<s>
As	O
applied	O
in	O
the	O
field	O
of	O
computer	B-Application
vision	I-Application
,	O
graph	B-Algorithm
cut	I-Algorithm
optimization	I-Algorithm
can	O
be	O
employed	O
to	O
efficiently	O
solve	O
a	O
wide	O
variety	O
of	O
low-level	O
computer	B-Application
vision	I-Application
problems	O
(	O
early	O
vision	O
)	O
,	O
such	O
as	O
image	O
smoothing	B-Application
,	O
the	O
stereo	B-Algorithm
correspondence	I-Algorithm
problem	O
,	O
image	B-Algorithm
segmentation	I-Algorithm
,	O
object	B-Algorithm
co-segmentation	I-Algorithm
,	O
and	O
many	O
other	O
computer	B-Application
vision	I-Application
problems	O
that	O
can	O
be	O
formulated	O
in	O
terms	O
of	O
energy	O
minimization	O
.	O
</s>
<s>
Many	O
of	O
these	O
energy	O
minimization	O
problems	O
can	O
be	O
approximated	O
by	O
solving	O
a	O
maximum	B-Algorithm
flow	I-Algorithm
problem	I-Algorithm
in	O
a	O
graph	O
(	O
and	O
thus	O
,	O
by	O
the	O
max-flow	B-Algorithm
min-cut	I-Algorithm
theorem	I-Algorithm
,	O
define	O
a	O
minimal	B-Algorithm
cut	I-Algorithm
of	O
the	O
graph	O
)	O
.	O
</s>
<s>
Under	O
most	O
formulations	O
of	O
such	O
problems	O
in	O
computer	B-Application
vision	I-Application
,	O
the	O
minimum	O
energy	O
solution	O
corresponds	O
to	O
the	O
maximum	B-General_Concept
a	I-General_Concept
posteriori	I-General_Concept
estimate	I-General_Concept
of	O
a	O
solution	O
.	O
</s>
<s>
Although	O
many	O
computer	B-Application
vision	I-Application
algorithms	O
involve	O
cutting	O
a	O
graph	O
(	O
e.g.	O
,	O
normalized	O
cuts	O
)	O
,	O
the	O
term	O
"	O
graph	B-Algorithm
cuts	I-Algorithm
"	O
is	O
applied	O
specifically	O
to	O
those	O
models	O
which	O
employ	O
a	O
max-flow/min	O
-cut	O
optimization	O
(	O
other	O
graph	O
cutting	O
algorithms	O
may	O
be	O
considered	O
as	O
graph	O
partitioning	O
algorithms	O
)	O
.	O
</s>
<s>
"	O
Binary	O
"	O
problems	O
(	O
such	O
as	O
denoising	O
a	O
binary	B-Algorithm
image	I-Algorithm
)	O
can	O
be	O
solved	O
exactly	O
using	O
this	O
approach	O
;	O
problems	O
where	O
pixels	O
can	O
be	O
labeled	O
with	O
more	O
than	O
two	O
different	O
labels	O
(	O
such	O
as	O
stereo	B-Algorithm
correspondence	I-Algorithm
,	O
or	O
denoising	O
of	O
a	O
grayscale	O
image	O
)	O
cannot	O
be	O
solved	O
exactly	O
,	O
but	O
solutions	O
produced	O
are	O
usually	O
near	O
the	O
global	O
optimum	O
.	O
</s>
<s>
The	O
theory	O
of	O
graph	B-Algorithm
cuts	I-Algorithm
used	O
as	O
an	B-Algorithm
optimization	I-Algorithm
method	I-Algorithm
was	O
first	O
applied	O
in	O
computer	B-Application
vision	I-Application
in	O
the	O
seminal	O
paper	O
by	O
Greig	O
,	O
Porteous	O
and	O
Seheult	O
of	O
Durham	O
University	O
.	O
</s>
<s>
In	O
the	O
Bayesian	O
statistical	O
context	O
of	O
smoothing	B-Application
noisy	O
(	O
or	O
corrupted	O
)	O
images	O
,	O
they	O
showed	O
how	O
the	O
maximum	B-General_Concept
a	I-General_Concept
posteriori	I-General_Concept
estimate	I-General_Concept
of	O
a	O
binary	B-Algorithm
image	I-Algorithm
can	O
be	O
obtained	O
exactly	O
by	O
maximizing	O
the	O
flow	B-Algorithm
through	O
an	O
associated	O
image	O
network	O
,	O
involving	O
the	O
introduction	O
of	O
a	O
source	O
and	O
sink	O
.	O
</s>
<s>
Prior	O
to	O
this	O
result	O
,	O
approximate	O
techniques	O
such	O
as	O
simulated	B-Algorithm
annealing	I-Algorithm
(	O
as	O
proposed	O
by	O
the	O
Geman	O
brothers	O
)	O
,	O
or	O
iterated	B-Algorithm
conditional	I-Algorithm
modes	I-Algorithm
(	O
a	O
type	O
of	O
greedy	B-Algorithm
algorithm	I-Algorithm
as	O
suggested	O
by	O
Julian	O
Besag	O
)	O
were	O
used	O
to	O
solve	O
such	O
image	O
smoothing	B-Application
problems	O
.	O
</s>
<s>
Although	O
the	O
general	O
-colour	O
problem	O
remains	O
unsolved	O
for	O
the	O
approach	O
of	O
Greig	O
,	O
Porteous	O
and	O
Seheult	O
has	O
turned	O
out	O
to	O
have	O
wide	O
applicability	O
in	O
general	O
computer	B-Application
vision	I-Application
problems	O
.	O
</s>
<s>
proposed	O
a	O
general	O
image	B-Algorithm
segmentation	I-Algorithm
framework	O
,	O
called	O
the	O
"	O
Power	O
Watershed	B-Algorithm
"	O
,	O
that	O
minimized	O
a	O
real-valued	O
indicator	O
function	O
from	O
 [ 0 , 1 ] 	O
over	O
a	O
graph	O
,	O
constrained	O
by	O
user	O
seeds	O
(	O
or	O
unary	O
terms	O
)	O
set	O
to	O
0	O
or	O
1	O
,	O
in	O
which	O
the	O
minimization	O
of	O
the	O
indicator	O
function	O
over	O
the	O
graph	O
is	O
optimized	O
with	O
respect	O
to	O
an	O
exponent	O
.	O
</s>
<s>
When	O
,	O
the	O
Power	O
Watershed	B-Algorithm
is	O
optimized	O
by	O
graph	B-Algorithm
cuts	I-Algorithm
,	O
when	O
the	O
Power	O
Watershed	B-Algorithm
is	O
optimized	O
by	O
shortest	O
paths	O
,	O
is	O
optimized	O
by	O
the	O
Random	B-Algorithm
walker	I-Algorithm
algorithm	I-Algorithm
and	O
is	O
optimized	O
by	O
the	O
Watershed	B-Algorithm
(	O
image	O
processing	O
)	O
algorithm	O
.	O
</s>
<s>
In	O
this	O
way	O
,	O
the	O
Power	O
Watershed	B-Algorithm
may	O
be	O
viewed	O
as	O
a	O
generalization	O
of	O
graph	B-Algorithm
cuts	I-Algorithm
that	O
provides	O
a	O
straightforward	O
connection	O
with	O
other	O
energy	O
optimization	O
segmentation/clustering	O
algorithms	O
.	O
</s>
<s>
Output	O
:	O
Segmentation	B-Algorithm
(	O
also	O
called	O
opacity	O
)	O
(	O
soft	O
segmentation	B-Algorithm
)	O
.	O
</s>
<s>
Optimization	O
:	O
The	O
segmentation	B-Algorithm
can	O
be	O
estimated	O
as	O
a	O
global	O
minimum	O
over	O
S	O
:	O
</s>
<s>
Standard	O
Graph	B-Algorithm
cuts	I-Algorithm
:	O
optimize	O
energy	O
function	O
over	O
the	O
segmentation	B-Algorithm
(	O
unknown	O
S	O
value	O
)	O
.	O
</s>
<s>
Iterated	O
Graph	B-Algorithm
cuts	I-Algorithm
:	O
</s>
<s>
Second	O
step	O
performs	O
the	O
usual	O
graph	B-Algorithm
cuts	I-Algorithm
algorithm	O
.	O
</s>
<s>
Standard	O
Markov	O
random	O
field	O
:	O
Associate	O
a	O
penalty	O
to	O
disagreeing	O
pixels	O
by	O
evaluating	O
the	O
difference	O
between	O
their	O
segmentation	B-Algorithm
label	O
(	O
crude	O
measure	O
of	O
the	O
length	O
of	O
the	O
boundaries	O
)	O
.	O
</s>
<s>
Conditional	B-General_Concept
random	I-General_Concept
field	I-General_Concept
:	O
If	O
the	O
color	O
is	O
very	O
different	O
,	O
it	O
might	O
be	O
a	O
good	O
place	O
to	O
put	O
a	O
boundary	O
.	O
</s>
<s>
Graph	B-Algorithm
cuts	I-Algorithm
methods	O
have	O
become	O
popular	O
alternatives	O
to	O
the	O
level	O
set-based	O
approaches	O
for	O
optimizing	O
the	O
location	O
of	O
a	O
contour	O
(	O
see	O
for	O
an	O
extensive	O
comparison	O
)	O
.	O
</s>
<s>
However	O
,	O
graph	O
cut	B-Algorithm
approaches	O
have	O
been	O
criticized	O
in	O
the	O
literature	O
for	O
several	O
issues	O
:	O
</s>
<s>
Metrication	O
artifacts	O
:	O
When	O
an	O
image	O
is	O
represented	O
by	O
a	O
4-connected	O
lattice	O
,	O
graph	B-Algorithm
cuts	I-Algorithm
methods	O
can	O
exhibit	O
unwanted	O
"	O
blockiness	O
"	O
artifacts	O
.	O
</s>
<s>
Various	O
methods	O
have	O
been	O
proposed	O
for	O
addressing	O
this	O
issue	O
,	O
such	O
as	O
using	O
additional	O
edges	O
or	O
by	O
formulating	O
the	O
max-flow	B-Algorithm
problem	I-Algorithm
in	O
continuous	O
space	O
.	O
</s>
<s>
Shrinking	O
bias	O
:	O
Since	O
graph	B-Algorithm
cuts	I-Algorithm
finds	O
a	O
minimum	O
cut	B-Algorithm
,	O
the	O
algorithm	O
can	O
be	O
biased	O
toward	O
producing	O
a	O
small	O
contour	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
algorithm	O
is	O
not	O
well-suited	O
for	O
segmentation	B-Algorithm
of	O
thin	O
objects	O
like	O
blood	O
vessels	O
(	O
see	O
for	O
a	O
proposed	O
fix	O
)	O
.	O
</s>
<s>
Multiple	O
labels	O
:	O
Graph	B-Algorithm
cuts	I-Algorithm
is	O
only	O
able	O
to	O
find	O
a	O
global	O
optimum	O
for	O
binary	O
labeling	O
(	O
i.e.	O
,	O
two	O
labels	O
)	O
problems	O
,	O
such	O
as	O
foreground/background	O
image	B-Algorithm
segmentation	I-Algorithm
.	O
</s>
<s>
Extensions	O
have	O
been	O
proposed	O
that	O
can	O
find	O
approximate	O
solutions	O
for	O
multilabel	O
graph	B-Algorithm
cuts	I-Algorithm
problems	O
.	O
</s>
<s>
Memory	O
:	O
the	O
memory	O
usage	O
of	O
graph	B-Algorithm
cuts	I-Algorithm
increases	O
quickly	O
as	O
the	O
image	O
size	O
increases	O
.	O
</s>
<s>
As	O
an	O
illustration	O
,	O
the	O
Boykov-Kolmogorov	O
max-flow	B-Algorithm
algorithm	O
v2.2	O
allocates	O
bytes	O
(	O
and	O
are	O
respectively	O
the	O
number	O
of	O
nodes	O
and	O
edges	O
in	O
the	O
graph	O
)	O
.	O
</s>
<s>
Nevertheless	O
,	O
some	O
amount	O
of	O
work	O
has	O
been	O
recently	O
done	O
in	O
this	O
direction	O
for	O
reducing	O
the	O
graphs	O
before	O
the	O
maximum-flow	O
computation	O
.	O
</s>
<s>
Minimization	O
is	O
done	O
using	O
a	O
standard	O
minimum	O
cut	B-Algorithm
algorithm	O
.	O
</s>
<s>
Due	O
to	O
the	O
Max-flow	B-Algorithm
min-cut	I-Algorithm
theorem	I-Algorithm
we	O
can	O
solve	O
energy	O
minimization	O
by	O
maximizing	O
the	O
flow	B-Algorithm
over	O
the	O
network	O
.	O
</s>
<s>
The	O
Max	B-Algorithm
Flow	I-Algorithm
problem	I-Algorithm
consists	O
of	O
a	O
directed	O
graph	O
with	O
edges	O
labeled	O
with	O
capacities	O
,	O
and	O
there	O
are	O
two	O
distinct	O
nodes	O
:	O
the	O
source	O
and	O
the	O
sink	O
.	O
</s>
<s>
Intuitively	O
,	O
it	O
is	O
easy	O
to	O
see	O
that	O
the	O
maximum	B-Algorithm
flow	I-Algorithm
is	O
determined	O
by	O
the	O
bottleneck	O
.	O
</s>
<s>
The	O
Boykov-Kolmogorov	O
algorithm	O
is	O
an	O
efficient	O
way	O
to	O
compute	O
the	O
max-flow	B-Algorithm
for	O
computer	B-Application
vision	I-Application
related	O
graph	O
.	O
</s>
<s>
The	O
Sim	O
Cut	B-Algorithm
algorithm	O
approximates	O
the	O
graph	O
cut	B-Algorithm
.	O
</s>
<s>
This	O
is	O
the	O
approach	O
suggested	O
by	O
Cederbaum	B-Algorithm
's	I-Algorithm
maximum	I-Algorithm
flow	I-Algorithm
theorem	I-Algorithm
.	O
</s>
<s>
—	O
An	O
implementation	O
of	O
the	O
Sim	O
Cut	B-Algorithm
;	O
an	O
algorithm	O
for	O
computing	O
an	O
approximate	O
solution	O
of	O
the	O
minimum	O
s-t	B-Algorithm
cut	I-Algorithm
in	O
a	O
massively	O
parallel	O
manner	O
.	O
</s>
