<s>
Chessboards	O
arise	O
frequently	O
in	O
computer	B-Application
vision	I-Application
theory	O
and	O
practice	O
because	O
their	O
highly	O
structured	O
geometry	O
is	O
well-suited	O
for	O
algorithmic	O
detection	O
and	O
processing	O
.	O
</s>
<s>
The	O
appearance	O
of	O
chessboards	O
in	O
computer	B-Application
vision	I-Application
can	O
be	O
divided	O
into	O
two	O
main	O
areas	O
:	O
camera	B-Algorithm
calibration	I-Algorithm
and	O
feature	B-Algorithm
extraction	I-Algorithm
.	O
</s>
<s>
A	O
classical	O
problem	O
in	O
computer	B-Application
vision	I-Application
is	O
three-dimensional	O
(	O
3D	O
)	O
reconstruction	O
,	O
where	O
one	O
seeks	O
to	O
infer	O
3D	O
structure	O
about	O
a	O
scene	O
from	O
two-dimensional	O
(	O
2D	O
)	O
images	O
of	O
it	O
.	O
</s>
<s>
Practical	O
cameras	O
are	O
complex	O
devices	O
,	O
and	O
photogrammetry	B-Application
is	O
needed	O
to	O
model	O
the	O
relationship	O
between	O
image	O
sensor	O
measurements	O
and	O
the	O
3D	O
world	O
.	O
</s>
<s>
In	O
this	O
setting	O
,	O
camera	B-Algorithm
calibration	I-Algorithm
is	O
the	O
process	O
of	O
estimating	O
the	O
parameters	O
of	O
the	O
matrix	O
of	O
the	O
perspective	O
model	O
.	O
</s>
<s>
Camera	B-Algorithm
calibration	I-Algorithm
is	O
an	O
important	O
step	O
in	O
the	O
computer	B-Application
vision	I-Application
pipeline	O
because	O
many	O
subsequent	O
algorithms	O
require	O
knowledge	O
of	O
camera	O
parameters	O
as	O
input	O
.	O
</s>
<s>
Chessboards	O
are	O
often	O
used	O
during	O
camera	B-Algorithm
calibration	I-Algorithm
because	O
they	O
are	O
simple	O
to	O
construct	O
,	O
and	O
their	O
planar	O
grid	O
structure	O
defines	O
many	O
natural	O
interest	O
points	O
in	O
an	O
image	O
.	O
</s>
<s>
Direct	B-General_Concept
linear	I-General_Concept
transformation	I-General_Concept
(	O
DLT	O
)	O
calibration	O
uses	O
correspondences	O
between	O
world	O
points	O
and	O
camera	O
image	O
points	O
to	O
estimate	O
camera	O
parameters	O
.	O
</s>
<s>
In	O
particular	O
,	O
DLT	O
calibration	O
exploits	O
the	O
fact	O
that	O
the	O
perspective	O
pinhole	B-Algorithm
camera	I-Algorithm
model	I-Algorithm
defines	O
a	O
set	O
of	O
similarity	O
relations	O
that	O
can	O
be	O
solved	O
via	O
the	O
direct	B-General_Concept
linear	I-General_Concept
transformation	I-General_Concept
algorithm	O
.	O
</s>
<s>
A	O
common	O
way	O
to	O
achieve	O
this	O
is	O
to	O
construct	O
a	O
camera	B-Algorithm
calibration	I-Algorithm
rig	O
(	O
example	O
below	O
)	O
built	O
from	O
three	O
mutually	O
perpendicular	O
chessboards	O
.	O
</s>
<s>
However	O
,	O
the	O
practical	O
use	O
of	O
DLT	O
calibration	O
is	O
limited	O
by	O
the	O
necessity	O
of	O
a	O
3D	O
calibration	O
rig	O
and	O
the	O
fact	O
that	O
extremely	O
accurate	O
3D	O
coordinates	O
are	O
required	O
to	O
avoid	O
numerical	B-Algorithm
instability	I-Algorithm
.	O
</s>
<s>
Multiplane	B-Algorithm
calibration	I-Algorithm
is	O
a	O
variant	O
of	O
camera	B-Algorithm
auto-calibration	I-Algorithm
that	O
allows	O
one	O
to	O
compute	O
the	O
parameters	O
of	O
a	O
camera	O
from	O
two	O
or	O
more	O
views	O
of	O
a	O
planar	O
surface	O
.	O
</s>
<s>
The	O
seminal	O
work	O
in	O
multiplane	B-Algorithm
calibration	I-Algorithm
is	O
due	O
to	O
Zhang	O
.	O
</s>
<s>
The	O
following	O
figures	O
demonstrate	O
a	O
practical	O
application	O
of	O
multiplane	O
camera	B-Algorithm
calibration	I-Algorithm
from	O
multiple	O
views	O
of	O
a	O
chessboard	O
.	O
</s>
<s>
The	O
second	O
context	O
in	O
which	O
chessboards	O
arise	O
in	O
computer	B-Application
vision	I-Application
is	O
to	O
demonstrate	O
several	O
canonical	O
feature	B-Algorithm
extraction	I-Algorithm
algorithms	O
.	O
</s>
<s>
In	O
feature	B-Algorithm
extraction	I-Algorithm
,	O
one	O
seeks	O
to	O
identify	O
image	O
interest	O
points	O
,	O
which	O
summarize	O
the	O
semantic	O
content	O
of	O
an	O
image	O
and	O
,	O
hence	O
,	O
offer	O
a	O
reduced	B-Algorithm
dimensionality	I-Algorithm
representation	O
of	O
one	O
's	O
data	O
.	O
</s>
<s>
Chessboards	O
-	O
in	O
particular	O
-	O
are	O
often	O
used	O
to	O
demonstrate	O
feature	B-Algorithm
extraction	I-Algorithm
algorithms	O
because	O
their	O
regular	O
geometry	O
naturally	O
exhibits	O
local	O
image	B-Algorithm
features	I-Algorithm
like	O
edges	O
,	O
lines	O
,	O
and	O
corners	O
.	O
</s>
<s>
The	O
following	O
sections	O
demonstrate	O
the	O
application	O
of	O
common	O
feature	B-Algorithm
extraction	I-Algorithm
algorithms	O
to	O
a	O
chessboard	O
image	O
.	O
</s>
<s>
Corners	O
are	O
a	O
natural	O
local	O
image	B-Algorithm
feature	I-Algorithm
exploited	O
in	O
many	O
computer	B-Application
vision	I-Application
systems	O
.	O
</s>
<s>
A	O
variety	O
of	O
corner	B-Algorithm
detection	I-Algorithm
algorithms	O
exist	O
that	O
formalize	O
this	O
notion	O
into	O
concrete	O
algorithms	O
.	O
</s>
<s>
Corners	O
are	O
a	O
useful	O
image	B-Algorithm
feature	I-Algorithm
because	O
they	O
are	O
necessarily	O
distinct	O
from	O
their	O
neighboring	O
pixels	O
.	O
</s>
<s>
The	O
Harris	O
corner	B-Algorithm
detector	I-Algorithm
is	O
a	O
standard	O
algorithm	O
for	O
corner	B-Algorithm
detection	I-Algorithm
in	O
computer	B-Application
vision	I-Application
.	O
</s>
<s>
The	O
algorithm	O
works	O
by	O
analyzing	O
the	O
eigenvalues	O
of	O
the	O
2D	O
discrete	O
structure	B-Algorithm
tensor	I-Algorithm
matrix	O
at	O
each	O
image	O
pixel	O
and	O
flagging	O
a	O
pixel	O
as	O
a	O
corner	O
when	O
the	O
eigenvalues	O
of	O
its	O
structure	B-Algorithm
tensor	I-Algorithm
are	O
sufficiently	O
large	O
.	O
</s>
<s>
Intuitively	O
,	O
the	O
eigenvalues	O
of	O
the	O
structure	B-Algorithm
tensor	I-Algorithm
matrix	O
associated	O
with	O
a	O
given	O
pixel	O
describe	O
the	O
gradient	O
strength	O
in	O
a	O
neighborhood	O
of	O
that	O
pixel	O
.	O
</s>
<s>
As	O
such	O
,	O
a	O
structure	B-Algorithm
tensor	I-Algorithm
matrix	O
with	O
large	O
eigenvalues	O
corresponds	O
to	O
an	O
image	O
neighborhood	O
with	O
large	O
gradients	O
in	O
orthogonal	O
directions	O
-	O
i.e.	O
,	O
a	O
corner	O
.	O
</s>
<s>
A	O
chessboard	O
contains	O
natural	O
corners	O
at	O
the	O
boundaries	O
between	O
board	O
squares	O
,	O
so	O
one	O
would	O
expect	O
corner	B-Algorithm
detection	I-Algorithm
algorithms	O
to	O
successfully	O
detect	O
them	O
in	O
practice	O
.	O
</s>
<s>
Indeed	O
,	O
the	O
following	O
figure	O
demonstrates	O
Harris	O
corner	B-Algorithm
detection	I-Algorithm
applied	O
to	O
a	O
perspective-transformed	O
chessboard	O
image	O
.	O
</s>
<s>
Lines	O
are	O
another	O
natural	O
local	O
image	B-Algorithm
feature	I-Algorithm
exploited	O
in	O
many	O
computer	B-Application
vision	I-Application
systems	O
.	O
</s>
<s>
The	O
discrete	O
Hough	B-Algorithm
transform	I-Algorithm
exploits	O
this	O
idea	O
by	O
transforming	O
a	O
spatial	O
image	O
into	O
a	O
matrix	O
in	O
-space	O
whose	O
-th	O
entry	O
counts	O
the	O
number	O
of	O
image	O
edge	O
points	O
that	O
lie	O
on	O
the	O
line	O
parametrized	O
by	O
.	O
</s>
<s>
As	O
such	O
,	O
one	O
can	O
detect	O
lines	O
in	O
an	O
image	O
by	O
simply	O
searching	O
for	O
local	O
maxima	O
of	O
its	O
discrete	O
Hough	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
Clearly	O
,	O
the	O
Hough	B-Algorithm
transform	I-Algorithm
is	O
able	O
to	O
accurately	O
detect	O
the	O
lines	O
induced	O
by	O
the	O
board	O
squares	O
.	O
</s>
<s>
The	O
following	O
MATLAB	B-Language
code	O
generates	O
the	O
above	O
images	O
using	O
the	O
:	O
</s>
