<s>
In	O
mathematics	O
,	O
the	O
discrete	B-Algorithm
Laplace	I-Algorithm
operator	I-Algorithm
is	O
an	O
analog	O
of	O
the	O
continuous	O
Laplace	O
operator	O
,	O
defined	O
so	O
that	O
it	O
has	O
meaning	O
on	O
a	O
graph	O
or	O
a	O
discrete	O
grid	O
.	O
</s>
<s>
For	O
the	O
case	O
of	O
a	O
finite-dimensional	O
graph	O
(	O
having	O
a	O
finite	O
number	O
of	O
edges	O
and	O
vertices	O
)	O
,	O
the	O
discrete	B-Algorithm
Laplace	I-Algorithm
operator	I-Algorithm
is	O
more	O
commonly	O
called	O
the	O
Laplacian	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
discrete	B-Algorithm
Laplace	I-Algorithm
operator	I-Algorithm
occurs	O
in	O
physics	O
problems	O
such	O
as	O
the	O
Ising	O
model	O
and	O
loop	O
quantum	O
gravity	O
,	O
as	O
well	O
as	O
in	O
the	O
study	O
of	O
discrete	O
dynamical	O
systems	O
.	O
</s>
<s>
It	O
is	O
also	O
used	O
in	O
numerical	B-General_Concept
analysis	I-General_Concept
as	O
a	O
stand-in	O
for	O
the	O
continuous	O
Laplace	O
operator	O
.	O
</s>
<s>
Common	O
applications	O
include	O
image	B-Algorithm
processing	I-Algorithm
,	O
where	O
it	O
is	O
known	O
as	O
the	O
Laplace	B-Algorithm
filter	I-Algorithm
,	O
and	O
in	O
machine	O
learning	O
for	O
clustering	B-Algorithm
and	O
semi-supervised	B-General_Concept
learning	I-General_Concept
on	O
neighborhood	O
graphs	O
.	O
</s>
<s>
The	O
traditional	O
definition	O
of	O
the	O
graph	B-Algorithm
Laplacian	I-Algorithm
,	O
given	O
below	O
,	O
corresponds	O
to	O
the	O
negative	O
continuous	O
Laplacian	O
on	O
a	O
domain	O
with	O
a	O
free	O
boundary	O
.	O
</s>
<s>
where	O
is	O
the	O
graph	O
distance	O
between	O
vertices	O
w	O
and	O
v	O
.	O
Thus	O
,	O
this	O
sum	O
is	O
over	O
the	O
nearest	O
neighbors	O
of	O
the	O
vertex	O
v	O
.	O
For	O
a	O
graph	O
with	O
a	O
finite	O
number	O
of	O
edges	O
and	O
vertices	O
,	O
this	O
definition	O
is	O
identical	O
to	O
that	O
of	O
the	O
Laplacian	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
That	O
is	O
,	O
can	O
be	O
written	O
as	O
a	O
column	O
vector	O
;	O
and	O
so	O
is	O
the	O
product	O
of	O
the	O
column	O
vector	O
and	O
the	O
Laplacian	B-Algorithm
matrix	I-Algorithm
,	O
while	O
is	O
just	O
the	O
vth	O
entry	O
of	O
the	O
product	O
vector	O
.	O
</s>
<s>
Approximations	O
of	O
the	O
Laplacian	O
,	O
obtained	O
by	O
the	O
finite-difference	B-Algorithm
method	I-Algorithm
or	O
by	O
the	O
finite-element	B-Application
method	I-Application
,	O
can	O
also	O
be	O
called	O
discrete	O
Laplacians	O
.	O
</s>
<s>
f(x, y )	O
given	O
on	O
the	O
boundary	O
of	O
the	O
grid	O
(	O
aka	O
,	O
Dirichlet	O
boundary	O
condition	O
)	O
,	O
is	O
rarely	O
used	O
for	O
graph	B-Algorithm
Laplacians	I-Algorithm
,	O
but	O
is	O
common	O
in	O
other	O
applications	O
.	O
</s>
<s>
Multidimensional	O
discrete	O
Laplacians	O
on	O
rectangular	O
cuboid	O
regular	O
grids	O
have	O
very	O
special	O
properties	O
,	O
e.g.	O
,	O
they	O
are	O
Kronecker	O
sums	O
of	O
one-dimensional	O
discrete	O
Laplacians	O
,	O
see	O
Kronecker	B-Algorithm
sum	I-Algorithm
of	I-Algorithm
discrete	I-Algorithm
Laplacians	I-Algorithm
,	O
in	O
which	O
case	O
all	O
its	O
eigenvalues	O
and	O
eigenvectors	O
can	O
be	O
explicitly	O
calculated	O
.	O
</s>
<s>
Discrete	B-Algorithm
Laplace	I-Algorithm
operator	I-Algorithm
is	O
often	O
used	O
in	O
image	B-Algorithm
processing	I-Algorithm
e.g.	O
</s>
<s>
in	O
edge	B-Algorithm
detection	I-Algorithm
and	O
motion	O
estimation	O
applications	O
.	O
</s>
<s>
The	O
smoothing	O
filter	O
and	O
Laplace	B-Algorithm
filter	I-Algorithm
are	O
often	O
combined	O
into	O
a	O
single	O
filter	O
.	O
</s>
<s>
For	O
one-	O
,	O
two	O
-	O
and	O
three-dimensional	O
signals	O
,	O
the	O
discrete	O
Laplacian	O
can	O
be	O
given	O
as	O
convolution	B-Language
with	O
the	O
following	O
kernels	B-Algorithm
:	O
</s>
<s>
corresponds	O
to	O
the	O
(	O
Five-point	B-Algorithm
stencil	I-Algorithm
)	O
finite-difference	O
formula	O
seen	O
previously	O
.	O
</s>
<s>
where	O
is	O
the	O
position	O
(	O
either	O
,	O
or	O
)	O
of	O
the	O
element	O
in	O
the	O
kernel	B-Algorithm
in	O
the	O
-th	O
direction	O
,	O
and	O
is	O
the	O
number	O
of	O
directions	O
for	O
which	O
.	O
</s>
<s>
These	O
kernels	B-Algorithm
are	O
deduced	O
by	O
using	O
discrete	O
differential	O
quotients	O
.	O
</s>
<s>
On	O
a	O
uniform	O
grid	O
,	O
such	O
as	O
images	O
,	O
and	O
for	O
bandlimited	O
functions	O
,	O
interpolation	O
functions	O
are	O
shift	O
invariant	O
amounting	O
to	O
with	O
being	O
an	O
appropriately	O
dilated	O
sinc	B-Algorithm
function	I-Algorithm
defined	O
in	O
-dimensions	O
i.e.	O
</s>
<s>
which	O
in	O
turn	O
is	O
a	O
convolution	B-Language
with	O
the	O
Laplacian	O
of	O
the	O
interpolation	O
function	O
on	O
the	O
uniform	O
(	O
image	O
)	O
grid	O
.	O
</s>
<s>
The	O
resulting	O
filtering	O
can	O
be	O
implemented	O
by	O
separable	O
filters	O
and	O
decimation	B-Algorithm
(	O
signal	O
processing	O
)	O
/pyramid	O
(	O
image	B-Algorithm
processing	I-Algorithm
)	O
representations	O
for	O
further	O
computational	O
efficiency	O
in	O
-dimensions	O
.	O
</s>
<s>
Structure	B-Algorithm
Tensor	I-Algorithm
,	O
and	O
Generalized	B-Algorithm
Structure	I-Algorithm
Tensor	I-Algorithm
which	O
are	O
used	O
in	O
pattern	O
recognition	O
for	O
their	O
total	O
least-square	O
optimality	O
in	O
orientation	O
estimation	O
,	O
can	O
be	O
realized	O
.	O
</s>
<s>
This	O
definition	O
of	O
the	O
Laplacian	O
is	O
commonly	O
used	O
in	O
numerical	B-General_Concept
analysis	I-General_Concept
and	O
in	O
image	B-Algorithm
processing	I-Algorithm
.	O
</s>
<s>
In	O
image	B-Algorithm
processing	I-Algorithm
,	O
it	O
is	O
considered	O
to	O
be	O
a	O
type	O
of	O
digital	O
filter	O
,	O
more	O
specifically	O
an	O
edge	B-Algorithm
filter	I-Algorithm
,	O
called	O
the	O
Laplace	B-Algorithm
filter	I-Algorithm
.	O
</s>
<s>
Notice	O
that	O
this	O
equation	O
takes	O
the	O
same	O
form	O
as	O
the	O
heat	O
equation	O
,	O
where	O
the	O
matrix	O
−L	O
is	O
replacing	O
the	O
Laplacian	O
operator	O
;	O
hence	O
,	O
the	O
"	O
graph	B-Algorithm
Laplacian	I-Algorithm
"	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
the	O
equilibrium	O
state	O
of	O
the	O
system	O
is	O
determined	O
completely	O
by	O
the	O
kernel	B-Algorithm
of	O
.	O
</s>
<s>
Since	O
by	O
definition	O
,	O
,	O
the	O
vector	O
of	O
all	O
ones	O
is	O
in	O
the	O
kernel	B-Algorithm
.	O
</s>
<s>
thumb|This	O
GIF	O
shows	O
the	O
progression	O
of	O
diffusion	O
,	O
as	O
solved	O
by	O
the	O
graph	B-Algorithm
laplacian	I-Algorithm
technique	O
.	O
</s>
<s>
It	O
shows	O
the	O
process	O
of	O
specifying	O
initial	O
conditions	O
,	O
projecting	O
these	O
initial	O
conditions	O
onto	O
the	O
eigenvalues	O
of	O
the	O
Laplacian	B-Algorithm
Matrix	I-Algorithm
,	O
and	O
simulating	O
the	O
exponential	O
decay	O
of	O
these	O
projected	O
initial	O
conditions	O
.	O
</s>
