Game Development Reference
In-Depth Information
Distorted projection approach with imaging-distortion model
This approach is used for the projection purpose.
Similarly, assume that the camera distortion is subject to the imaging-distortion
model. Substituting equation 5 into equation 10 gives:
,
(35)
x u
ˆ
im
w
T
x
~
w
where
ˆ
im
x
ˆ
im
y
ˆ
im
;
D
is a matrix with 3 rows and 120 columns.
is
=
1
()()()
i
j
k
a column vector with 120 elements in the form
ˆ
w
ˆ
w
ˆ
w
(
i
,
j
,
k
=
0
7
x
y
z
and
i
+
j
+
k
≤
7
).
[
]
T
If all world points
=
x
belong to the same plane, it can be
assumed that without loss of generality all
w
w
w
w
x
y
z
z
w
equal
0
. If at the same time only
Im
1
Im
Im
Im
Im
2
distortion coe
ffi
cients
k
,
P
,
P
,
s
, and
s
are considered, then the
~
w
dimension of
D
decreases to
3 ×
10
and that of
to
10 ×
1
.
ˆ
~
),
D
can be calculated from equation 35. From
D
, the projection of any world point
into the current camera can be computed by means of equation 35 in a linear
fashion.
Again, if we have enough pairs of
im
and
co
rresponding
x
w
(and thus
w
Perspectivity
x
[
]
T
For an arbitrary point with coordinates
in a world plane
Π
,
w
w
w
=
x
y
1
1
1
1
Π
its coordinates in the WCS, whose
x
and
y
axes are the same as those of
,
1
x
[
]
~
T
are obviously
. Following the same procedure as the one
leading to equation 33, it can be derived that:
w
w
w
=
x
y
1
1
1
1
xEu
w
≅
1
ˆ
im
,
(36)
1
x
[
]
~
T
w
w
w
where
=
x
y
1
,
E
1
is a matrix with 3 rows and 36 columns,
1
1
1
x
[
]
T
w
im
im
is the projection of
onto the image plane of the current camera,
x
ˆ
y
ˆ
1
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