Game Development Reference
In-Depth Information
ˆ
()()
i
j
im
and
is a column vector with 36 elements in the form
x
ˆ
im
y
ˆ
im
(
i
,
j
=
0
7
i
+
j
≤
7
and
).
x
T
w
w
w
At the same time, if a point with coordinate
=
x
y
in another plane
2
2
2
T
Π
im
im
projects onto the image plane of the current camera also at
x
ˆ
y
ˆ
, we
2
similarly obtain that:
w
im
xEu
≅
2
ˆ
,
(37)
2
x
T
where
w
=
x
w
y
w
1
,
E
2
is also a matrix with 3 rows and 36 columns.
2
2
2
T
im
im
Since they have the same projection
x
ˆ
y
ˆ
, a special relation called
x
x
w
w
perspectivity
(Hartley & Zisserman, 2000) should exist between
and
.
1
2
Π
Π
This perspectivity relation between all corresponding pairs in
and
can be
1
2
matrix
C
as:
described by a
33
×
w
w
xCx
.
(38)
2
1
Of course
C
has to fulfill some constraints for it to be a perspective transforma-
tion (Wei & Ma, 1994). Otherwise, it is just a plane-to-plane homography.
Combining equation 38 with equations 36 and 37 yields:
ECE
≅
.
(39)
2
1
Therefore, instead of recovering
E
1
and
E
2
separately,
E
1
and
~
can be
calculated first. Then equation 39 is employed to recover
E
2
. Thus, it is ensured
that the perspective constraint is satisfied.
Discussion
By considering the distortion implicitly, as was done in this section, linear
estimation techniques can be utilized for calibration. However, several problems
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