<s>
Epipolar	B-Algorithm
geometry	I-Algorithm
is	O
the	O
geometry	O
of	O
stereo	O
vision	O
.	O
</s>
<s>
These	O
relations	O
are	O
derived	O
based	O
on	O
the	O
assumption	O
that	O
the	O
cameras	O
can	O
be	O
approximated	O
by	O
the	O
pinhole	B-Algorithm
camera	I-Algorithm
model	I-Algorithm
.	O
</s>
<s>
This	O
conversion	O
from	O
3D	O
to	O
2D	O
is	O
referred	O
to	O
as	O
a	O
perspective	B-Architecture
projection	I-Architecture
and	O
is	O
described	O
by	O
the	O
pinhole	B-Algorithm
camera	I-Algorithm
model	I-Algorithm
.	O
</s>
<s>
These	O
two	O
image	O
points	O
,	O
denoted	O
by	O
eL	O
and	O
eR	O
,	O
are	O
called	O
epipoles	O
or	O
epipolar	B-Algorithm
points	O
.	O
</s>
<s>
That	O
line	O
(	O
eR	O
–	O
xR	O
)	O
in	O
the	O
right	O
camera	O
is	O
called	O
an	O
epipolar	B-Algorithm
line	O
.	O
</s>
<s>
Symmetrically	O
,	O
the	O
line	O
OR	O
–	O
X	O
is	O
seen	O
by	O
the	O
right	O
camera	O
as	O
a	O
point	O
and	O
is	O
seen	O
as	O
epipolar	B-Algorithm
line	O
eL	O
–	O
xLby	O
the	O
left	O
camera	O
.	O
</s>
<s>
An	O
epipolar	B-Algorithm
line	O
is	O
a	O
function	O
of	O
the	O
position	O
of	O
point	O
X	O
in	O
the	O
3D	O
space	O
,	O
i.e.	O
</s>
<s>
as	O
X	O
varies	O
,	O
a	O
set	O
of	O
epipolar	B-Algorithm
lines	O
is	O
generated	O
in	O
both	O
images	O
.	O
</s>
<s>
OL	O
–	O
X	O
passes	O
through	O
the	O
optical	O
center	O
of	O
the	O
lens	O
OL	O
,	O
the	O
corresponding	O
epipolar	B-Algorithm
line	O
in	O
the	O
right	O
image	O
must	O
pass	O
through	O
the	O
epipole	O
eR	O
(	O
and	O
correspondingly	O
for	O
epipolar	B-Algorithm
lines	O
in	O
the	O
left	O
image	O
)	O
.	O
</s>
<s>
All	O
epipolar	B-Algorithm
lines	O
in	O
one	O
image	O
contain	O
the	O
epipolar	B-Algorithm
point	O
of	O
that	O
image	O
.	O
</s>
<s>
In	O
fact	O
,	O
any	O
line	O
which	O
contains	O
the	O
epipolar	B-Algorithm
point	O
is	O
an	O
epipolar	B-Algorithm
line	O
since	O
it	O
can	O
be	O
derived	O
from	O
some	O
3D	O
point	O
X	O
.	O
</s>
<s>
As	O
an	O
alternative	O
visualization	O
,	O
consider	O
the	O
points	O
X	O
,	O
OL	O
&	O
OR	O
that	O
form	O
a	O
plane	O
called	O
the	O
epipolar	B-Algorithm
plane	I-Algorithm
.	O
</s>
<s>
The	O
epipolar	B-Algorithm
plane	I-Algorithm
intersects	O
each	O
camera	O
's	O
image	O
plane	O
where	O
it	O
forms	O
lines	O
—	O
the	O
epipolar	B-Algorithm
lines	O
.	O
</s>
<s>
All	O
epipolar	B-Algorithm
planes	I-Algorithm
and	O
epipolar	B-Algorithm
lines	O
intersect	O
the	O
epipole	O
regardless	O
of	O
where	O
X	O
is	O
located	O
.	O
</s>
<s>
Assume	O
the	O
projection	O
point	O
xL	O
is	O
known	O
,	O
and	O
the	O
epipolar	B-Algorithm
line	O
eR	O
–	O
xR	O
is	O
known	O
and	O
the	O
point	O
X	O
projects	O
into	O
the	O
right	O
image	O
,	O
on	O
a	O
point	O
xR	O
which	O
must	O
lie	O
on	O
this	O
particular	O
epipolar	B-Algorithm
line	O
.	O
</s>
<s>
This	O
means	O
that	O
for	O
each	O
point	O
observed	O
in	O
one	O
image	O
the	O
same	O
point	O
must	O
be	O
observed	O
in	O
the	O
other	O
image	O
on	O
a	O
known	O
epipolar	B-Algorithm
line	O
.	O
</s>
<s>
This	O
provides	O
an	O
epipolar	B-Algorithm
constraint	I-Algorithm
:	O
the	O
projection	O
of	O
X	O
on	O
the	O
right	O
camera	O
plane	O
xR	O
must	O
be	O
contained	O
in	O
the	O
eR	O
–	O
xR	O
epipolar	B-Algorithm
line	O
.	O
</s>
<s>
It	O
means	O
that	O
it	O
is	O
possible	O
to	O
test	O
if	O
two	O
points	O
correspond	B-Algorithm
to	O
the	O
same	O
3D	O
point	O
.	O
</s>
<s>
Epipolar	B-Algorithm
constraints	I-Algorithm
can	O
also	O
be	O
described	O
by	O
the	O
essential	B-General_Concept
matrix	I-General_Concept
or	O
the	O
fundamental	B-General_Concept
matrix	I-General_Concept
between	O
the	O
two	O
cameras	O
.	O
</s>
<s>
If	O
the	O
two	O
image	O
points	O
correspond	B-Algorithm
to	O
the	O
same	O
3D	O
point	O
X	O
the	O
projection	O
lines	O
must	O
intersect	O
precisely	O
at	O
X	O
.	O
</s>
<s>
This	O
means	O
that	O
X	O
can	O
be	O
calculated	O
from	O
the	O
coordinates	O
of	O
the	O
two	O
image	O
points	O
,	O
a	O
process	O
called	O
triangulation	B-Algorithm
.	O
</s>
<s>
The	O
epipolar	B-Algorithm
geometry	I-Algorithm
is	O
simplified	O
if	O
the	O
two	O
camera	O
image	O
planes	O
coincide	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
the	O
epipolar	B-Algorithm
lines	O
also	O
coincide	O
(	O
eL	O
–	O
XL	O
=	O
eR	O
–	O
XR	O
)	O
.	O
</s>
<s>
Furthermore	O
,	O
the	O
epipolar	B-Algorithm
lines	O
are	O
parallel	O
to	O
the	O
line	O
OL	O
–	O
OR	O
between	O
the	O
centers	O
of	O
projection	O
,	O
and	O
can	O
in	O
practice	O
be	O
aligned	O
with	O
the	O
horizontal	O
axes	O
of	O
the	O
two	O
images	O
.	O
</s>
<s>
This	O
means	O
that	O
for	O
each	O
point	O
in	O
one	O
image	O
,	O
its	O
corresponding	B-Algorithm
point	I-Algorithm
in	O
the	O
other	O
image	O
can	O
be	O
found	O
by	O
looking	O
only	O
along	O
a	O
horizontal	O
line	O
.	O
</s>
<s>
This	O
process	O
is	O
called	O
image	B-Algorithm
rectification	I-Algorithm
.	O
</s>
<s>
Epipolar	B-Algorithm
geometry	I-Algorithm
of	O
this	O
sensor	O
is	O
quite	O
different	O
from	O
that	O
of	O
pinhole	B-Algorithm
projection	I-Algorithm
cameras	O
.	O
</s>
<s>
First	O
,	O
the	O
epipolar	B-Algorithm
line	O
of	O
pushbroom	O
sensor	O
is	O
not	O
straight	O
,	O
but	O
hyperbola-like	O
curve	O
.	O
</s>
<s>
Second	O
,	O
epipolar	B-Algorithm
'	O
curve	O
 '	O
pair	O
does	O
not	O
exist	O
.	O
</s>
<s>
However	O
,	O
in	O
some	O
special	O
conditions	O
,	O
the	O
epipolar	B-Algorithm
geometry	I-Algorithm
of	O
the	O
satellite	O
images	O
could	O
be	O
considered	O
as	O
a	O
linear	O
model	O
.	O
</s>
