<s>
The	O
intrinsic	B-Algorithm
dimension	I-Algorithm
for	O
a	O
data	O
set	O
can	O
be	O
thought	O
of	O
as	O
the	O
number	O
of	O
variables	O
needed	O
in	O
a	O
minimal	O
representation	O
of	O
the	O
data	O
.	O
</s>
<s>
Similarly	O
,	O
in	O
signal	O
processing	O
of	O
multidimensional	O
signals	O
,	O
the	O
intrinsic	B-Algorithm
dimension	I-Algorithm
of	O
the	O
signal	O
describes	O
how	O
many	O
variables	O
are	O
needed	O
to	O
generate	O
a	O
good	O
approximation	O
of	O
the	O
signal	O
.	O
</s>
<s>
When	O
estimating	O
intrinsic	B-Algorithm
dimension	I-Algorithm
,	O
however	O
,	O
a	O
slightly	O
broader	O
definition	O
based	O
on	O
manifold	O
dimension	O
is	O
often	O
used	O
,	O
where	O
a	O
representation	O
in	O
the	O
intrinsic	B-Algorithm
dimension	I-Algorithm
does	O
only	O
need	O
to	O
exist	O
locally	O
.	O
</s>
<s>
Such	O
intrinsic	B-Algorithm
dimension	I-Algorithm
estimation	O
methods	O
can	O
thus	O
handle	O
data	O
sets	O
with	O
different	O
intrinsic	B-Algorithm
dimensions	I-Algorithm
in	O
different	O
parts	O
of	O
the	O
data	O
set	O
.	O
</s>
<s>
The	O
intrinsic	B-Algorithm
dimension	I-Algorithm
can	O
be	O
used	O
as	O
a	O
lower	O
bound	O
of	O
what	O
dimension	O
it	O
is	O
possible	O
to	O
compress	O
a	O
data	O
set	O
into	O
through	O
dimension	O
reduction	O
,	O
but	O
it	O
can	O
also	O
be	O
used	O
as	O
a	O
measure	O
of	O
the	O
complexity	O
of	O
the	O
data	O
set	O
or	O
signal	O
.	O
</s>
<s>
For	O
a	O
data	O
set	O
or	O
signal	O
of	O
N	O
variables	O
,	O
its	O
intrinsic	B-Algorithm
dimension	I-Algorithm
M	O
satisfies	O
0	O
≤	O
M	O
≤	O
N	O
,	O
although	O
estimators	O
may	O
yield	O
higher	O
values	O
.	O
</s>
<s>
It	O
is	O
only	O
necessary	O
to	O
know	O
the	O
value	O
of	O
one	O
,	O
namely	O
the	O
first	O
,	O
variable	O
in	O
order	O
to	O
determine	O
the	O
value	O
of	O
f	O
.	O
Hence	O
,	O
it	O
is	O
a	O
two-variable	O
function	O
but	O
its	O
intrinsic	B-Algorithm
dimension	I-Algorithm
is	O
one	O
.	O
</s>
<s>
Since	O
the	O
variation	O
in	O
f	O
can	O
be	O
described	O
by	O
the	O
single	O
variable	O
y1	O
its	O
intrinsic	B-Algorithm
dimension	I-Algorithm
is	O
one	O
.	O
</s>
<s>
For	O
the	O
case	O
that	O
f	O
is	O
constant	O
,	O
its	O
intrinsic	B-Algorithm
dimension	I-Algorithm
is	O
zero	O
since	O
no	O
variable	O
is	O
needed	O
to	O
describe	O
variation	O
.	O
</s>
<s>
For	O
the	O
general	O
case	O
,	O
when	O
the	O
intrinsic	B-Algorithm
dimension	I-Algorithm
of	O
the	O
two-variable	O
function	O
f	O
is	O
neither	O
zero	O
or	O
one	O
,	O
it	O
is	O
two	O
.	O
</s>
<s>
In	O
the	O
literature	O
,	O
functions	O
which	O
are	O
of	O
intrinsic	B-Algorithm
dimension	I-Algorithm
zero	O
,	O
one	O
,	O
or	O
two	O
are	O
sometimes	O
referred	O
to	O
as	O
i0D	O
,	O
i1D	O
or	O
i2D	O
,	O
respectively	O
.	O
</s>
<s>
then	O
the	O
intrinsic	B-Algorithm
dimension	I-Algorithm
of	O
f	O
is	O
M	O
.	O
</s>
<s>
The	O
intrinsic	B-Algorithm
dimension	I-Algorithm
is	O
a	O
characterization	O
of	O
f	O
,	O
it	O
is	O
not	O
an	O
unambiguous	O
characterization	O
of	O
g	O
nor	O
of	O
A	O
.	O
</s>
<s>
An	O
N	O
variable	O
function	O
which	O
has	O
intrinsic	B-Algorithm
dimension	I-Algorithm
M	O
<	O
N	O
has	O
a	O
characteristic	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
Intuitively	O
,	O
since	O
this	O
type	O
of	O
function	O
is	O
constant	O
along	O
one	O
or	O
several	O
dimensions	O
its	O
Fourier	B-Algorithm
transform	I-Algorithm
must	O
appear	O
like	O
an	O
impulse	O
(	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
a	O
constant	O
)	O
along	O
the	O
same	O
dimension	O
in	O
the	O
frequency	O
domain	O
.	O
</s>
<s>
Here	O
G	O
is	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
g	O
(	O
both	O
are	O
one-variable	O
functions	O
)	O
,	O
δ	O
is	O
the	O
Dirac	O
impulse	O
function	O
and	O
m	O
is	O
a	O
normalized	O
vector	O
in	O
perpendicular	O
to	O
n	O
.	O
This	O
means	O
that	O
F	O
vanishes	O
everywhere	O
except	O
on	O
a	O
line	O
which	O
passes	O
through	O
the	O
origin	O
of	O
the	O
frequency	O
domain	O
and	O
is	O
parallel	O
to	O
m	O
.	O
Along	O
this	O
line	O
F	O
varies	O
according	O
to	O
G	O
.	O
</s>
<s>
Its	O
Fourier	B-Algorithm
transform	I-Algorithm
F	O
can	O
then	O
be	O
described	O
as	O
follows	O
:	O
</s>
<s>
The	O
type	O
of	O
intrinsic	B-Algorithm
dimension	I-Algorithm
described	O
above	O
assumes	O
that	O
a	O
linear	B-Architecture
transformation	I-Architecture
is	O
applied	O
to	O
the	O
coordinates	O
of	O
the	O
N-variable	O
function	O
f	O
to	O
produce	O
the	O
M	O
variables	O
which	O
are	O
necessary	O
to	O
represent	O
every	O
value	O
of	O
f	O
.	O
This	O
means	O
that	O
f	O
is	O
constant	O
along	O
lines	O
,	O
planes	O
,	O
or	O
hyperplanes	O
,	O
depending	O
on	O
N	O
and	O
M	O
.	O
</s>
<s>
For	O
the	O
general	O
case	O
,	O
a	O
simple	O
description	O
of	O
either	O
the	O
point	O
sets	O
for	O
which	O
f	O
is	O
constant	O
or	O
its	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
usually	O
not	O
possible	O
.	O
</s>
<s>
After	O
Shepard	O
introduced	O
non-metric	O
multidimensional	O
scaling	O
in	O
1962	O
one	O
of	O
the	O
major	O
research	O
areas	O
within	O
multi-dimensional	O
scaling	O
(	O
MDS	O
)	O
was	O
estimation	O
of	O
the	O
intrinsic	B-Algorithm
dimension	I-Algorithm
.	O
</s>
<s>
The	O
topic	O
was	O
also	O
studied	O
in	O
information	O
theory	O
,	O
pioneered	O
by	O
Bennet	O
in	O
1965	O
who	O
coined	O
the	O
term	O
"	O
intrinsic	B-Algorithm
dimension	I-Algorithm
"	O
and	O
wrote	O
a	O
computer	O
program	O
to	O
estimate	O
it	O
.	O
</s>
<s>
Estimating	O
intrinsic	B-Algorithm
dimension	I-Algorithm
of	O
sets	O
and	O
probability	O
measures	O
has	O
also	O
been	O
extensively	O
studied	O
since	O
around	O
1980	O
in	O
the	O
field	O
of	O
dynamical	O
systems	O
,	O
where	O
dimensions	O
of	O
(	O
strange	O
)	O
attractors	O
have	O
been	O
the	O
subject	O
of	O
interest	O
.	O
</s>
<s>
In	O
the	O
2000s	O
the	O
"	O
curse	O
of	O
dimensionality	O
"	O
has	O
been	O
exploited	O
to	O
estimate	O
intrinsic	B-Algorithm
dimension	I-Algorithm
.	O
</s>
<s>
The	O
case	O
of	O
a	O
two-variable	O
signal	O
which	O
is	O
i1D	O
appears	O
frequently	O
in	O
computer	B-Application
vision	I-Application
and	O
image	B-Algorithm
processing	I-Algorithm
and	O
captures	O
the	O
idea	O
of	O
local	O
image	O
regions	O
which	O
contain	O
lines	O
or	O
edges	O
.	O
</s>
<s>
The	O
analysis	O
of	O
such	O
regions	O
has	O
a	O
long	O
history	O
,	O
but	O
it	O
was	O
not	O
until	O
a	O
more	O
formal	O
and	O
theoretical	O
treatment	O
of	O
such	O
operations	O
began	O
that	O
the	O
concept	O
of	O
intrinsic	B-Algorithm
dimension	I-Algorithm
was	O
established	O
,	O
even	O
though	O
the	O
name	O
has	O
varied	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
concept	O
which	O
here	O
is	O
referred	O
to	O
as	O
an	O
image	O
neighborhood	O
of	O
intrinsic	B-Algorithm
dimension	I-Algorithm
1	O
or	O
i1D	O
neighborhood	O
is	O
called	O
1-dimensional	O
by	O
Knutsson	O
(	O
1982	O
)	O
,	O
linear	O
symmetric	O
by	O
Bigün	O
&	O
Granlund	O
(	O
1987	O
)	O
and	O
simple	O
neighborhood	O
in	O
Granlund	O
&	O
Knutsson	O
(	O
1995	O
)	O
.	O
</s>
