<s>
The	O
random	B-Algorithm
walker	I-Algorithm
algorithm	I-Algorithm
is	O
an	O
algorithm	O
for	O
image	B-Algorithm
segmentation	I-Algorithm
.	O
</s>
<s>
The	O
unlabeled	O
pixels	O
are	O
each	O
imagined	O
to	O
release	O
a	O
random	B-Algorithm
walker	I-Algorithm
,	O
and	O
the	O
probability	O
is	O
computed	O
that	O
each	O
pixel	O
's	O
random	B-Algorithm
walker	I-Algorithm
first	O
arrives	O
at	O
a	O
seed	O
bearing	O
each	O
label	O
,	O
i.e.	O
,	O
if	O
a	O
user	O
places	O
K	O
seeds	O
,	O
each	O
with	O
a	O
different	O
label	O
,	O
then	O
it	O
is	O
necessary	O
to	O
compute	O
,	O
for	O
each	O
pixel	O
,	O
the	O
probability	O
that	O
a	O
random	B-Algorithm
walker	I-Algorithm
leaving	O
the	O
pixel	O
will	O
first	O
arrive	O
at	O
each	O
seed	O
.	O
</s>
<s>
After	O
computing	O
these	O
probabilities	O
for	O
each	O
pixel	O
,	O
the	O
pixel	O
is	O
assigned	O
to	O
the	O
label	O
for	O
which	O
it	O
is	O
most	O
likely	O
to	O
send	O
a	O
random	B-Algorithm
walker	I-Algorithm
.	O
</s>
<s>
Although	O
the	O
initial	O
algorithm	O
was	O
formulated	O
as	O
an	O
interactive	O
method	O
for	O
image	B-Algorithm
segmentation	I-Algorithm
,	O
it	O
has	O
been	O
extended	O
to	O
be	O
a	O
fully	O
automatic	O
algorithm	O
,	O
given	O
a	O
data	O
fidelity	O
term	O
(	O
e.g.	O
,	O
an	O
intensity	O
prior	O
)	O
.	O
</s>
<s>
Although	O
the	O
algorithm	O
was	O
described	O
in	O
terms	O
of	O
random	O
walks	O
,	O
the	O
probability	O
that	O
each	O
node	O
sends	O
a	O
random	B-Algorithm
walker	I-Algorithm
to	O
the	O
seeds	O
may	O
be	O
calculated	O
analytically	O
by	O
solving	O
a	O
sparse	O
,	O
positive-definite	O
system	O
of	O
linear	O
equations	O
with	O
the	O
graph	B-Algorithm
Laplacian	I-Algorithm
matrix	O
,	O
which	O
we	O
may	O
represent	O
with	O
the	O
variable	O
.	O
</s>
<s>
The	O
nodes	O
,	O
edges	O
and	O
weights	O
can	O
then	O
be	O
used	O
to	O
construct	O
the	O
graph	B-Algorithm
Laplacian	I-Algorithm
matrix	O
.	O
</s>
<s>
where	O
the	O
subscripts	O
are	O
used	O
to	O
indicate	O
the	O
portion	O
of	O
the	O
graph	B-Algorithm
Laplacian	I-Algorithm
matrix	O
indexed	O
by	O
the	O
respective	O
sets	O
.	O
</s>
<s>
The	O
random	B-Algorithm
walker	I-Algorithm
algorithm	I-Algorithm
was	O
initially	O
motivated	O
by	O
labelling	O
a	O
pixel	O
as	O
object/background	O
based	O
on	O
the	O
probability	O
that	O
a	O
random	B-Algorithm
walker	I-Algorithm
dropped	O
at	O
the	O
pixel	O
would	O
first	O
reach	O
an	O
object	O
(	O
foreground	O
)	O
seed	O
or	O
a	O
background	O
seed	O
.	O
</s>
<s>
Consequently	O
,	O
the	O
random	B-Algorithm
walker	I-Algorithm
algorithm	I-Algorithm
has	O
two	O
different	O
interpretations	O
in	O
terms	O
of	O
an	O
electric	O
circuit	O
.	O
</s>
<s>
The	O
steady-state	O
electrical	O
circuit	O
potentials	O
established	O
at	O
each	O
node	O
by	O
this	O
circuit	O
configuration	O
will	O
exactly	O
equal	O
the	O
random	B-Algorithm
walker	I-Algorithm
probabilities	O
.	O
</s>
<s>
Specifically	O
,	O
the	O
electrical	O
potential	O
,	O
at	O
node	O
will	O
equal	O
the	O
probability	O
that	O
a	O
random	B-Algorithm
walker	I-Algorithm
dropped	O
at	O
node	O
will	O
reach	O
an	O
object/foreground	O
node	O
before	O
reaching	O
a	O
background	O
node	O
.	O
</s>
<s>
In	O
the	O
second	O
interpretation	O
,	O
labeling	O
a	O
node	O
as	O
object	O
or	O
background	O
by	O
thresholding	O
the	O
random	B-Algorithm
walker	I-Algorithm
probability	O
at	O
0.5	O
is	O
equivalent	O
to	O
labeling	O
a	O
node	O
as	O
object	O
or	O
background	O
based	O
on	O
the	O
relative	O
effective	O
conductance	O
between	O
the	O
node	O
and	O
the	O
object	O
or	O
background	O
seeds	O
.	O
</s>
<s>
The	O
traditional	O
random	B-Algorithm
walker	I-Algorithm
algorithm	I-Algorithm
described	O
above	O
has	O
been	O
extended	O
in	O
several	O
ways	O
:	O
</s>
<s>
Beyond	O
image	B-Algorithm
segmentation	I-Algorithm
,	O
the	O
random	B-Algorithm
walker	I-Algorithm
algorithm	I-Algorithm
or	O
its	O
extensions	O
has	O
been	O
additionally	O
applied	O
to	O
several	O
problems	O
in	O
computer	O
vision	O
and	O
graphics	O
:	O
</s>
