Game Development Reference
In-Depth Information
The relation between the camera coordinates (in CCS) and the metric projection
coordinates (in PCS) is inferred from the principle of lens projection (modeled
as a pinhole, see Figure 1b). This perspective transformation is a kind of
projective mapping (ref. Figure 1a).
In Figure 1a, the
optical center
, denoted by
O
, is the center of the focus of
projection. The distance between the image plane and
O
is the
focal length
,
which is a camera constant and denoted by
f
. The line going through
O
that is
perpendicular to the image plane is called the
optical axis
. The intersection of
the optical axis and the image plane is denoted by
o
, and is termed the
principal
point
or
image center
. The plane going through
O
that is parallel to the image
plane is called the
focal plane
.
The perspective projection from the 3-D space (in CCS) onto the image plane
(in IMCS) through the PCS can be formulated as:
im
m
c
x
1
/
s
0
x
x
−
f
0
x
x
x
0
x
0
z
c
y
im
=
z
c
0
1
/
s
y
y
m
=
0
−
f
y
y
c
,
(3)
y
0
y
0
c
1
0
0
1
1
0
0
1
z
[
]
T
where
f
x
=
f / s
x
and
f
y
=
f / s
y
.
0
is the pixel coordinate of the principal
point with respect to the IMCS, and
s
and
s
are the effective pixel distances
in the horizontal (x-axis) and vertical (y-axis) directions, determined by the
sampling rates for Vidicon cameras or the sensitive distance for CCD and CID
cameras. Most CCD cameras do not have square pixels, but rectangular pixels
with an
aspect ratio s
y
/ s
x
of about 0.9 to 1.1.
f
,
x
0
,
y
0
, and are called the
intrinsic
parameters
of the camera.
Substituting equation 2 into equation 3 we obtain:
x
y
0
w
x
−
im
x
f
0
x
x
0
w
y
() ()
T
T
im
w
w
w
y
≅
0
−
f
y
R
−
R
⋅
t
y
0
c
c
c
w
z
,
1
0
0
1
1
where
means “equal up to a non-zero scale factor”, which is the symbol of
equality in the
projective space
(Faugeras, 1993).
≅
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