<s>
The	O
pinhole	B-Algorithm
camera	I-Algorithm
model	I-Algorithm
describes	O
the	O
mathematical	O
relationship	O
between	O
the	O
coordinates	O
of	O
a	O
point	O
in	O
three-dimensional	O
space	O
and	O
its	O
projection	B-Algorithm
onto	O
the	O
image	O
plane	O
of	O
an	O
ideal	O
pinhole	B-Algorithm
camera	I-Algorithm
,	O
where	O
the	O
camera	O
aperture	B-General_Concept
is	O
described	O
as	O
a	O
point	O
and	O
no	O
lenses	O
are	O
used	O
to	O
focus	O
light	O
.	O
</s>
<s>
The	O
model	O
does	O
not	O
include	O
,	O
for	O
example	O
,	O
geometric	O
distortions	O
or	O
blurring	O
of	O
unfocused	O
objects	O
caused	O
by	O
lenses	O
and	O
finite	O
sized	O
apertures	B-General_Concept
.	O
</s>
<s>
This	O
means	O
that	O
the	O
pinhole	B-Algorithm
camera	I-Algorithm
model	I-Algorithm
can	O
only	O
be	O
used	O
as	O
a	O
first	O
order	O
approximation	O
of	O
the	O
mapping	O
from	O
a	O
3D	O
scene	O
to	O
a	O
2D	O
image	O
.	O
</s>
<s>
Some	O
of	O
the	O
effects	O
that	O
the	O
pinhole	B-Algorithm
camera	I-Algorithm
model	I-Algorithm
does	O
not	O
take	O
into	O
account	O
can	O
be	O
compensated	O
,	O
for	O
example	O
by	O
applying	O
suitable	O
coordinate	O
transformations	O
on	O
the	O
image	O
coordinates	O
;	O
other	O
effects	O
are	O
sufficiently	O
small	O
to	O
be	O
neglected	O
if	O
a	O
high	O
quality	O
camera	O
is	O
used	O
.	O
</s>
<s>
This	O
means	O
that	O
the	O
pinhole	B-Algorithm
camera	I-Algorithm
model	I-Algorithm
often	O
can	O
be	O
used	O
as	O
a	O
reasonable	O
description	O
of	O
how	O
a	O
camera	O
depicts	O
a	O
3D	O
scene	O
,	O
for	O
example	O
in	O
computer	B-Application
vision	I-Application
and	O
computer	O
graphics	O
.	O
</s>
<s>
The	O
geometry	O
related	O
to	O
the	O
mapping	O
of	O
a	O
pinhole	B-Algorithm
camera	I-Algorithm
is	O
illustrated	O
in	O
the	O
figure	O
.	O
</s>
<s>
This	O
is	O
also	O
where	O
the	O
camera	O
aperture	B-General_Concept
is	O
located	O
.	O
</s>
<s>
An	O
image	O
plane	O
,	O
where	O
the	O
3D	O
world	O
is	O
projected	O
through	O
the	O
aperture	B-General_Concept
of	O
the	O
camera	O
.	O
</s>
<s>
The	O
image	O
plane	O
is	O
parallel	O
to	O
axes	O
X1	O
and	O
X2	O
and	O
is	O
located	O
at	O
distance	O
from	O
the	O
origin	O
O	O
in	O
the	O
negative	O
direction	O
of	O
the	O
X3	O
axis	O
,	O
where	O
f	O
is	O
the	O
focal	B-Algorithm
length	I-Algorithm
of	O
the	O
pinhole	B-Algorithm
camera	I-Algorithm
.	O
</s>
<s>
A	O
practical	O
implementation	O
of	O
a	O
pinhole	B-Algorithm
camera	I-Algorithm
implies	O
that	O
the	O
image	O
plane	O
is	O
located	O
such	O
that	O
it	O
intersects	O
the	O
X3	O
axis	O
at	O
coordinate	O
-f	O
where	O
f	O
>	O
0	O
.	O
</s>
<s>
The	O
projection	B-Algorithm
line	O
of	O
point	O
P	O
into	O
the	O
camera	O
.	O
</s>
<s>
The	O
projection	B-Algorithm
of	O
point	O
P	O
onto	O
the	O
image	O
plane	O
,	O
denoted	O
Q	O
.	O
</s>
<s>
This	O
point	O
is	O
given	O
by	O
the	O
intersection	O
of	O
the	O
projection	B-Algorithm
line	O
(	O
green	O
)	O
and	O
the	O
image	O
plane	O
.	O
</s>
<s>
The	O
pinhole	O
aperture	B-General_Concept
of	O
the	O
camera	O
,	O
through	O
which	O
all	O
projection	B-Algorithm
lines	O
must	O
pass	O
,	O
is	O
assumed	O
to	O
be	O
infinitely	O
small	O
,	O
a	O
point	O
.	O
</s>
<s>
In	O
this	O
figure	O
we	O
see	O
two	O
similar	O
triangles	O
,	O
both	O
having	O
parts	O
of	O
the	O
projection	B-Algorithm
line	O
(	O
green	O
)	O
as	O
their	O
hypotenuses	O
.	O
</s>
<s>
The	O
mapping	O
from	O
3D	O
to	O
2D	O
coordinates	O
described	O
by	O
a	O
pinhole	B-Algorithm
camera	I-Algorithm
is	O
a	O
perspective	B-Architecture
projection	I-Architecture
followed	O
by	O
a	O
180°	O
rotation	O
in	O
the	O
image	O
plane	O
.	O
</s>
<s>
This	O
corresponds	O
to	O
how	O
a	O
real	O
pinhole	B-Algorithm
camera	I-Algorithm
operates	O
;	O
the	O
resulting	O
image	O
is	O
rotated	O
180°	O
and	O
the	O
relative	O
size	O
of	O
projected	O
objects	O
depends	O
on	O
their	O
distance	O
to	O
the	O
focal	O
point	O
and	O
the	O
overall	O
size	O
of	O
the	O
image	O
depends	O
on	O
the	O
distance	O
f	O
between	O
the	O
image	O
plane	O
and	O
the	O
focal	O
point	O
.	O
</s>
<s>
This	O
is	O
the	O
way	O
any	O
practical	O
implementation	O
of	O
a	O
pinhole	B-Algorithm
camera	I-Algorithm
would	O
solve	O
the	O
problem	O
;	O
for	O
a	O
photographic	O
camera	O
we	O
rotate	O
the	O
image	O
before	O
looking	O
at	O
it	O
,	O
and	O
for	O
a	O
digital	O
camera	O
we	O
read	O
out	O
the	O
pixels	O
in	O
such	O
an	O
order	O
that	O
it	O
becomes	O
rotated	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
means	O
equality	O
between	O
elements	O
of	O
projective	O
spaces	O
.	O
</s>
<s>
This	O
description	O
of	O
the	O
pinhole	B-Algorithm
camera	I-Algorithm
mapping	O
,	O
as	O
a	O
linear	O
transformation	O
instead	O
of	O
as	O
a	O
fraction	O
of	O
two	O
linear	O
expressions	O
,	O
makes	O
it	O
possible	O
to	O
simplify	O
many	O
derivations	O
of	O
relations	O
between	O
3D	O
and	O
2D	O
coordinates	O
.	O
</s>
