<s>
Image	B-Algorithm
rectification	I-Algorithm
is	O
a	O
transformation	O
process	O
used	O
to	O
project	O
images	O
onto	O
a	O
common	O
image	O
plane	O
.	O
</s>
<s>
Image	B-Algorithm
rectification	I-Algorithm
is	O
used	O
in	O
computer	B-Algorithm
stereo	I-Algorithm
vision	I-Algorithm
to	O
simplify	O
the	O
problem	O
of	O
finding	O
matching	O
points	O
between	O
images	O
(	O
i.e.	O
</s>
<s>
the	O
correspondence	B-Algorithm
problem	I-Algorithm
)	O
,	O
and	O
in	O
geographic	B-Application
information	I-Application
systems	I-Application
to	O
merge	O
images	O
taken	O
from	O
multiple	O
perspectives	O
into	O
a	O
common	O
map	O
coordinate	O
system	O
.	O
</s>
<s>
Computer	B-Algorithm
stereo	I-Algorithm
vision	I-Algorithm
takes	O
two	O
or	O
more	O
images	O
with	O
known	O
relative	O
camera	O
positions	O
that	O
show	O
an	O
object	O
from	O
different	O
viewpoints	O
.	O
</s>
<s>
pixels	O
showing	O
the	O
same	O
scene	O
point	O
)	O
in	O
the	O
other	O
image(s )	O
and	O
then	O
applying	O
triangulation	B-Algorithm
to	O
the	O
found	O
matches	O
to	O
determine	O
their	O
depth	O
.	O
</s>
<s>
Finding	O
matches	O
in	O
stereo	B-Algorithm
vision	I-Algorithm
is	O
restricted	O
by	O
epipolar	B-Algorithm
geometry	I-Algorithm
:	O
Each	O
pixel	O
's	O
match	O
in	O
another	O
image	O
can	O
only	O
be	O
found	O
on	O
a	O
line	O
called	O
the	O
epipolar	B-Algorithm
line	O
.	O
</s>
<s>
they	O
were	O
taken	O
such	O
that	O
the	O
right	O
camera	O
is	O
only	O
offset	O
horizontally	O
compared	O
to	O
the	O
left	O
camera	O
(	O
not	O
being	O
moved	O
towards	O
the	O
object	O
or	O
rotated	O
)	O
,	O
then	O
each	O
pixel	O
's	O
epipolar	B-Algorithm
line	O
is	O
horizontal	O
and	O
at	O
the	O
same	O
vertical	O
position	O
as	O
that	O
pixel	O
.	O
</s>
<s>
However	O
,	O
in	O
general	O
settings	O
(	O
the	O
camera	O
does	O
move	O
towards	O
the	O
object	O
or	O
rotate	O
)	O
the	O
epipolar	B-Algorithm
lines	O
are	O
slanted	O
.	O
</s>
<s>
Image	B-Algorithm
rectification	I-Algorithm
warps	O
both	O
images	O
such	O
that	O
they	O
appear	O
as	O
if	O
they	O
have	O
been	O
taken	O
with	O
only	O
a	O
horizontal	O
displacement	O
and	O
as	O
a	O
consequence	O
all	O
epipolar	B-Algorithm
lines	O
are	O
horizontal	O
,	O
which	O
slightly	O
simplifies	O
the	O
stereo	O
matching	O
process	O
.	O
</s>
<s>
Image	B-Algorithm
rectification	I-Algorithm
is	O
also	O
an	O
equivalent	O
(	O
and	O
more	O
often	O
used	O
)	O
alternative	O
to	O
perfect	O
camera	O
coplanarity	O
.	O
</s>
<s>
Even	O
with	O
high-precision	O
equipment	O
,	O
image	B-Algorithm
rectification	I-Algorithm
is	O
usually	O
performed	O
because	O
it	O
may	O
be	O
impractical	O
to	O
maintain	O
perfect	O
coplanarity	O
between	O
cameras	O
.	O
</s>
<s>
Image	B-Algorithm
rectification	I-Algorithm
can	O
only	O
be	O
performed	O
with	O
two	O
images	O
at	O
a	O
time	O
and	O
simultaneous	O
rectification	O
of	O
more	O
than	O
two	O
images	O
is	O
generally	O
impossible	O
.	O
</s>
<s>
If	O
the	O
images	O
to	O
be	O
rectified	O
are	O
taken	O
from	O
camera	O
pairs	O
without	O
geometric	O
distortion	O
,	O
this	O
calculation	O
can	O
easily	O
be	O
made	O
with	O
a	O
linear	B-Architecture
transformation	I-Architecture
.	O
</s>
<s>
In	O
performing	O
the	O
transform	O
,	O
if	O
the	O
cameras	O
themselves	O
are	O
calibrated	O
for	O
internal	O
parameters	O
,	O
an	O
essential	B-General_Concept
matrix	I-General_Concept
provides	O
the	O
relationship	O
between	O
the	O
cameras	O
.	O
</s>
<s>
The	O
more	O
general	O
case	O
(	O
without	O
camera	O
calibration	O
)	O
is	O
represented	O
by	O
the	O
fundamental	B-General_Concept
matrix	I-General_Concept
.	O
</s>
<s>
If	O
the	O
fundamental	B-General_Concept
matrix	I-General_Concept
is	O
not	O
known	O
,	O
it	O
is	O
necessary	O
to	O
find	O
preliminary	O
point	O
correspondences	O
between	O
stereo	O
images	O
to	O
facilitate	O
its	O
extraction	O
.	O
</s>
<s>
There	O
are	O
three	O
main	O
categories	O
for	O
image	B-Algorithm
rectification	I-Algorithm
algorithms	O
:	O
planar	O
rectification	O
,	O
cylindrical	O
rectification	O
and	O
polar	O
rectification	O
.	O
</s>
<s>
All	O
rectified	B-Algorithm
images	I-Algorithm
satisfy	O
the	O
following	O
two	O
properties	O
:	O
</s>
<s>
All	O
epipolar	B-Algorithm
lines	O
are	O
parallel	O
to	O
the	O
horizontal	O
axis	O
.	O
</s>
<s>
Corresponding	B-Algorithm
points	I-Algorithm
have	O
identical	O
vertical	O
coordinates	O
.	O
</s>
<s>
In	O
order	O
to	O
transform	O
the	O
original	O
image	O
pair	O
into	O
a	O
rectified	B-Algorithm
image	I-Algorithm
pair	O
,	O
it	O
is	O
necessary	O
to	O
find	O
a	O
projective	B-Algorithm
transformation	I-Algorithm
H	O
.	O
Constraints	O
are	O
placed	O
on	O
H	O
to	O
satisfy	O
the	O
two	O
properties	O
above	O
.	O
</s>
<s>
For	O
example	O
,	O
constraining	O
the	O
epipolar	B-Algorithm
lines	O
to	O
be	O
parallel	O
with	O
the	O
horizontal	O
axis	O
means	O
that	O
epipoles	O
must	O
be	O
mapped	O
to	O
the	O
infinite	O
point	O
[1,0,0]T	O
in	O
homogeneous	O
coordinates	O
.	O
</s>
<s>
Poor	O
choices	O
of	O
H	O
and	O
H	O
 '	O
can	O
result	O
in	O
rectified	B-Algorithm
images	I-Algorithm
that	O
are	O
dramatically	O
changed	O
in	O
scale	O
or	O
severely	O
distorted	O
.	O
</s>
<s>
There	O
are	O
many	O
different	O
strategies	O
for	O
choosing	O
a	O
projective	B-Algorithm
transform	I-Algorithm
H	O
for	O
each	O
image	O
from	O
all	O
possible	O
solutions	O
.	O
</s>
<s>
One	O
advanced	O
method	O
is	O
minimizing	O
the	O
disparity	O
or	O
least-square	O
difference	O
of	O
corresponding	B-Algorithm
points	I-Algorithm
on	O
the	O
horizontal	O
axis	O
of	O
the	O
rectified	B-Algorithm
image	I-Algorithm
pair	O
.	O
</s>
<s>
Another	O
method	O
is	O
separating	O
H	O
into	O
a	O
specialized	O
projective	B-Algorithm
transform	I-Algorithm
,	O
similarity	O
transform	O
,	O
and	O
shearing	O
transform	O
to	O
minimize	O
image	O
distortion	O
.	O
</s>
<s>
We	O
will	O
briefly	O
outline	O
and	O
depict	O
the	O
results	O
for	O
a	O
simple	O
approach	O
to	O
find	O
a	O
H	O
and	O
H	O
 '	O
projective	B-Algorithm
transformation	I-Algorithm
that	O
rectify	O
the	O
image	O
pair	O
from	O
the	O
example	O
scene	O
.	O
</s>
<s>
Second	O
,	O
we	O
find	O
a	O
projective	B-Algorithm
transformation	I-Algorithm
H1	O
that	O
rotates	O
our	O
first	O
image	O
to	O
be	O
parallel	O
to	O
the	O
baseline	O
connecting	O
O	O
and	O
O	O
 '	O
(	O
row	O
2	O
,	O
column	O
1	O
of	O
2D	O
image	O
set	O
)	O
.	O
</s>
<s>
Next	O
,	O
we	O
find	O
the	O
projective	B-Algorithm
transformation	I-Algorithm
H2	O
that	O
takes	O
the	O
rotated	O
image	O
and	O
twists	O
it	O
so	O
that	O
the	O
horizontal	O
axis	O
aligns	O
with	O
the	O
baseline	O
.	O
</s>
<s>
Finally	O
,	O
define	O
as	O
the	O
projective	B-Algorithm
transformation	I-Algorithm
for	O
rectifying	O
the	O
first	O
image	O
.	O
</s>
<s>
In	O
this	O
example	O
,	O
we	O
simply	O
define	O
H	O
 '	O
using	O
the	O
rotation	O
matrix	O
R	O
and	O
initial	O
projective	B-Algorithm
transformation	I-Algorithm
H	O
as	O
.	O
</s>
<s>
Image	B-Algorithm
rectification	I-Algorithm
in	O
GIS	B-Application
converts	O
images	O
to	O
a	O
standard	O
map	O
coordinate	O
system	O
.	O
</s>
<s>
The	O
maps	O
that	O
are	O
used	O
with	O
rectified	B-Algorithm
images	I-Algorithm
are	O
non-topographical	O
.	O
</s>
<s>
Image	B-Algorithm
rectification	I-Algorithm
is	O
a	O
standard	O
feature	O
available	O
with	O
GIS	B-Application
software	I-Application
packages	O
.	O
</s>
