<s>
The	O
eight-point	B-General_Concept
algorithm	I-General_Concept
is	O
an	O
algorithm	O
used	O
in	O
computer	B-Application
vision	I-Application
to	O
estimate	O
the	O
essential	B-General_Concept
matrix	I-General_Concept
or	O
the	O
fundamental	B-General_Concept
matrix	I-General_Concept
related	O
to	O
a	O
stereo	O
camera	O
pair	O
from	O
a	O
set	O
of	O
corresponding	O
image	O
points	O
.	O
</s>
<s>
It	O
was	O
introduced	O
by	O
Christopher	O
Longuet-Higgins	O
in	O
1981	O
for	O
the	O
case	O
of	O
the	O
essential	B-General_Concept
matrix	I-General_Concept
.	O
</s>
<s>
In	O
theory	O
,	O
this	O
algorithm	O
can	O
be	O
used	O
also	O
for	O
the	O
fundamental	B-General_Concept
matrix	I-General_Concept
,	O
but	O
in	O
practice	O
the	O
normalized	O
eight-point	B-General_Concept
algorithm	I-General_Concept
,	O
described	O
by	O
Richard	O
Hartley	O
in	O
1997	O
,	O
is	O
better	O
suited	O
for	O
this	O
case	O
.	O
</s>
<s>
The	O
algorithm	O
's	O
name	O
derives	O
from	O
the	O
fact	O
that	O
it	O
estimates	O
the	O
essential	B-General_Concept
matrix	I-General_Concept
or	O
the	O
fundamental	B-General_Concept
matrix	I-General_Concept
from	O
a	O
set	O
of	O
eight	O
(	O
or	O
more	O
)	O
corresponding	O
image	O
points	O
.	O
</s>
<s>
One	O
may	O
express	O
the	O
epipolar	B-Algorithm
geometry	I-Algorithm
of	O
two	O
cameras	O
and	O
a	O
point	O
in	O
space	O
with	O
an	O
algebraic	O
equation	O
.	O
</s>
<s>
The	O
product	O
is	O
often	O
called	O
essential	B-General_Concept
matrix	I-General_Concept
and	O
denoted	O
with	O
.	O
</s>
<s>
The	O
basic	O
eight-point	B-General_Concept
algorithm	I-General_Concept
is	O
here	O
described	O
for	O
the	O
case	O
of	O
estimating	O
the	O
essential	B-General_Concept
matrix	I-General_Concept
.	O
</s>
<s>
that	O
is	O
,	O
represents	O
the	O
essential	B-General_Concept
matrix	I-General_Concept
in	O
the	O
form	O
of	O
a	O
9-dimensional	O
vector	O
and	O
this	O
vector	O
must	O
be	O
orthogonal	O
to	O
the	O
vector	O
which	O
can	O
be	O
seen	O
as	O
a	O
vector	O
representation	O
of	O
the	O
matrix	O
.	O
</s>
<s>
Another	O
consequence	O
of	O
dealing	O
with	O
noisy	O
image	O
coordinates	O
is	O
that	O
the	O
resulting	O
matrix	O
may	O
not	O
satisfy	O
the	O
internal	O
constraint	O
of	O
the	O
essential	B-General_Concept
matrix	I-General_Concept
,	O
that	O
is	O
,	O
two	O
of	O
its	O
singular	O
values	O
are	O
equal	O
and	O
nonzero	O
and	O
the	O
other	O
is	O
zero	O
.	O
</s>
<s>
The	O
matrix	O
is	O
the	O
resulting	O
estimate	O
of	O
the	O
essential	B-General_Concept
matrix	I-General_Concept
provided	O
by	O
the	O
algorithm	O
.	O
</s>
<s>
The	O
basic	O
eight-point	B-General_Concept
algorithm	I-General_Concept
can	O
in	O
principle	O
be	O
used	O
also	O
for	O
estimating	O
the	O
fundamental	B-General_Concept
matrix	I-General_Concept
.	O
</s>
<s>
In	O
practice	O
,	O
however	O
,	O
the	O
resulting	O
fundamental	B-General_Concept
matrix	I-General_Concept
may	O
not	O
be	O
useful	O
for	O
determining	O
epipolar	B-Algorithm
constraints	I-Algorithm
.	O
</s>
<s>
The	O
problem	O
is	O
that	O
the	O
resulting	O
often	O
is	O
ill-conditioned	B-Algorithm
.	O
</s>
<s>
This	O
means	O
that	O
it	O
is	O
possible	O
to	O
use	O
the	O
normalized	O
homogeneous	O
image	O
coordinates	O
to	O
estimate	O
the	O
transformed	O
fundamental	B-General_Concept
matrix	I-General_Concept
using	O
the	O
basic	O
eight-point	B-General_Concept
algorithm	I-General_Concept
described	O
above	O
.	O
</s>
<s>
The	O
purpose	O
of	O
the	O
normalization	O
transformations	O
is	O
that	O
the	O
matrix	O
,	O
constructed	O
from	O
the	O
normalized	O
image	O
coordinates	O
,	O
in	O
general	O
,	O
has	O
a	O
better	O
condition	B-Algorithm
number	I-Algorithm
than	O
has	O
.	O
</s>
<s>
In	O
general	O
,	O
this	O
estimate	O
of	O
the	O
fundamental	B-General_Concept
matrix	I-General_Concept
is	O
a	O
better	O
one	O
than	O
would	O
have	O
been	O
obtained	O
by	O
estimating	O
from	O
the	O
un-normalized	O
coordinates	O
.	O
</s>
<s>
David	O
Nister	O
proposed	O
an	O
efficient	O
solution	O
to	O
estimate	O
the	O
essential	B-General_Concept
matrix	I-General_Concept
from	O
set	O
of	O
five	O
paired	O
points	O
,	O
known	O
as	O
the	O
five-point	O
algorithm	O
.	O
</s>
