metric	O
space	O
is	O
one	O
of	O
the	O
fundamental	O
operations	O
since	O
being	O
able	O
to	O
cluster	O
and	O
recognize	O
anatomical	O
configurations	O
often	O
requires	O
a	O
metric	O
of	O
close	O
and	O
far	O
between	O
shapes	O
.	O
</s>
<s>
The	O
diffeomorphometry	B-Algorithm
metric	O
of	O
computational	B-Algorithm
anatomy	I-Algorithm
measures	O
how	O
far	O
two	O
diffeomorphic	O
changes	O
of	O
coordinates	O
are	O
from	O
each	O
other	O
,	O
which	O
in	O
turn	O
induces	O
a	O
metric	O
on	O
the	O
shapes	O
and	O
images	O
indexed	O
to	O
them	O
.	O
</s>
<s>
The	O
models	O
of	O
metric	O
pattern	B-General_Concept
theory	I-General_Concept
,	O
in	O
particular	O
group	O
action	O
on	O
the	O
orbit	O
of	O
shapes	O
and	O
forms	O
is	O
a	O
central	O
tool	O
to	O
the	O
formal	O
definitions	O
in	O
computational	B-Algorithm
anatomy	I-Algorithm
.	O
</s>
<s>
Computational	B-Algorithm
anatomy	I-Algorithm
is	O
the	O
study	O
of	O
shape	O
and	O
form	O
at	O
the	O
morphome	O
or	O
gross	O
anatomy	O
millimeter	O
,	O
or	O
morphology	O
scale	O
,	O
focusing	O
on	O
the	O
study	O
of	O
sub-manifolds	O
of	O
points	O
,	O
curves	O
surfaces	O
and	O
subvolumes	O
of	O
human	O
anatomy	O
.	O
</s>
<s>
Jean	O
Talairach	O
's	O
publication	O
of	O
Talairach	O
coordinates	O
is	O
an	O
important	O
milestone	O
at	O
the	O
morphome	O
scale	O
demonstrating	O
the	O
fundamental	O
basis	O
of	O
local	O
coordinate	O
systems	O
in	O
studying	O
neuroanatomy	O
and	O
therefore	O
the	O
clear	O
link	O
to	O
charts	B-Language
of	I-Language
differential	I-Language
geometry	I-Language
.	O
</s>
<s>
Concurrently	O
,	O
virtual	O
mapping	O
in	O
computational	B-Algorithm
anatomy	I-Algorithm
across	O
high	O
resolution	O
dense	O
image	O
coordinates	O
was	O
already	O
happening	O
in	O
Ruzena	O
Bajcy	O
's	O
and	O
Fred	O
Bookstein	O
's	O
earliest	O
developments	O
based	O
on	O
computed	O
axial	O
tomography	O
and	O
magnetic	O
resonance	O
imagery	O
.	O
</s>
<s>
The	O
earliest	O
introduction	O
of	O
the	O
use	O
of	O
flows	O
of	O
diffeomorphisms	O
for	O
transformation	O
of	O
coordinate	O
systems	O
in	O
image	O
analysis	O
and	O
medical	B-Application
imaging	I-Application
was	O
by	O
Christensen	O
,	O
Joshi	O
,	O
Miller	O
,	O
and	O
Rabbitt	O
.	O
</s>
<s>
The	O
first	O
formalization	O
of	O
computational	B-Algorithm
anatomy	I-Algorithm
as	O
an	O
orbit	O
of	O
exemplar	O
templates	O
under	O
diffeomorphism	O
group	O
action	O
was	O
in	O
the	O
original	O
lecture	O
given	O
by	O
Grenander	O
and	O
Miller	O
with	O
that	O
title	O
in	O
May	O
1997	O
at	O
the	O
50th	O
Anniversary	O
of	O
the	O
Division	O
of	O
Applied	O
Mathematics	O
at	O
Brown	O
University	O
,	O
and	O
subsequent	O
publication	O
.	O
</s>
<s>
This	O
was	O
the	O
basis	O
for	O
the	O
strong	O
departure	O
from	O
much	O
of	O
the	O
previous	O
work	O
on	O
advanced	O
methods	O
for	O
spatial	B-Algorithm
normalization	I-Algorithm
and	O
image	B-Algorithm
registration	I-Algorithm
which	O
were	O
historically	O
built	O
on	O
notions	O
of	O
addition	O
and	O
basis	O
expansion	O
.	O
</s>
<s>
The	O
structure	O
preserving	O
transformations	O
central	O
to	O
the	O
modern	O
field	O
of	O
Computational	B-Algorithm
Anatomy	I-Algorithm
,	O
homeomorphisms	O
and	O
diffeomorphisms	O
carry	O
smooth	O
submanifolds	O
smoothly	O
.	O
</s>
<s>
The	O
original	O
model	O
of	O
computational	B-Algorithm
anatomy	I-Algorithm
was	O
as	O
the	O
triple	O
,	O
the	O
group	O
,	O
the	O
orbit	O
of	O
shapes	O
and	O
forms	O
,	O
and	O
the	O
probability	O
laws	O
which	O
encode	O
the	O
variations	O
of	O
the	O
objects	O
in	O
the	O
orbit	O
.	O
</s>
<s>
The	O
Lagrangian	O
and	O
Hamiltonian	O
formulations	O
of	O
the	O
equations	O
of	O
motion	O
of	O
computational	B-Algorithm
anatomy	I-Algorithm
took	O
off	O
post	O
1997	O
with	O
several	O
pivotal	O
meetings	O
including	O
the	O
1997	O
Luminy	O
meeting	O
organized	O
by	O
the	O
Azencott	O
school	O
at	O
Ecole-Normale	O
Cachan	O
on	O
the	O
"	O
Mathematics	O
of	O
Shape	O
Recognition	O
"	O
and	O
the	O
1998	O
Trimestre	O
at	O
Institute	O
Henri	O
Poincaré	O
organized	O
by	O
David	O
Mumford	O
"	O
Questions	O
Mathématiques	O
en	O
Traitement	O
du	O
Signal	O
et	O
de	O
l'Image	O
"	O
which	O
catalyzed	O
the	O
Hopkins-Brown-ENS	O
Cachan	O
groups	O
and	O
subsequent	O
developments	O
and	O
connections	O
of	O
computational	B-Algorithm
anatomy	I-Algorithm
to	O
developments	O
in	O
global	O
analysis	O
.	O
</s>
<s>
The	O
developments	O
in	O
computational	B-Algorithm
anatomy	I-Algorithm
included	O
the	O
establishment	O
of	O
the	O
Sobolev	O
smoothness	O
conditions	O
on	O
the	O
diffeomorphometry	B-Algorithm
metric	O
to	O
insure	O
existence	O
of	O
solutions	O
of	O
variational	B-Algorithm
problems	I-Algorithm
in	O
the	O
space	O
of	O
diffeomorphisms	O
,	O
the	O
derivation	O
of	O
the	O
Euler-Lagrange	O
equations	O
characterizing	O
geodesics	O
through	O
the	O
group	O
and	O
associated	O
conservation	O
laws	O
,	O
the	O
demonstration	O
of	O
the	O
metric	O
properties	O
of	O
the	O
right	O
invariant	O
metric	O
,	O
the	O
demonstration	O
that	O
the	O
Euler-Lagrange	O
equations	O
have	O
a	O
well-posed	O
initial	O
value	O
problem	O
with	O
unique	O
solutions	O
for	O
all	O
time	O
,	O
and	O
with	O
the	O
first	O
results	O
on	O
sectional	O
curvatures	O
for	O
the	O
diffeomorphometry	B-Algorithm
metric	O
in	O
landmarked	O
spaces	O
.	O
</s>
<s>
Following	O
the	O
Los	O
Alamos	O
meeting	O
in	O
2002	O
,	O
Joshi	O
's	O
original	O
large	O
deformation	O
singular	O
Landmark	O
solutions	O
in	O
computational	B-Algorithm
anatomy	I-Algorithm
were	O
connected	O
to	O
peaked	O
solitons	O
or	O
peakons	O
as	O
solutions	O
for	O
the	O
Camassa-Holm	O
equation	O
.	O
</s>
<s>
Subsequently	O
,	O
connections	O
were	O
made	O
between	O
computational	B-Algorithm
anatomy	I-Algorithm
's	O
Euler-Lagrange	O
equations	O
for	O
momentum	B-Algorithm
densities	O
for	O
the	O
right-invariant	O
metric	O
satisfying	O
Sobolev	O
smoothness	O
to	O
Vladimir	O
Arnold	O
's	O
characterization	O
of	O
the	O
Euler	O
equation	O
for	O
incompressible	O
flows	O
as	O
describing	O
geodesics	O
in	O
the	O
group	O
of	O
volume	O
preserving	O
diffeomorphisms	O
.	O
</s>
<s>
The	O
first	O
algorithms	O
,	O
generally	O
termed	O
LDDMM	O
for	O
large	O
deformation	O
diffeomorphic	O
mapping	O
for	O
computing	O
connections	O
between	O
landmarks	O
in	O
volumes	O
and	O
spherical	O
manifolds	B-Architecture
,	O
curves	O
,	O
currents	O
and	O
surfaces	O
,	O
volumes	O
,	O
tensors	O
,	O
varifolds	O
,	O
and	O
time-series	O
have	O
followed	O
.	O
</s>
<s>
These	O
contributions	O
of	O
computational	B-Algorithm
anatomy	I-Algorithm
to	O
the	O
global	O
analysis	O
associated	O
to	O
the	O
infinite	O
dimensional	O
manifolds	B-Architecture
of	O
subgroups	O
of	O
the	O
diffeomorphism	O
group	O
is	O
far	O
from	O
trivial	O
.	O
</s>
<s>
The	O
original	O
idea	O
of	O
doing	O
differential	B-Language
geometry	I-Language
,	O
curvature	O
and	O
geodesics	O
on	O
infinite	O
dimensional	O
manifolds	B-Architecture
goes	O
back	O
to	O
Bernhard	O
Riemann	O
's	O
Habilitation	O
(	O
Ueber	O
die	O
Hypothesen	O
,	O
welche	O
der	O
Geometrie	O
zu	O
Grunde	O
liegen	O
)	O
;	O
the	O
key	O
modern	O
book	O
laying	O
the	O
foundations	O
of	O
such	O
ideas	O
in	O
global	O
analysis	O
are	O
from	O
Michor	O
.	O
</s>
<s>
The	O
applications	O
within	O
medical	B-Application
imaging	I-Application
of	O
computational	B-Algorithm
anatomy	I-Algorithm
continued	O
to	O
flourish	O
after	O
two	O
organized	O
meetings	O
at	O
the	O
Institute	O
for	O
Pure	O
and	O
Applied	O
Mathematics	O
conferences	O
at	O
University	O
of	O
California	O
,	O
Los	O
Angeles	O
.	O
</s>
<s>
Computational	B-Algorithm
anatomy	I-Algorithm
has	O
been	O
useful	O
in	O
creating	O
accurate	O
models	O
of	O
the	O
atrophy	O
of	O
the	O
human	O
brain	O
at	O
the	O
morphome	O
scale	O
,	O
as	O
well	O
as	O
Cardiac	O
templates	O
,	O
as	O
well	O
as	O
in	O
modeling	O
biological	O
systems	O
.	O
</s>
<s>
Since	O
the	O
late	O
1990s	O
,	O
computational	B-Algorithm
anatomy	I-Algorithm
has	O
become	O
an	O
important	O
part	O
of	O
developing	O
emerging	O
technologies	O
for	O
the	O
field	O
of	O
medical	B-Application
imaging	I-Application
.	O
</s>
<s>
Voxel-based	B-Algorithm
morphometry	I-Algorithm
is	O
an	O
important	O
technology	O
built	O
on	O
many	O
of	O
these	O
principles	O
.	O
</s>
<s>
Deformable	O
template	O
models	O
have	O
been	O
central	O
to	O
Grenander	O
's	O
metric	O
pattern	B-General_Concept
theory	I-General_Concept
,	O
accounting	O
for	O
typicality	O
via	O
templates	O
,	O
and	O
accounting	O
for	O
variability	O
via	O
transformation	O
of	O
the	O
template	O
.	O
</s>
<s>
An	O
orbit	O
under	O
group	O
action	O
as	O
the	O
representation	O
of	O
the	O
deformable	O
template	O
is	O
a	O
classic	O
formulation	O
from	O
differential	B-Language
geometry	I-Language
.	O
</s>
<s>
The	O
orbit	O
model	O
of	O
computational	B-Algorithm
anatomy	I-Algorithm
is	O
an	O
abstract	O
algebra	O
-	O
to	O
be	O
compared	O
to	O
linear	B-Language
algebra	I-Language
-	O
since	O
the	O
groups	O
act	O
nonlinearly	O
on	O
the	O
shapes	O
.	O
</s>
<s>
This	O
is	O
a	O
generalization	O
of	O
the	O
classical	O
models	O
of	O
linear	B-Language
algebra	I-Language
,	O
in	O
which	O
the	O
set	O
of	O
finite	O
dimensional	O
vectors	O
are	O
replaced	O
by	O
the	O
finite-dimensional	O
anatomical	O
submanifolds	O
(	O
points	O
,	O
curves	O
,	O
surfaces	O
and	O
volumes	O
)	O
and	O
images	O
of	O
them	O
,	O
and	O
the	O
matrices	O
of	O
linear	B-Language
algebra	I-Language
are	O
replaced	O
by	O
coordinate	O
transformations	O
based	O
on	O
linear	O
and	O
affine	O
groups	O
and	O
the	O
more	O
general	O
high-dimensional	O
diffeomorphism	O
groups	O
.	O
</s>
<s>
The	O
central	O
objects	O
are	O
shapes	O
or	O
forms	O
in	O
computational	B-Algorithm
anatomy	I-Algorithm
,	O
one	O
set	O
of	O
examples	O
being	O
the	O
0	O
,	O
1	O
,	O
2	O
,	O
3-dimensional	O
submanifolds	O
of	O
,	O
a	O
second	O
set	O
of	O
examples	O
being	O
images	O
generated	O
via	O
medical	B-Application
imaging	I-Application
such	O
as	O
via	O
magnetic	O
resonance	O
imaging	O
(	O
MRI	O
)	O
and	O
functional	B-Algorithm
magnetic	I-Algorithm
resonance	I-Algorithm
imaging	I-Algorithm
.	O
</s>
<s>
The	O
0-dimensional	O
manifolds	B-Architecture
are	O
landmarks	O
or	O
fiducial	B-Architecture
points	O
;	O
1-dimensional	O
manifolds	B-Architecture
are	O
curves	O
such	O
as	O
sulcul	O
and	O
gyral	O
curves	O
in	O
the	O
brain	O
;	O
2-dimensional	O
manifolds	B-Architecture
correspond	O
to	O
boundaries	O
of	O
substructures	O
in	O
anatomy	O
such	O
as	O
the	O
subcortical	O
structures	O
of	O
the	O
midbrain	O
or	O
the	O
gyral	O
surface	O
of	O
the	O
neocortex	O
;	O
subvolumes	O
correspond	O
to	O
subregions	O
of	O
the	O
human	O
body	O
,	O
the	O
heart	O
,	O
the	O
thalamus	O
,	O
the	O
kidney	O
.	O
</s>
<s>
The	O
landmarks	O
are	O
a	O
collections	O
of	O
points	O
with	O
no	O
other	O
structure	O
,	O
delineating	O
important	O
fiducials	B-Architecture
within	O
human	O
shape	O
and	O
form	O
(	O
see	O
associated	O
landmarked	O
image	O
)	O
.	O
</s>
<s>
The	O
sub-manifold	O
shapes	O
such	O
as	O
surfaces	O
are	O
collections	O
of	O
points	O
modeled	O
as	O
parametrized	O
by	O
a	O
local	O
chart	O
or	O
immersion	O
,	O
(	O
see	O
Figure	O
showing	O
shapes	O
as	O
mesh	O
surfaces	O
)	O
.	O
</s>
<s>
Groups	O
and	O
group	O
actions	O
are	O
familiar	O
to	O
the	O
Engineering	O
community	O
with	O
the	O
universal	O
popularization	O
and	O
standardization	O
of	O
linear	B-Language
algebra	I-Language
as	O
a	O
basic	O
model	O
for	O
analyzing	O
signals	O
and	O
systems	O
in	O
mechanical	O
engineering	O
,	O
electrical	O
engineering	O
and	O
applied	O
mathematics	O
.	O
</s>
<s>
In	O
linear	B-Language
algebra	I-Language
the	O
matrix	B-Algorithm
groups	I-Algorithm
(	O
matrices	O
with	O
inverses	O
)	O
are	O
the	O
central	O
structure	O
,	O
with	O
group	O
action	O
defined	O
by	O
the	O
usual	O
definition	O
of	O
as	O
an	O
matrix	O
,	O
acting	O
on	O
as	O
vectors	O
;	O
the	O
orbit	O
in	O
linear-algebra	O
is	O
the	O
set	O
of	O
-vectors	O
given	O
by	O
,	O
which	O
is	O
a	O
group	O
action	O
of	O
the	O
matrices	O
through	O
the	O
orbit	O
of	O
.	O
</s>
<s>
The	O
central	O
group	O
in	O
computational	B-Algorithm
anatomy	I-Algorithm
defined	O
on	O
volumes	O
in	O
are	O
the	O
diffeomorphisms	O
which	O
are	O
mappings	O
with	O
3-components	O
,	O
law	O
of	O
composition	O
of	O
functions	O
,	O
with	O
inverse	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
matrix	B-Algorithm
groups	I-Algorithm
are	O
low-dimensional	O
mappings	O
,	O
which	O
are	O
diffeomorphisms	O
that	O
provide	O
one-to-one	O
correspondences	O
between	O
coordinate	O
systems	O
,	O
with	O
a	O
smooth	O
inverse	O
.	O
</s>
<s>
The	O
matrix	B-Algorithm
group	I-Algorithm
of	O
rotations	O
and	O
scales	O
can	O
be	O
generated	O
via	O
a	O
closed	O
form	O
finite-dimensional	O
matrices	O
which	O
are	O
solution	O
of	O
simple	O
ordinary	O
differential	O
equations	O
with	O
solutions	O
given	O
by	O
the	O
matrix	O
exponential	O
.	O
</s>
<s>
For	O
the	O
study	O
of	O
deformable	O
shape	O
in	O
computational	B-Algorithm
anatomy	I-Algorithm
,	O
a	O
more	O
general	O
diffeomorphism	O
group	O
has	O
been	O
the	O
group	O
of	O
choice	O
,	O
which	O
is	O
the	O
infinite	O
dimensional	O
analogue	O
.	O
</s>
<s>
The	O
high-dimensional	O
differeomorphism	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
as	O
first	O
introduced	O
in.	O
,	O
satisfying	O
the	O
ordinary	O
differential	O
equation	O
:	O
</s>
<s>
The	O
vector	O
fields	O
are	O
functions	O
in	O
a	O
function	O
space	O
,	O
modelled	O
as	O
a	O
smooth	O
Hilbert	O
space	O
of	O
high-dimension	O
,	O
with	O
the	O
Jacobian	O
of	O
the	O
flow	O
a	O
high-dimensional	O
field	O
in	O
a	O
function	O
space	O
as	O
well	O
,	O
rather	O
than	O
a	O
low-dimensional	O
matrix	O
as	O
in	O
the	O
matrix	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
Flows	O
were	O
first	O
introduced	O
for	O
large	O
deformations	O
in	O
image	B-Algorithm
matching	I-Algorithm
;	O
is	O
the	O
instantaneous	O
velocity	O
of	O
particle	O
at	O
time	O
.	O
</s>
<s>
For	O
diffeomorphisms	O
on	O
,	O
vector	O
fields	O
are	O
modelled	O
as	O
elements	O
of	O
the	O
Hilbert	O
space	O
using	O
the	O
Sobolev	O
embedding	O
theorems	O
so	O
that	O
each	O
element	O
has	O
strictly	O
greater	O
than	O
2	O
generalized	O
square-integrable	B-Algorithm
spatial	O
derivatives	O
(	O
thus	O
is	O
sufficient	O
)	O
,	O
yielding	O
1-time	O
continuously	O
differentiable	O
functions	O
.	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>
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
the	O
metric	O
length	O
is	O
the	O
shortest	O
length	O
of	O
the	O
flow	O
which	O
carries	O
one	O
coordinate	O
system	O
into	O
the	O
other	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>
this	O
is	O
the	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>
The	O
Euler-Lagrange	O
equation	O
in	O
computational	B-Algorithm
anatomy	I-Algorithm
describes	O
the	O
geodesic	O
shortest	O
path	O
flows	O
between	O
coordinate	O
systems	O
of	O
the	O
diffeomorphism	O
metric	O
.	O
</s>
<s>
In	O
computational	B-Algorithm
anatomy	I-Algorithm
the	O
generalized	O
coordinates	O
are	O
the	O
flow	O
of	O
the	O
diffeomorphism	O
and	O
its	O
Lagrangian	O
velocity	O
,	O
the	O
two	O
related	O
via	O
the	O
Eulerian	O
velocity	O
.	O
</s>
<s>
In	O
computational	B-Algorithm
anatomy	I-Algorithm
,	O
was	O
first	O
called	O
the	O
Eulerian	O
or	O
diffeomorphic	O
shape	O
momentum	B-Algorithm
since	O
when	O
integrated	O
against	O
Eulerian	O
velocity	O
gives	O
energy	O
density	O
,	O
and	O
since	O
there	O
is	O
a	O
conservation	O
of	O
diffeomorphic	O
shape	O
momentum	B-Algorithm
which	O
holds	O
.	O
</s>
<s>
Defining	O
the	O
adjoint	O
then	O
the	O
first	O
order	O
variation	O
gives	O
the	O
Eulerian	O
shape	O
momentum	B-Algorithm
satisfying	O
the	O
generalized	O
equation	O
:	O
</s>
<s>
Computational	B-Algorithm
anatomy	I-Algorithm
is	O
the	O
study	O
of	O
the	O
motions	O
of	O
submanifolds	O
,	O
points	O
,	O
curves	O
,	O
surfaces	O
and	O
volumes	O
.	O
</s>
<s>
Momentum	B-Algorithm
associated	O
to	O
points	O
,	O
curves	O
and	O
surfaces	O
are	O
all	O
singular	O
,	O
implying	O
the	O
momentum	B-Algorithm
is	O
concentrated	O
on	O
subsets	O
of	O
which	O
are	O
dimension	O
in	O
Lebesgue	O
measure	O
.	O
</s>
<s>
In	O
such	O
cases	O
,	O
the	O
energy	O
is	O
still	O
well	O
defined	O
since	O
although	O
is	O
a	O
generalized	O
function	O
,	O
the	O
vector	O
fields	O
are	O
smooth	O
and	O
the	O
Eulerian	O
momentum	B-Algorithm
is	O
understood	O
via	O
its	O
action	O
on	O
smooth	O
functions	O
.	O
</s>
<s>
For	O
the	O
momentum	B-Algorithm
density	O
case	O
,	O
then	O
Euler	O
–	O
Lagrange	O
equation	O
has	O
a	O
classical	O
solution:The	O
Euler-Lagrange	O
equation	O
on	O
diffeomorphisms	O
,	O
classically	O
defined	O
for	O
momentum	B-Algorithm
densities	O
first	O
appeared	O
in	O
for	O
medical	B-Application
image	I-Application
analysis	O
.	O
</s>
<s>
In	O
medical	B-Application
imaging	I-Application
and	O
computational	B-Algorithm
anatomy	I-Algorithm
,	O
positioning	O
and	O
coordinatizing	O
shapes	O
are	O
fundamental	O
operations	O
;	O
the	O
system	O
for	O
positioning	O
anatomical	O
coordinates	O
and	O
shapes	O
built	O
on	O
the	O
metric	O
and	O
the	O
Euler-Lagrange	O
equation	O
a	O
geodesic	O
positioning	O
system	O
as	O
first	O
explicated	O
in	O
Miller	O
Trouve	O
and	O
Younes	O
.	O
</s>
<s>
reparameterizes	O
the	O
momentum	B-Algorithm
distribution	O
in	O
terms	O
of	O
the	O
conjugate	O
momentum	B-Algorithm
or	O
canonical	O
momentum	B-Algorithm
,	O
introduced	O
as	O
a	O
Lagrange	O
multiplier	O
constraining	O
the	O
Lagrangian	O
velocity	O
.accordingly	O
:	O
</s>
<s>
The	O
Lagrange	O
multiplier	O
in	O
its	O
action	O
as	O
a	O
linear	O
form	O
has	O
its	O
own	O
inner	O
product	O
of	O
the	O
canonical	O
momentum	B-Algorithm
acting	O
on	O
the	O
velocity	O
of	O
the	O
flow	O
which	O
is	O
dependent	O
on	O
the	O
shape	O
,	O
e.g.	O
</s>
<s>
In	O
all	O
cases	O
the	O
Greens	O
kernels	O
carry	O
weights	O
which	O
are	O
the	O
canonical	O
momentum	B-Algorithm
evolving	O
according	O
to	O
an	O
ordinary	O
differential	O
equation	O
which	O
corresponds	O
to	O
EL	O
but	O
is	O
the	O
geodesic	O
reparameterization	O
in	O
canonical	O
momentum	B-Algorithm
.	O
</s>
<s>
Whereas	O
the	O
vector	O
fields	O
are	O
extended	O
across	O
the	O
entire	O
background	O
space	O
of	O
,	O
the	O
geodesic	O
flows	O
associated	O
to	O
the	O
submanifolds	O
has	O
Eulerian	O
shape	O
momentum	B-Algorithm
which	O
evolves	O
as	O
a	O
generalized	O
function	O
concentrated	O
to	O
the	O
submanifolds	O
.	O
</s>
<s>
For	O
landmarks	O
the	O
geodesics	O
have	O
Eulerian	O
shape	O
momentum	B-Algorithm
which	O
are	O
a	O
superposition	O
of	O
delta	O
distributions	O
travelling	O
with	O
the	O
finite	O
numbers	O
of	O
particles	O
;	O
the	O
diffeomorphic	O
flow	O
of	O
coordinates	O
have	O
velocities	O
in	O
the	O
range	O
of	O
weighted	O
Green	O
's	O
Kernels	O
.	O
</s>
<s>
For	O
surfaces	O
,	O
the	O
momentum	B-Algorithm
is	O
a	O
surface	O
integral	O
of	O
delta	O
distributions	O
travelling	O
with	O
the	O
surface	O
.	O
</s>
<s>
The	O
stationarity	O
of	O
the	O
Hamiltonian	O
demonstrates	O
the	O
interpretation	O
of	O
the	O
Lagrange	O
multiplier	O
as	O
momentum	B-Algorithm
;	O
integrated	O
against	O
velocity	O
gives	O
energy	O
density	O
.	O
</s>
<s>
The	O
canonical	O
momentum	B-Algorithm
has	O
many	O
names	O
.	O
</s>
<s>
In	O
optimal	O
control	O
,	O
the	O
flows	O
is	O
interpreted	O
as	O
the	O
state	O
,	O
and	O
is	O
interpreted	O
as	O
conjugate	O
state	O
,	O
or	O
conjugate	O
momentum	B-Algorithm
.	O
</s>
<s>
The	O
geodesi	O
of	O
EL	O
implies	O
specification	O
of	O
the	O
vector	O
fields	O
or	O
Eulerian	O
momentum	B-Algorithm
at	O
,	O
or	O
specification	O
of	O
canonical	O
momentum	B-Algorithm
determines	O
the	O
flow	O
.	O
</s>
<s>
In	O
computational	B-Algorithm
anatomy	I-Algorithm
the	O
submanifolds	O
are	O
pointsets	O
,	O
curves	O
,	O
surfaces	O
and	O
subvolumes	O
which	O
are	O
the	O
basic	O
primitives	O
.	O
</s>
<s>
The	O
geodesic	O
flows	O
between	O
the	O
submanifolds	O
determine	O
the	O
distance	O
,	O
and	O
form	O
the	O
basic	O
measuring	O
and	O
transporting	O
tools	O
of	O
diffeomorphometry	B-Algorithm
.	O
</s>
<s>
At	O
the	O
geodesic	O
has	O
vector	O
field	O
determined	O
by	O
the	O
conjugate	O
momentum	B-Algorithm
and	O
the	O
Green	O
's	O
kernel	O
of	O
the	O
inertial	O
operator	O
defining	O
the	O
Eulerian	O
momentum	B-Algorithm
.	O
</s>
<s>
Given	O
the	O
least-action	O
there	O
is	O
a	O
natural	O
definition	O
of	O
momentum	B-Algorithm
associated	O
to	O
generalized	O
coordinates	O
;	O
the	O
quantity	O
acting	O
against	O
velocity	O
gives	O
energy	O
.	O
</s>
<s>
The	O
field	O
has	O
studied	O
two	O
forms	O
,	O
the	O
momentum	B-Algorithm
associated	O
to	O
the	O
Eulerian	O
vector	O
field	O
termed	O
Eulerian	O
diffeomorphic	O
shape	O
momentum	B-Algorithm
,	O
and	O
the	O
momentum	B-Algorithm
associated	O
to	O
the	O
initial	O
coordinates	O
or	O
canonical	O
coordinates	O
termed	O
canonical	O
diffeomorphic	O
shape	O
momentum	B-Algorithm
.	O
</s>
<s>
The	O
conservation	O
of	O
momentum	B-Algorithm
goes	O
hand	O
in	O
hand	O
with	O
the	O
.	O
</s>
<s>
In	O
computational	B-Algorithm
anatomy	I-Algorithm
,	O
is	O
the	O
Eulerian	O
Momentum	B-Algorithm
since	O
when	O
integrated	O
against	O
Eulerian	O
velocity	O
gives	O
energy	O
density	O
;	O
operator	O
the	O
generalized	O
moment	O
of	O
inertia	O
or	O
inertial	O
operator	O
which	O
acting	O
on	O
the	O
Eulerian	O
velocity	O
gives	O
momentum	B-Algorithm
which	O
is	O
conserved	O
along	O
the	O
geodesic	O
:	O
</s>
<s>
LDDMM	O
has	O
been	O
solved	O
for	O
landmarks	O
with	O
and	O
without	O
correspondence	O
and	O
for	O
dense	O
image	B-Algorithm
matchings	I-Algorithm
.	O
</s>
<s>
Existence	O
of	O
solutions	O
were	O
examined	O
for	O
image	B-Algorithm
matching	I-Algorithm
.	O
</s>
<s>
The	O
solution	O
of	O
the	O
variational	B-Algorithm
problem	I-Algorithm
satisfies	O
the	O
for	O
with	O
boundary	O
condition	O
.	O
</s>
<s>
The	O
matching	O
problem	O
explicitly	O
indexed	O
to	O
initial	O
condition	O
is	O
called	O
shooting	O
,	O
which	O
can	O
also	O
be	O
reparamerized	O
via	O
the	O
conjugate	O
momentum	B-Algorithm
.	O
</s>
<s>
Dense	O
image	B-Algorithm
matching	I-Algorithm
has	O
a	O
long	O
history	O
now	O
with	O
the	O
earliest	O
efforts	O
exploiting	O
a	O
small	O
deformation	O
framework	O
.	O
</s>
<s>
Large	O
deformations	O
began	O
in	O
the	O
early	O
1990s	O
,	O
with	O
the	O
first	O
existence	O
to	O
solutions	O
to	O
the	O
variational	B-Algorithm
problem	I-Algorithm
for	O
flows	O
of	O
diffeomorphisms	O
for	O
dense	O
image	B-Algorithm
matching	I-Algorithm
established	O
in	O
.	O
</s>
<s>
Beg	O
solved	O
via	O
one	O
of	O
the	O
earliest	O
LDDMM	O
algorithms	O
based	O
on	O
solving	O
the	O
variational	B-Algorithm
matching	O
with	O
endpoint	O
defined	O
by	O
the	O
dense	O
imagery	O
with	O
respect	O
to	O
the	O
vector	O
fields	O
,	O
taking	O
variations	O
with	O
respect	O
to	O
the	O
vector	O
fields	O
.	O
</s>
<s>
Another	O
solution	O
for	O
dense	O
image	B-Algorithm
matching	I-Algorithm
reparameterizes	O
the	O
optimization	O
problem	O
in	O
terms	O
of	O
the	O
state	O
giving	O
the	O
solution	O
in	O
terms	O
of	O
the	O
infinitesimal	O
action	O
defined	O
by	O
the	O
advection	O
equation	O
.	O
</s>
<s>
The	O
iterative	O
algorithm	O
is	O
given	O
in	O
Beg	O
's	O
LDDMM	O
algorithm	O
for	O
dense	O
image	B-Algorithm
matching	I-Algorithm
.	O
</s>
<s>
Dense	O
LDDMM	O
tensor	O
matching	O
takes	O
the	O
images	O
as	O
3x1	O
vectors	O
and	O
3x3	O
tensors	O
solving	O
the	O
variational	B-Algorithm
problem	I-Algorithm
matching	O
between	O
coordinate	O
system	O
based	O
on	O
the	O
principle	O
eigenvectors	O
of	O
the	O
diffusion	B-Algorithm
tensor	I-Algorithm
MRI	I-Algorithm
image	O
(	O
DTI	O
)	O
denoted	O
consisting	O
of	O
the	O
-tensor	O
at	O
every	O
voxel	O
.	O
</s>
<s>
LDDMM	O
Tensor	O
Image	B-Algorithm
Matching	I-Algorithm
.	O
</s>
<s>
High	O
angular	O
resolution	O
diffusion	B-Algorithm
imaging	I-Algorithm
(	O
HARDI	O
)	O
addresses	O
the	O
well-known	O
limitation	O
of	O
DTI	O
,	O
that	O
is	O
,	O
DTI	O
can	O
only	O
reveal	O
one	O
dominant	O
fiber	O
orientation	O
at	O
each	O
location	O
.	O
</s>
<s>
Dense	O
LDDMM	O
ODF	O
matching	O
takes	O
the	O
HARDI	O
data	O
as	O
ODF	O
at	O
each	O
voxel	O
and	O
solves	O
the	O
LDDMM	O
variational	B-Algorithm
problem	I-Algorithm
in	O
the	O
space	O
of	O
ODF	O
.	O
</s>
<s>
In	O
the	O
field	O
of	O
information	O
geometry	O
,	O
the	O
space	O
of	O
ODF	O
forms	O
a	O
Riemannian	B-Architecture
manifold	I-Architecture
with	O
the	O
Fisher-Rao	O
metric	O
.	O
</s>
<s>
The	O
variational	B-Algorithm
problem	I-Algorithm
for	O
matching	O
assumes	O
that	O
two	O
ODF	O
volumes	O
can	O
be	O
generated	O
from	O
one	O
to	O
another	O
via	O
flows	O
of	O
diffeomorphisms	O
,	O
which	O
are	O
solutions	O
of	O
ordinary	O
differential	O
equations	O
starting	O
from	O
the	O
identity	O
map	O
.	O
</s>
<s>
Regression	O
analysis	O
on	O
ODF	O
is	O
developed	O
in	O
the	O
ODF	O
manifold	B-Architecture
space	O
in	O
.	O
</s>
<s>
For	O
setting	O
in	O
which	O
pairs	O
of	O
images	O
are	O
not	O
related	O
by	O
diffeomorphisms	O
but	O
have	O
photometric	O
variation	O
or	O
image	O
variation	O
not	O
represented	O
by	O
the	O
template	O
,	O
active	O
appearance	O
modelling	O
has	O
been	O
introduced	O
,	O
originally	O
by	O
Edwards-Cootes-Taylor	O
and	O
in	O
3D	O
medical	B-Application
imaging	I-Application
in	O
.	O
</s>
<s>
In	O
the	O
context	O
of	O
computational	B-Algorithm
anatomy	I-Algorithm
in	O
which	O
metrics	O
on	O
the	O
anatomical	O
orbit	O
has	O
been	O
studied	O
,	O
metamorphosis	O
for	O
modelling	O
structures	O
such	O
as	O
tumors	O
and	O
photometric	O
changes	O
which	O
are	O
not	O
resident	O
in	O
the	O
template	O
was	O
introduced	O
in	O
for	O
Magnetic	O
Resonance	O
image	O
models	O
,	O
with	O
many	O
subsequent	O
developments	O
extending	O
the	O
metamorphosis	O
framework	O
.	O
</s>
<s>
For	O
image	B-Algorithm
matching	I-Algorithm
the	O
image	O
metamorphosis	O
framework	O
enlarges	O
the	O
action	O
so	O
that	O
with	O
action	O
.	O
</s>
<s>
In	O
this	O
setting	O
metamorphosis	O
combines	O
both	O
the	O
diffeomorphic	O
coordinate	O
system	O
transformation	O
of	O
computational	B-Algorithm
anatomy	I-Algorithm
as	O
well	O
as	O
the	O
early	O
morphing	O
technologies	O
which	O
only	O
faded	O
or	O
modified	O
the	O
photometric	O
or	O
image	O
intensity	O
alone	O
.	O
</s>
<s>
Transforming	O
coordinate	O
systems	O
based	O
on	O
Landmark	B-General_Concept
point	I-General_Concept
or	O
fiducial	B-Architecture
marker	I-Architecture
features	O
dates	O
back	O
to	O
Bookstein	O
's	O
early	O
work	O
on	O
small	O
deformation	O
spline	O
methods	O
for	O
interpolating	O
correspondences	O
defined	O
by	O
fiducial	B-Architecture
points	O
to	O
the	O
two-dimensional	O
or	O
three-dimensional	O
background	O
space	O
in	O
which	O
the	O
fiducials	B-Architecture
are	O
defined	O
.	O
</s>
<s>
Matching	O
geometrical	O
objects	O
like	O
unlabelled	O
point	O
distributions	O
,	O
curves	O
or	O
surfaces	O
is	O
another	O
common	O
problem	O
in	O
computational	B-Algorithm
anatomy	I-Algorithm
.	O
</s>
<s>
From	O
the	O
theoretical	O
point	O
of	O
view	O
,	O
while	O
any	O
submanifold	O
in	O
,	O
can	O
be	O
parameterized	O
in	O
local	O
charts	O
,	O
all	O
reparametrizations	O
of	O
these	O
charts	O
give	O
geometrically	O
the	O
same	O
manifold	B-Architecture
.	O
</s>
<s>
Therefore	O
,	O
early	O
on	O
in	O
computational	B-Algorithm
anatomy	I-Algorithm
,	O
investigators	O
have	O
identified	O
the	O
necessity	O
of	O
parametrization	O
invariant	O
representations	O
.	O
</s>
<s>
The	O
geodesic	O
Eulerian	O
momentum	B-Algorithm
is	O
a	O
generalized	O
function	O
,	O
supported	O
on	O
the	O
landmarked	O
set	O
in	O
the	O
variational	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
The	O
endpoint	O
condition	O
with	O
conservation	O
implies	O
the	O
initial	O
momentum	B-Algorithm
at	O
the	O
identity	O
of	O
the	O
group	O
:	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>
Shown	O
in	O
the	O
Figure	O
on	O
the	O
right	O
the	O
cartoon	O
orbit	O
,	O
are	O
a	O
random	O
spray	O
of	O
the	O
subcortical	O
manifolds	B-Architecture
generated	O
by	O
randomizing	O
the	O
vector	O
fields	O
supported	O
over	O
the	O
submanifolds	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>
Shape	O
in	O
computational	B-Algorithm
anatomy	I-Algorithm
is	O
a	O
local	O
theory	O
,	O
indexing	O
shapes	O
and	O
structures	O
to	O
templates	O
to	O
which	O
they	O
are	O
bijectively	O
mapped	O
.	O
</s>
<s>
Statistical	O
shape	O
in	O
computational	B-Algorithm
anatomy	I-Algorithm
is	O
the	O
empirical	O
study	O
of	O
diffeomorphic	O
correspondences	O
between	O
populations	O
and	O
common	O
template	O
coordinate	O
systems	O
.	O
</s>
<s>
This	O
is	O
a	O
strong	O
departure	O
from	O
Procrustes	O
Analyses	O
and	O
shape	O
theories	O
pioneered	O
by	O
David	O
G	O
.	O
Kendall	O
in	O
that	O
the	O
central	O
group	O
of	O
Kendall	O
's	O
theories	O
are	O
the	O
finite-dimensional	O
Lie	O
groups	O
,	O
whereas	O
the	O
theories	O
of	O
shape	O
in	O
computational	B-Algorithm
anatomy	I-Algorithm
have	O
focused	O
on	O
the	O
diffeomorphism	O
group	O
,	O
which	O
to	O
first	O
order	O
via	O
the	O
Jacobian	O
can	O
be	O
thought	O
of	O
as	O
a	O
field	O
–	O
thus	O
infinite	O
dimensional	O
–	O
of	O
low-dimensional	O
Lie	O
groups	O
of	O
scale	O
and	O
rotations	O
.	O
</s>
<s>
The	O
random	O
orbit	O
model	O
provides	O
the	O
natural	O
setting	O
to	O
understand	O
empirical	O
shape	O
and	O
shape	O
statistics	O
within	O
computational	B-Algorithm
anatomy	I-Algorithm
since	O
the	O
non-linearity	O
of	O
the	O
induced	O
probability	O
law	O
on	O
anatomical	O
shapes	O
and	O
forms	O
is	O
induced	O
via	O
the	O
reduction	O
to	O
the	O
vector	O
fields	O
at	O
the	O
tangent	O
space	O
at	O
the	O
identity	O
of	O
the	O
diffeomorphism	O
group	O
.	O
</s>
<s>
Since	O
both	O
the	O
vector	O
fields	O
and	O
the	O
Eulerian	O
momentum	B-Algorithm
are	O
in	O
a	O
Hilbert	O
space	O
the	O
natural	O
model	O
is	O
one	O
of	O
a	O
Gaussian	O
random	O
field	O
,	O
so	O
that	O
given	O
test	O
function	O
,	O
then	O
the	O
inner-products	O
with	O
the	O
test	O
functions	O
are	O
Gaussian	O
distributed	O
with	O
mean	O
and	O
covariance	O
.	O
</s>
<s>
There	O
are	O
several	O
important	O
methods	O
for	O
generating	O
templates	O
including	O
methods	O
based	O
on	O
Frechet	O
averaging	O
,	O
and	O
statistical	O
approaches	O
based	O
on	O
the	O
expectation-maximization	B-Algorithm
algorithm	I-Algorithm
and	O
the	O
Bayes	O
Random	O
orbit	O
models	O
of	O
computational	B-Algorithm
anatomy	I-Algorithm
.	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>
