<s>
In	O
computer	B-Application
vision	I-Application
,	O
triangulation	B-Algorithm
refers	O
to	O
the	O
process	O
of	O
determining	O
a	O
point	O
in	O
3D	O
space	O
given	O
its	O
projections	O
onto	O
two	O
,	O
or	O
more	O
,	O
images	O
.	O
</s>
<s>
In	O
order	O
to	O
solve	O
this	O
problem	O
it	O
is	O
necessary	O
to	O
know	O
the	O
parameters	O
of	O
the	O
camera	O
projection	O
function	O
from	O
3D	O
to	O
2D	O
for	O
the	O
cameras	O
involved	O
,	O
in	O
the	O
simplest	O
case	O
represented	O
by	O
the	O
camera	B-General_Concept
matrices	I-General_Concept
.	O
</s>
<s>
Triangulation	B-Algorithm
is	O
sometimes	O
also	O
referred	O
to	O
as	O
reconstruction	O
or	O
intersection	O
.	O
</s>
<s>
The	O
triangulation	B-Algorithm
problem	O
is	O
in	O
principle	O
trivial	O
.	O
</s>
<s>
If	O
a	O
pair	O
of	O
corresponding	B-Algorithm
points	I-Algorithm
in	O
two	O
,	O
or	O
more	O
images	O
,	O
can	O
be	O
found	O
it	O
must	O
be	O
the	O
case	O
that	O
they	O
are	O
the	O
projection	O
of	O
a	O
common	O
3D	O
point	O
x	O
.	O
</s>
<s>
In	O
the	O
following	O
,	O
it	O
is	O
assumed	O
that	O
triangulation	B-Algorithm
is	O
made	O
on	O
corresponding	O
image	O
points	O
from	O
two	O
views	O
generated	O
by	O
pinhole	B-Algorithm
cameras	I-Algorithm
.	O
</s>
<s>
The	O
image	O
to	O
the	O
left	O
illustrates	O
the	O
epipolar	B-Algorithm
geometry	I-Algorithm
of	O
a	O
pair	O
of	O
stereo	O
cameras	O
of	O
pinhole	B-Algorithm
model	I-Algorithm
.	O
</s>
<s>
Using	O
basic	O
linear	B-Language
algebra	I-Language
that	O
intersection	O
point	O
can	O
be	O
determined	O
in	O
a	O
straightforward	O
way	O
.	O
</s>
<s>
Geometric	O
distortion	O
,	O
for	O
example	O
lens	O
distortion	O
,	O
which	O
means	O
that	O
the	O
3D	O
to	O
2D	O
mapping	O
of	O
the	O
camera	O
deviates	O
from	O
the	O
pinhole	B-Algorithm
camera	I-Algorithm
model	I-Algorithm
.	O
</s>
<s>
The	O
image	O
points	O
y1	O
 '	O
and	O
y2	O
 '	O
used	O
for	O
triangulation	B-Algorithm
are	O
often	O
found	O
using	O
various	O
types	O
of	O
feature	O
extractors	O
,	O
for	O
example	O
of	O
corners	O
or	O
interest	O
points	O
in	O
general	O
.	O
</s>
<s>
There	O
is	O
an	O
inherent	O
localization	O
error	O
for	O
any	O
type	O
of	O
feature	O
extraction	O
based	O
on	O
neighborhood	B-Algorithm
operations	I-Algorithm
.	O
</s>
<s>
In	O
fact	O
,	O
these	O
lines	O
intersect	O
if	O
and	O
only	O
if	O
and	O
satisfy	O
the	O
epipolar	B-Algorithm
constraint	I-Algorithm
defined	O
by	O
the	O
fundamental	B-General_Concept
matrix	I-General_Concept
.	O
</s>
<s>
Given	O
the	O
measurement	O
noise	O
in	O
and	O
it	O
is	O
rather	O
likely	O
that	O
the	O
epipolar	B-Algorithm
constraint	I-Algorithm
is	O
not	O
satisfied	O
and	O
the	O
projection	O
lines	O
do	O
not	O
intersect	O
.	O
</s>
<s>
This	O
observation	O
leads	O
to	O
the	O
problem	O
which	O
is	O
solved	O
in	O
triangulation	B-Algorithm
.	O
</s>
<s>
All	O
triangulation	B-Algorithm
methods	O
produce	O
xest	O
=	O
x	O
in	O
the	O
case	O
that	O
and	O
,	O
that	O
is	O
,	O
when	O
the	O
epipolar	B-Algorithm
constraint	I-Algorithm
is	O
satisfied	O
(	O
except	O
for	O
singular	O
points	O
,	O
see	O
below	O
)	O
.	O
</s>
<s>
where	O
are	O
the	O
homogeneous	O
coordinates	O
of	O
the	O
detected	O
image	O
points	O
and	O
are	O
the	O
camera	B-General_Concept
matrices	I-General_Concept
.	O
</s>
<s>
Which	O
triangulation	B-Algorithm
method	O
is	O
chosen	O
for	O
a	O
particular	O
problem	O
depends	O
to	O
some	O
extent	O
on	O
these	O
characteristics	O
.	O
</s>
<s>
A	O
point	O
in	O
this	O
subset	O
is	O
then	O
a	O
singularity	O
of	O
the	O
triangulation	B-Algorithm
method	O
.	O
</s>
<s>
In	O
some	O
applications	O
,	O
it	O
is	O
desirable	O
that	O
the	O
triangulation	B-Algorithm
is	O
independent	O
of	O
the	O
coordinate	O
system	O
used	O
to	O
represent	O
3D	O
points	O
;	O
if	O
the	O
triangulation	B-Algorithm
problem	O
is	O
formulated	O
in	O
one	O
coordinate	O
system	O
and	O
then	O
transformed	O
into	O
another	O
the	O
resulting	O
estimate	O
xest	O
should	O
transform	O
in	O
the	O
same	O
way	O
.	O
</s>
<s>
Not	O
every	O
triangulation	B-Algorithm
method	O
assures	O
invariance	O
,	O
at	O
least	O
not	O
for	O
general	O
types	O
of	O
coordinate	O
transformations	O
.	O
</s>
<s>
For	O
each	O
triangulation	B-Algorithm
method	O
,	O
it	O
can	O
be	O
determined	O
if	O
this	O
last	O
relation	O
is	O
valid	O
.	O
</s>
<s>
Each	O
of	O
the	O
two	O
image	O
points	O
and	O
has	O
a	O
corresponding	O
projection	O
line	O
(	O
blue	O
in	O
the	O
right	O
image	O
above	O
)	O
,	O
here	O
denoted	O
as	O
and	O
,	O
which	O
can	O
be	O
determined	O
given	O
the	O
camera	B-General_Concept
matrices	I-General_Concept
.	O
</s>
<s>
If	O
the	O
essential	B-General_Concept
matrix	I-General_Concept
is	O
known	O
and	O
the	O
corresponding	O
rotation	O
and	O
translation	O
transformations	O
have	O
been	O
determined	O
,	O
this	O
algorithm	O
(	O
described	O
in	O
Longuet-Higgins	O
'	O
paper	O
)	O
provides	O
a	O
solution	O
.	O
</s>
<s>
A	O
final	O
remark	O
relates	O
to	O
the	O
fact	O
that	O
if	O
the	O
essential	B-General_Concept
matrix	I-General_Concept
is	O
determined	O
from	O
corresponding	O
image	O
coordinate	O
,	O
which	O
often	O
is	O
the	O
case	O
when	O
3D	O
points	O
are	O
determined	O
in	O
this	O
way	O
,	O
the	O
translation	O
vector	O
is	O
known	O
only	O
up	O
to	O
an	O
unknown	O
positive	O
scaling	O
.	O
</s>
