<s>
Active	B-General_Concept
contour	I-General_Concept
model	I-General_Concept
,	O
also	O
called	O
snakes	B-General_Concept
,	O
is	O
a	O
framework	O
in	O
computer	B-Application
vision	I-Application
introduced	O
by	O
Michael	O
Kass	O
,	O
Andrew	O
Witkin	O
,	O
and	O
Demetri	O
Terzopoulos	O
for	O
delineating	O
an	O
object	O
outline	O
from	O
a	O
possibly	O
noisy	B-Algorithm
2D	O
image	O
.	O
</s>
<s>
The	O
snakes	B-General_Concept
model	O
is	O
popular	O
in	O
computer	B-Application
vision	I-Application
,	O
and	O
snakes	B-General_Concept
are	O
widely	O
used	O
in	O
applications	O
like	O
object	O
tracking	O
,	O
shape	O
recognition	O
,	O
segmentation	B-Algorithm
,	O
edge	B-Algorithm
detection	I-Algorithm
and	O
stereo	O
matching	O
.	O
</s>
<s>
A	O
snake	B-General_Concept
is	O
an	O
energy	O
minimizing	O
,	O
deformable	O
spline	B-Algorithm
influenced	O
by	O
constraint	O
and	O
image	O
forces	O
that	O
pull	O
it	O
towards	O
object	O
contours	O
and	O
internal	O
forces	O
that	O
resist	O
deformation	O
.	O
</s>
<s>
Snakes	B-General_Concept
may	O
be	O
understood	O
as	O
a	O
special	O
case	O
of	O
the	O
general	O
technique	O
of	O
matching	O
a	O
deformable	O
model	O
to	O
an	O
image	O
by	O
means	O
of	O
energy	O
minimization	O
.	O
</s>
<s>
In	O
two	O
dimensions	O
,	O
the	O
active	B-General_Concept
shape	I-General_Concept
model	I-General_Concept
represents	O
a	O
discrete	O
version	O
of	O
this	O
approach	O
,	O
taking	O
advantage	O
of	O
the	O
point	B-General_Concept
distribution	I-General_Concept
model	I-General_Concept
to	O
restrict	O
the	O
shape	O
range	O
to	O
an	O
explicit	O
domain	O
learnt	O
from	O
a	O
training	O
set	O
.	O
</s>
<s>
Snakes	B-General_Concept
do	O
not	O
solve	O
the	O
entire	O
problem	O
of	O
finding	O
contours	O
in	O
images	O
,	O
since	O
the	O
method	O
requires	O
knowledge	O
of	O
the	O
desired	O
contour	O
shape	O
beforehand	O
.	O
</s>
<s>
Rather	O
,	O
they	O
depend	O
on	O
other	O
mechanisms	O
such	O
as	O
interaction	O
with	O
a	O
user	O
,	O
interaction	O
with	O
some	O
higher	O
level	O
image	B-Application
understanding	I-Application
process	O
,	O
or	O
information	O
from	O
image	O
data	O
adjacent	O
in	O
time	O
or	O
space	O
.	O
</s>
<s>
In	O
computer	B-Application
vision	I-Application
,	O
contour	O
models	O
describe	O
the	O
boundaries	O
of	O
shapes	O
in	O
an	O
image	O
.	O
</s>
<s>
Snakes	B-General_Concept
in	O
particular	O
are	O
designed	O
to	O
solve	O
problems	O
where	O
the	O
approximate	O
shape	O
of	O
the	O
boundary	O
is	O
known	O
.	O
</s>
<s>
By	O
being	O
a	O
deformable	O
model	O
,	O
snakes	B-General_Concept
can	O
adapt	O
to	O
differences	O
and	O
noise	O
in	O
stereo	O
matching	O
and	O
motion	O
tracking	O
.	O
</s>
<s>
Compared	O
to	O
classical	O
feature	O
extraction	O
techniques	O
,	O
snakes	B-General_Concept
have	O
multiple	O
advantages	O
:	O
</s>
<s>
External	O
image	O
forces	O
act	O
upon	O
the	O
snake	B-General_Concept
in	O
an	O
intuitive	O
manner	O
.	O
</s>
<s>
A	O
simple	O
elastic	O
snake	B-General_Concept
is	O
defined	O
by	O
a	O
set	O
of	O
n	O
points	O
for	O
,	O
the	O
internal	O
elastic	O
energy	O
term	O
,	O
and	O
the	O
external	O
edge-based	O
energy	O
term	O
.	O
</s>
<s>
The	O
purpose	O
of	O
the	O
internal	O
energy	O
term	O
is	O
to	O
control	O
the	O
deformations	O
made	O
to	O
the	O
snake	B-General_Concept
,	O
and	O
the	O
purpose	O
of	O
the	O
external	O
energy	O
term	O
is	O
to	O
control	O
the	O
fitting	O
of	O
the	O
contour	O
onto	O
the	O
image	O
.	O
</s>
<s>
The	O
internal	O
energy	O
of	O
the	O
snake	B-General_Concept
is	O
composed	O
of	O
the	O
continuity	O
of	O
the	O
contour	O
and	O
the	O
smoothness	O
of	O
the	O
contour	O
.	O
</s>
<s>
where	O
and	O
are	O
user-defined	O
weights	O
;	O
these	O
control	O
the	O
internal	O
energy	O
function	O
's	O
sensitivity	O
to	O
the	O
amount	O
of	O
stretch	O
in	O
the	O
snake	B-General_Concept
and	O
the	O
amount	O
of	O
curvature	O
in	O
the	O
snake	B-General_Concept
,	O
respectively	O
,	O
and	O
thereby	O
control	O
the	O
number	O
of	O
constraints	O
on	O
the	O
shape	O
of	O
the	O
snake	B-General_Concept
.	O
</s>
<s>
A	O
snake	B-General_Concept
originating	O
far	O
from	O
the	O
desired	O
object	O
contour	O
may	O
erroneously	O
converge	O
to	O
some	O
local	O
minimum	O
.	O
</s>
<s>
This	O
is	O
achieved	O
by	O
using	O
a	O
blur	O
filter	O
on	O
the	O
image	O
and	O
reducing	O
the	O
amount	O
of	O
blur	O
as	O
the	O
calculation	O
progresses	O
to	O
refine	O
the	O
fit	O
of	O
the	O
snake	B-General_Concept
.	O
</s>
<s>
Minima	O
of	O
this	O
function	O
fall	O
on	O
the	O
zero-crossings	B-Algorithm
of	O
which	O
define	O
edges	O
as	O
per	O
Marr	B-Algorithm
–	I-Algorithm
Hildreth	I-Algorithm
theory	O
.	O
</s>
<s>
Some	O
systems	O
,	O
including	O
the	O
original	O
snakes	B-General_Concept
implementation	O
,	O
allowed	O
for	O
user	O
interaction	O
to	O
guide	O
the	O
snakes	B-General_Concept
,	O
not	O
only	O
in	O
initial	O
placement	O
but	O
also	O
in	O
their	O
energy	O
terms	O
.	O
</s>
<s>
Such	O
constraint	O
energy	O
can	O
be	O
used	O
to	O
interactively	O
guide	O
the	O
snakes	B-General_Concept
towards	O
or	O
away	O
from	O
particular	O
features	O
.	O
</s>
<s>
Given	O
an	O
initial	O
guess	O
for	O
a	O
snake	B-General_Concept
,	O
the	O
energy	O
function	O
of	O
the	O
snake	B-General_Concept
is	O
iteratively	O
minimized	O
.	O
</s>
<s>
Gradient	B-Algorithm
descent	I-Algorithm
minimization	O
is	O
one	O
of	O
the	O
simplest	O
optimizations	O
which	O
can	O
be	O
used	O
to	O
minimize	O
snake	B-General_Concept
energy	O
.	O
</s>
<s>
Where	O
is	O
the	O
force	O
on	O
the	O
snake	B-General_Concept
,	O
which	O
is	O
defined	O
by	O
the	O
negative	O
of	O
the	O
gradient	O
of	O
the	O
energy	O
field	O
.	O
</s>
<s>
As	O
such	O
,	O
discrete	O
approximations	O
must	O
be	O
made	O
for	O
practical	O
implementation	O
of	O
snakes	B-General_Concept
.	O
</s>
<s>
The	O
energy	O
function	O
of	O
the	O
snake	B-General_Concept
can	O
be	O
approximated	O
by	O
using	O
the	O
discrete	O
points	O
on	O
the	O
snake	B-General_Concept
.	O
</s>
<s>
Gradient	O
approximation	O
can	O
be	O
done	O
through	O
any	O
finite	O
approximation	O
method	O
with	O
respect	O
to	O
s	O
,	O
such	O
as	O
Finite	B-Algorithm
difference	I-Algorithm
.	O
</s>
<s>
The	O
introduction	O
of	O
discrete	O
time	O
into	O
the	O
algorithm	O
can	O
introduce	O
updates	O
which	O
the	O
snake	B-General_Concept
is	O
moved	O
past	O
the	O
minima	O
it	O
is	O
attracted	O
to	O
;	O
this	O
further	O
can	O
cause	O
oscillations	O
around	O
the	O
minima	O
or	O
lead	O
to	O
a	O
different	O
minima	O
being	O
found	O
.	O
</s>
<s>
Alternatively	O
,	O
the	O
image	O
forces	O
can	O
be	O
normalized	O
for	O
each	O
step	O
such	O
that	O
the	O
image	O
forces	O
only	O
update	O
the	O
snake	B-General_Concept
by	O
one	O
pixel	O
.	O
</s>
<s>
The	O
energies	O
in	O
a	O
continuous	O
image	O
may	O
have	O
zero-crossing	B-Algorithm
that	O
do	O
not	O
exist	O
as	O
a	O
pixel	O
in	O
the	O
image	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
a	O
point	O
in	O
the	O
snake	B-General_Concept
would	O
oscillate	O
between	O
the	O
two	O
pixels	O
that	O
neighbor	O
this	O
zero-crossing	B-Algorithm
.	O
</s>
<s>
The	O
default	O
method	O
of	O
snakes	B-General_Concept
has	O
various	O
limitation	O
and	O
corner	O
cases	O
where	O
the	O
convergence	O
performs	O
poorly	O
.	O
</s>
<s>
The	O
gradient	B-General_Concept
vector	I-General_Concept
flow	I-General_Concept
(	O
GVF	O
)	O
snake	B-General_Concept
model	O
addresses	O
two	O
issues	O
with	O
snakes	B-General_Concept
:	O
</s>
<s>
The	O
balloon	O
model	O
addresses	O
these	O
problems	O
with	O
the	O
default	O
active	B-General_Concept
contour	I-General_Concept
model	I-General_Concept
:	O
</s>
<s>
The	O
snake	B-General_Concept
is	O
not	O
attracted	O
to	O
distant	O
edges	O
.	O
</s>
<s>
The	O
snake	B-General_Concept
will	O
shrink	O
inwards	O
if	O
no	O
substantial	O
images	O
forces	O
are	O
acting	O
upon	O
it	O
.	O
</s>
<s>
a	O
snake	B-General_Concept
larger	O
than	O
the	O
minima	O
contour	O
will	O
eventually	O
shrink	O
into	O
it	O
,	O
but	O
a	O
snake	B-General_Concept
smaller	O
than	O
the	O
minima	O
contour	O
will	O
not	O
find	O
the	O
minima	O
and	O
instead	O
continue	O
to	O
shrink	O
.	O
</s>
<s>
should	O
have	O
the	O
same	O
magnitude	O
as	O
the	O
image	O
normalization	O
factor	O
and	O
be	O
smaller	O
in	O
value	O
than	O
to	O
allow	O
forces	O
at	O
image	B-Algorithm
edges	I-Algorithm
to	O
overcome	O
the	O
inflation	O
force	O
.	O
</s>
<s>
Instead	O
of	O
shrinking	O
,	O
the	O
snake	B-General_Concept
expands	O
into	O
the	O
minima	O
and	O
will	O
not	O
find	O
minima	O
contours	O
smaller	O
than	O
it	O
.	O
</s>
<s>
The	O
inflation	O
force	O
can	O
overpower	O
forces	O
from	O
weak	O
edges	O
,	O
amplifying	O
the	O
issue	O
with	O
snakes	B-General_Concept
ignoring	O
weaker	O
features	O
in	O
an	O
image	O
.	O
</s>
<s>
The	O
diffusion	O
snake	B-General_Concept
model	O
addresses	O
the	O
sensitivity	O
of	O
snakes	B-General_Concept
to	O
noise	O
,	O
clutter	O
,	O
and	O
occlusion	O
.	O
</s>
<s>
It	O
implements	O
a	O
modification	O
of	O
the	O
Mumford	B-Algorithm
–	I-Algorithm
Shah	I-Algorithm
functional	I-Algorithm
and	O
its	O
cartoon	O
limit	O
and	O
incorporates	O
statistical	O
shape	O
knowledge	O
.	O
</s>
<s>
where	O
are	O
quadratic	O
B-spline	O
basis	O
functions	O
and	O
are	O
the	O
control	O
points	O
of	O
the	O
splines	B-Algorithm
.	O
</s>
<s>
The	O
strength	O
of	O
this	O
method	O
relies	O
on	O
the	O
strength	O
of	O
the	O
training	O
data	O
as	O
well	O
as	O
the	O
tuning	O
of	O
the	O
modified	O
Mumford	B-Algorithm
–	I-Algorithm
Shah	I-Algorithm
functional	I-Algorithm
.	O
</s>
<s>
Different	O
snakes	B-General_Concept
will	O
require	O
different	O
training	O
data	O
sets	O
and	O
tunings	O
.	O
</s>
<s>
Geometric	O
active	B-General_Concept
contour	I-General_Concept
,	O
or	O
geodesic	O
active	B-General_Concept
contour	I-General_Concept
(	O
GAC	O
)	O
or	O
conformal	O
active	B-General_Concept
contours	I-General_Concept
employs	O
ideas	O
from	O
Euclidean	O
curve	O
shortening	O
evolution	O
.	O
</s>
<s>
These	O
models	O
are	O
largely	O
inspired	O
by	O
level	B-Algorithm
sets	I-Algorithm
,	O
and	O
have	O
been	O
extensively	O
employed	O
in	O
medical	B-Algorithm
image	I-Algorithm
computing	I-Algorithm
.	O
</s>
<s>
This	O
simple	O
yet	O
powerful	O
level-set	O
reformation	O
enables	O
active	B-General_Concept
contours	I-General_Concept
to	O
handle	O
topology	O
changes	O
during	O
the	O
gradient	B-Algorithm
descent	I-Algorithm
curve	O
evolution	O
.	O
</s>
<s>
It	O
has	O
inspired	O
tremendous	O
progress	O
in	O
the	O
related	O
fields	O
,	O
and	O
using	O
numerical	O
methods	O
to	O
solve	O
the	O
level-set	O
reformulation	O
is	O
now	O
commonly	O
known	O
as	O
the	O
level-set	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Although	O
the	O
level	B-Algorithm
set	I-Algorithm
method	I-Algorithm
has	O
become	O
quite	O
a	O
popular	O
tool	O
for	O
implementing	O
active	B-General_Concept
contours	I-General_Concept
,	O
Wang	O
and	O
Chan	O
argued	O
that	O
not	O
all	O
curve	O
evolution	O
equations	O
should	O
be	O
directly	O
solved	O
by	O
it	O
.	O
</s>
<s>
More	O
recent	O
developments	O
in	O
active	B-General_Concept
contours	I-General_Concept
address	O
modeling	O
of	O
regional	O
properties	O
,	O
incorporation	O
of	O
flexible	O
shape	O
priors	O
and	O
fully	O
automatic	O
segmentation	B-Algorithm
,	O
etc	O
.	O
</s>
<s>
Graph	B-Algorithm
cuts	I-Algorithm
,	O
or	O
max-flow/min	B-Algorithm
-cut	I-Algorithm
,	O
is	O
a	O
generic	O
method	O
for	O
minimizing	O
a	O
particular	O
form	O
of	O
energy	O
called	O
Markov	O
random	O
field	O
(	O
MRF	O
)	O
energy	O
.	O
</s>
<s>
The	O
Graph	B-Algorithm
cuts	I-Algorithm
method	O
has	O
been	O
applied	O
to	O
image	B-Algorithm
segmentation	I-Algorithm
as	O
well	O
,	O
and	O
it	O
sometimes	O
outperforms	O
the	O
level	B-Algorithm
set	I-Algorithm
method	I-Algorithm
when	O
the	O
model	O
is	O
MRF	O
or	O
can	O
be	O
approximated	O
by	O
MRF	O
.	O
</s>
