<s>
The	O
triple	B-Algorithm
correlation	I-Algorithm
of	O
an	O
ordinary	O
function	O
on	O
the	O
real	O
line	O
is	O
the	O
integral	O
of	O
the	O
product	O
of	O
that	O
function	O
with	O
two	O
independently	O
shifted	O
copies	O
of	O
itself	O
:	O
</s>
<s>
The	O
Fourier	O
transform	O
of	O
triple	B-Algorithm
correlation	I-Algorithm
is	O
the	O
bispectrum	O
.	O
</s>
<s>
The	O
triple	B-Algorithm
correlation	I-Algorithm
extends	O
the	O
concept	O
of	O
autocorrelation	O
,	O
which	O
correlates	O
a	O
function	O
with	O
a	O
single	O
shifted	O
copy	O
of	O
itself	O
and	O
thereby	O
enhances	O
its	O
latent	O
periodicities	O
.	O
</s>
<s>
The	O
theory	O
of	O
the	O
triple	B-Algorithm
correlation	I-Algorithm
was	O
first	O
investigated	O
by	O
statisticians	O
examining	O
the	O
cumulant	O
structure	O
of	O
non-Gaussian	O
random	O
processes	O
.	O
</s>
<s>
Hideya	O
Gamo	O
in	O
1963	O
described	O
an	O
apparatus	O
for	O
measuring	O
the	O
triple	B-Algorithm
correlation	I-Algorithm
of	O
a	O
laser	O
beam	O
,	O
and	O
also	O
showed	O
how	O
phase	O
information	O
can	O
be	O
recovered	O
from	O
the	O
real	O
part	O
of	O
the	O
bispectrum	O
—	O
up	O
to	O
sign	O
reversal	O
and	O
linear	O
offset	O
.	O
</s>
<s>
Yellott	O
&	O
Iverson	O
also	O
pointed	O
out	O
the	O
connection	O
between	O
triple	B-Algorithm
correlations	I-Algorithm
and	O
the	O
visual	O
texture	O
discrimination	O
theory	O
proposed	O
by	O
Bela	O
Julesz	O
.	O
</s>
<s>
Triple	B-Algorithm
correlation	I-Algorithm
methods	O
are	O
frequently	O
used	O
in	O
signal	O
processing	O
for	O
treating	O
signals	O
that	O
are	O
corrupted	O
by	O
additive	O
white	O
Gaussian	O
noise	O
;	O
in	O
particular	O
,	O
triple	B-Algorithm
correlation	I-Algorithm
techniques	O
are	O
suitable	O
when	O
multiple	O
observations	O
of	O
the	O
signal	O
are	O
available	O
and	O
the	O
signal	O
may	O
be	O
translating	O
in	O
between	O
the	O
observations	O
,	O
e.g.	O
,	O
a	O
sequence	O
of	O
images	O
of	O
an	O
object	O
translating	O
on	O
a	O
noisy	O
background	O
.	O
</s>
<s>
What	O
makes	O
the	O
triple	B-Algorithm
correlation	I-Algorithm
particularly	O
useful	O
for	O
such	O
tasks	O
are	O
three	O
properties	O
:	O
(	O
1	O
)	O
it	O
is	O
invariant	O
under	O
translation	O
of	O
the	O
underlying	O
signal	O
;	O
(	O
2	O
)	O
it	O
is	O
unbiased	O
in	O
additive	O
Gaussian	O
noise	O
;	O
and	O
(	O
3	O
)	O
it	O
retains	O
nearly	O
all	O
of	O
the	O
relevant	O
phase	O
information	O
in	O
the	O
underlying	O
signal	O
.	O
</s>
<s>
Properties	O
(	O
1	O
)	O
-(3 )	O
of	O
the	O
triple	B-Algorithm
correlation	I-Algorithm
extend	O
in	O
many	O
cases	O
to	O
functions	O
on	O
an	O
arbitrary	O
locally	O
compact	O
group	O
,	O
in	O
particular	O
to	O
the	O
groups	O
of	O
rotations	O
and	O
rigid	O
motions	O
of	O
euclidean	O
space	O
that	O
arise	O
in	O
computer	O
vision	O
and	O
signal	O
processing	O
.	O
</s>
<s>
The	O
triple	B-Algorithm
correlation	I-Algorithm
may	O
be	O
defined	O
for	O
any	O
locally	O
compact	O
group	O
by	O
using	O
the	O
group	O
's	O
left-invariant	O
Haar	O
measure	O
.	O
</s>
<s>
What	O
is	O
more	O
interesting	O
is	O
the	O
question	O
of	O
uniqueness	O
:	O
when	O
two	O
functions	O
have	O
the	O
same	O
triple	B-Algorithm
correlation	I-Algorithm
,	O
how	O
are	O
the	O
functions	O
related	O
?	O
</s>
<s>
For	O
many	O
cases	O
of	O
practical	O
interest	O
,	O
the	O
triple	B-Algorithm
correlation	I-Algorithm
of	O
a	O
function	O
on	O
an	O
abstract	O
group	O
uniquely	O
identifies	O
that	O
function	O
up	O
to	O
a	O
single	O
unknown	O
group	O
action	O
.	O
</s>
<s>
Algorithms	O
exist	O
for	O
recovering	O
bandlimited	O
functions	O
from	O
their	O
triple	B-Algorithm
correlation	I-Algorithm
on	O
Euclidean	O
space	O
,	O
as	O
well	O
as	O
rotation	O
groups	O
in	O
two	O
and	O
three	O
dimensions	O
.	O
</s>
<s>
There	O
is	O
also	O
an	O
interesting	O
link	O
with	O
Wiener	O
's	O
tauberian	O
theorem	O
:	O
any	O
function	O
whose	O
translates	O
are	O
dense	O
in	O
,	O
where	O
is	O
a	O
locally	O
compact	O
Abelian	O
group	O
,	O
is	O
also	O
uniquely	O
identified	O
by	O
its	O
triple	B-Algorithm
correlation	I-Algorithm
.	O
</s>
