<s>
In	O
computer	B-Application
vision	I-Application
a	O
camera	B-General_Concept
matrix	I-General_Concept
or	O
(	O
camera	O
)	O
projection	O
matrix	B-Architecture
is	O
a	O
matrix	B-Architecture
which	O
describes	O
the	O
mapping	O
of	O
a	O
pinhole	B-Algorithm
camera	I-Algorithm
from	O
3D	O
points	O
in	O
the	O
world	O
to	O
2D	O
points	O
in	O
an	O
image	O
.	O
</s>
<s>
Let	O
be	O
a	O
representation	O
of	O
a	O
3D	O
point	O
in	O
homogeneous	O
coordinates	O
(	O
a	O
4-dimensional	O
vector	O
)	O
,	O
and	O
let	O
be	O
a	O
representation	O
of	O
the	O
image	O
of	O
this	O
point	O
in	O
the	O
pinhole	B-Algorithm
camera	I-Algorithm
(	O
a	O
3-dimensional	O
vector	O
)	O
.	O
</s>
<s>
where	O
is	O
the	O
camera	B-General_Concept
matrix	I-General_Concept
and	O
the	O
sign	O
implies	O
that	O
the	O
left	O
and	O
right	O
hand	O
sides	O
are	O
equal	O
except	O
for	O
a	O
multiplication	O
by	O
a	O
non-zero	O
scalar	O
:	O
</s>
<s>
Since	O
the	O
camera	B-General_Concept
matrix	I-General_Concept
is	O
involved	O
in	O
the	O
mapping	O
between	O
elements	O
of	O
two	O
projective	O
spaces	O
,	O
it	O
too	O
can	O
be	O
regarded	O
as	O
a	O
projective	O
element	O
.	O
</s>
<s>
This	O
means	O
that	O
it	O
has	O
only	O
11	O
degrees	O
of	O
freedom	O
since	O
any	O
multiplication	O
by	O
a	O
non-zero	O
scalar	O
results	O
in	O
an	O
equivalent	O
camera	B-General_Concept
matrix	I-General_Concept
.	O
</s>
<s>
where	O
are	O
the	O
3D	O
coordinates	O
of	O
P	O
relative	O
to	O
a	O
camera	O
centered	O
coordinate	O
system	O
,	O
are	O
the	O
resulting	O
image	O
coordinates	O
,	O
and	O
f	O
is	O
the	O
camera	O
's	O
focal	B-Algorithm
length	I-Algorithm
for	O
which	O
we	O
assume	O
f	O
>	O
0	O
.	O
</s>
<s>
To	O
derive	O
the	O
camera	B-General_Concept
matrix	I-General_Concept
,	O
the	O
expression	O
above	O
is	O
rewritten	O
in	O
terms	O
of	O
homogeneous	O
coordinates	O
.	O
</s>
<s>
Finally	O
,	O
also	O
the	O
3D	O
coordinates	O
are	O
expressed	O
in	O
a	O
homogeneous	O
representation	O
and	O
this	O
is	O
how	O
the	O
camera	B-General_Concept
matrix	I-General_Concept
appears	O
:	O
</s>
<s>
The	O
camera	B-General_Concept
matrix	I-General_Concept
derived	O
here	O
may	O
appear	O
trivial	O
in	O
the	O
sense	O
that	O
it	O
contains	O
very	O
few	O
non-zero	O
elements	O
.	O
</s>
<s>
This	O
is	O
also	O
the	O
homogeneous	O
representation	O
of	O
the	O
3D	O
point	O
which	O
has	O
coordinates	O
(	O
0	O
,	O
0	O
,	O
0	O
)	O
,	O
that	O
is	O
,	O
the	O
"	O
camera	O
center	O
"	O
(	O
aka	O
the	O
entrance	O
pupil	O
;	O
the	O
position	O
of	O
the	O
pinhole	O
of	O
a	O
pinhole	B-Algorithm
camera	I-Algorithm
)	O
is	O
at	O
O	O
.	O
</s>
<s>
The	O
camera	B-General_Concept
matrix	I-General_Concept
derived	O
above	O
can	O
be	O
simplified	O
even	O
further	O
if	O
we	O
assume	O
that	O
f	O
=	O
1	O
:	O
</s>
<s>
where	O
here	O
denotes	O
a	O
identity	O
matrix	B-Architecture
.	O
</s>
<s>
Note	O
that	O
matrix	B-Architecture
here	O
is	O
divided	O
into	O
a	O
concatenation	O
of	O
a	O
matrix	B-Architecture
and	O
a	O
3-dimensional	O
vector	O
.	O
</s>
<s>
The	O
camera	B-General_Concept
matrix	I-General_Concept
is	O
sometimes	O
referred	O
to	O
as	O
a	O
canonical	O
form	O
.	O
</s>
<s>
So	O
far	O
all	O
points	O
in	O
the	O
3D	O
world	O
have	O
been	O
represented	O
in	O
a	O
camera	O
centered	O
coordinate	O
system	O
,	O
that	O
is	O
,	O
a	O
coordinate	O
system	O
which	O
has	O
its	O
origin	O
at	O
the	O
camera	O
center	O
(	O
the	O
location	O
of	O
the	O
pinhole	O
of	O
a	O
pinhole	B-Algorithm
camera	I-Algorithm
)	O
.	O
</s>
<s>
where	O
is	O
a	O
rotation	B-Algorithm
matrix	I-Algorithm
and	O
is	O
a	O
3-dimensional	O
translation	O
vector	O
.	O
</s>
<s>
When	O
the	O
first	O
matrix	B-Architecture
is	O
multiplied	O
onto	O
the	O
homogeneous	O
representation	O
of	O
a	O
3D	O
point	O
,	O
the	O
result	O
is	O
the	O
homogeneous	O
representation	O
of	O
the	O
rotated	O
point	O
,	O
and	O
the	O
second	O
matrix	B-Architecture
performs	O
instead	O
a	O
translation	O
.	O
</s>
<s>
a	O
concatenation	O
of	O
a	O
3D	O
rotation	B-Algorithm
matrix	I-Algorithm
and	O
a	O
3-dimensional	O
translation	O
vector	O
.	O
</s>
<s>
This	O
type	O
of	O
camera	B-General_Concept
matrix	I-General_Concept
is	O
referred	O
to	O
as	O
a	O
normalized	O
camera	B-General_Concept
matrix	I-General_Concept
,	O
it	O
assumes	O
focal	B-Algorithm
length	I-Algorithm
=	O
1	O
and	O
that	O
image	O
coordinates	O
are	O
measured	O
in	O
a	O
coordinate	O
system	O
where	O
the	O
origin	O
is	O
located	O
at	O
the	O
intersection	O
between	O
axis	O
X3	O
and	O
the	O
image	O
plane	O
and	O
has	O
the	O
same	O
units	O
as	O
the	O
3D	O
coordinate	O
system	O
.	O
</s>
<s>
This	O
implies	O
that	O
the	O
camera	O
center	O
(	O
in	O
its	O
homogeneous	O
representation	O
)	O
lies	O
in	O
the	O
null	B-Algorithm
space	I-Algorithm
of	O
the	O
camera	B-General_Concept
matrix	I-General_Concept
,	O
provided	O
that	O
it	O
is	O
represented	O
in	O
terms	O
of	O
3D	O
coordinates	O
relative	O
to	O
the	O
same	O
coordinate	O
system	O
as	O
the	O
camera	B-General_Concept
matrix	I-General_Concept
refers	O
to	O
.	O
</s>
<s>
Given	O
the	O
mapping	O
produced	O
by	O
a	O
normalized	O
camera	B-General_Concept
matrix	I-General_Concept
,	O
the	O
resulting	O
normalized	O
image	O
coordinates	O
can	O
be	O
transformed	O
by	O
means	O
of	O
an	O
arbitrary	O
2D	O
homography	B-Algorithm
.	O
</s>
<s>
Such	O
a	O
transformation	O
can	O
be	O
represented	O
as	O
a	O
matrix	B-Architecture
which	O
maps	O
the	O
homogeneous	O
normalized	O
image	O
coordinates	O
to	O
the	O
homogeneous	O
transformed	O
image	O
coordinates	O
:	O
</s>
