<s>
Phase	B-General_Concept
correlation	I-General_Concept
is	O
an	O
approach	O
to	O
estimate	O
the	O
relative	O
translative	B-Algorithm
offset	O
between	O
two	O
similar	O
images	O
(	O
digital	O
image	O
correlation	O
)	O
or	O
other	O
data	O
sets	O
.	O
</s>
<s>
It	O
is	O
commonly	O
used	O
in	O
image	B-Algorithm
registration	I-Algorithm
and	O
relies	O
on	O
a	O
frequency-domain	O
representation	O
of	O
the	O
data	O
,	O
usually	O
calculated	O
by	O
fast	O
Fourier	B-Algorithm
transforms	I-Algorithm
.	O
</s>
<s>
The	O
term	O
is	O
applied	O
particularly	O
to	O
a	O
subset	O
of	O
cross-correlation	O
techniques	O
that	O
isolate	O
the	O
phase	O
information	O
from	O
the	O
Fourier-space	O
representation	O
of	O
the	O
cross-correlogram	O
.	O
</s>
<s>
The	O
following	O
image	O
demonstrates	O
the	O
usage	O
of	O
phase	B-General_Concept
correlation	I-General_Concept
to	O
determine	O
relative	O
translative	B-Algorithm
movement	O
between	O
two	O
images	O
corrupted	O
by	O
independent	O
Gaussian	O
noise	O
.	O
</s>
<s>
Apply	O
a	O
window	B-Language
function	I-Language
(	O
e.g.	O
,	O
a	O
Hamming	O
window	O
)	O
on	O
both	O
images	O
to	O
reduce	O
edge	O
effects	O
(	O
this	O
may	O
be	O
optional	O
depending	O
on	O
the	O
image	O
characteristics	O
)	O
.	O
</s>
<s>
Then	O
,	O
calculate	O
the	O
discrete	O
2D	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
both	O
images	O
.	O
</s>
<s>
Calculate	O
the	O
cross-power	O
spectrum	O
by	O
taking	O
the	O
complex	O
conjugate	O
of	O
the	O
second	O
result	O
,	O
multiplying	O
the	O
Fourier	B-Algorithm
transforms	I-Algorithm
together	O
elementwise	O
,	O
and	O
normalizing	O
this	O
product	O
elementwise	O
.	O
</s>
<s>
Obtain	O
the	O
normalized	O
cross-correlation	O
by	O
applying	O
the	O
inverse	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
Commonly	O
,	O
interpolation	B-Algorithm
methods	O
are	O
used	O
to	O
estimate	O
the	O
peak	O
location	O
in	O
the	O
cross-correlogram	O
to	O
non-integer	O
values	O
,	O
despite	O
the	O
fact	O
that	O
the	O
data	O
are	O
discrete	O
,	O
and	O
this	O
procedure	O
is	O
often	O
termed	O
'	O
subpixel	O
registration	O
 '	O
.	O
</s>
<s>
A	O
large	O
variety	O
of	O
subpixel	O
interpolation	B-Algorithm
methods	O
are	O
given	O
in	O
the	O
technical	O
literature	O
.	O
</s>
<s>
Common	O
peak	O
interpolation	B-Algorithm
methods	O
such	O
as	O
parabolic	O
interpolation	B-Algorithm
have	O
been	O
used	O
,	O
and	O
the	O
OpenCV	B-Language
computer	O
vision	O
package	O
uses	O
a	O
centroid-based	O
method	O
,	O
though	O
these	O
generally	O
have	O
inferior	O
accuracy	O
compared	O
to	O
more	O
sophisticated	O
methods	O
.	O
</s>
<s>
Because	O
the	O
Fourier	O
representation	O
of	O
the	O
data	O
has	O
already	O
been	O
computed	O
,	O
it	O
is	O
especially	O
convenient	O
to	O
use	O
the	O
Fourier	O
shift	O
theorem	O
with	O
real-valued	O
(	O
sub-integer	O
)	O
shifts	O
for	O
this	O
purpose	O
,	O
which	O
essentially	O
interpolates	O
using	O
the	O
sinusoidal	O
basis	O
functions	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
Some	O
methods	O
shift	O
the	O
peak	O
in	O
Fourier	O
space	O
and	O
apply	O
non-linear	O
optimization	O
to	O
maximize	O
the	O
correlogram	B-Application
peak	O
,	O
but	O
these	O
tend	O
to	O
be	O
very	O
slow	O
since	O
they	O
must	O
apply	O
an	O
inverse	O
Fourier	B-Algorithm
transform	I-Algorithm
or	O
its	O
equivalent	O
in	O
the	O
objective	O
function	O
.	O
</s>
<s>
These	O
methods	O
usually	O
use	O
a	O
linear	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
(	O
LLS	O
)	O
fit	O
of	O
the	O
phase	O
angles	O
to	O
a	O
planar	O
model	O
.	O
</s>
<s>
The	O
long	O
latency	O
of	O
the	O
phase	B-Application
angle	I-Application
computation	O
in	O
these	O
methods	O
is	O
a	O
disadvantage	O
,	O
but	O
the	O
speed	O
can	O
sometimes	O
be	O
comparable	O
to	O
the	O
Foroosh	O
et	O
al	O
.	O
</s>
<s>
This	O
fact	O
also	O
may	O
limit	O
the	O
usefulness	O
of	O
high	O
numerical	O
accuracy	O
in	O
an	O
algorithm	O
,	O
since	O
the	O
uncertainty	O
due	O
to	O
interpolation	B-Algorithm
method	O
choice	O
may	O
be	O
larger	O
than	O
any	O
numerical	O
or	O
approximation	O
error	O
in	O
the	O
particular	O
method	O
.	O
</s>
<s>
Then	O
,	O
the	O
discrete	B-Algorithm
Fourier	I-Algorithm
transforms	I-Algorithm
of	O
the	O
images	O
will	O
be	O
shifted	O
relatively	O
in	O
phase	O
:	O
</s>
<s>
The	O
inverse	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
a	O
complex	O
exponential	O
is	O
a	O
Kronecker	O
delta	O
,	O
i.e.	O
</s>
<s>
The	O
advantage	O
of	O
this	O
method	O
is	O
that	O
the	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
and	O
its	O
inverse	O
can	O
be	O
performed	O
using	O
the	O
fast	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
which	O
is	O
much	O
faster	O
than	O
correlation	O
for	O
large	O
images	O
.	O
</s>
<s>
Unlike	O
many	O
spatial-domain	O
algorithms	O
,	O
the	O
phase	B-General_Concept
correlation	I-General_Concept
method	O
is	O
resilient	O
to	O
noise	O
,	O
occlusions	O
,	O
and	O
other	O
defects	O
typical	O
of	O
medical	O
or	O
satellite	O
images	O
.	O
</s>
<s>
Due	O
to	O
properties	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
the	O
rotation	O
and	O
scaling	O
parameters	O
can	O
be	O
determined	O
in	O
a	O
manner	O
invariant	O
to	O
translation	O
.	O
</s>
<s>
In	O
such	O
cases	O
,	O
a	O
window	B-Language
function	I-Language
(	O
such	O
as	O
a	O
Gaussian	O
or	O
Tukey	O
window	O
)	O
should	O
be	O
employed	O
during	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
to	O
reduce	O
edge	O
effects	O
,	O
or	O
the	O
images	O
should	O
be	O
zero	O
padded	O
so	O
that	O
the	O
edge	O
effects	O
can	O
be	O
ignored	O
.	O
</s>
<s>
For	O
periodic	O
images	O
(	O
such	O
as	O
a	O
chessboard	O
or	O
picket	O
fence	O
)	O
,	O
phase	B-General_Concept
correlation	I-General_Concept
may	O
yield	O
ambiguous	O
results	O
with	O
several	O
peaks	O
in	O
the	O
resulting	O
output	O
.	O
</s>
<s>
Phase	B-General_Concept
correlation	I-General_Concept
is	O
the	O
preferred	O
method	O
for	O
television	O
standards	O
conversion	O
,	O
as	O
it	O
leaves	O
the	O
fewest	O
artifacts	O
.	O
</s>
