Game Development Reference
In-Depth Information
2
2
()
()
()
()
T
T
1
w
1
2
w
2
qx
+
q
q x
+
q
N
∑
4
4
im
im
d
=
−
x
+
−
y
!
.
geo
i
i
T
T
3
w
3
3
w
3
qx
+
q
qx
+
q
i
=
1
4
4
"
#
Then the problem to be solved becomes:
Minimize d
geo
subject to the constraints in equation 27
.
(28)
After solving the optimization problem 28, all original camera parameters can be
recovered by applying equation 24.
Modified DLT
To decrease the number of constraint equations required in the nonlinear
optimization process (so as to increase its efficiency), one can incorporate the
constraints in equation 27 in an alternative way.
According to Hatze (1988), equation 17 can be rewritten as:
1
w
2
w
3
w
w
w
w
rx
+
r y
+
rz
+
t
ax
+
ay
+
az
+
a
im
1
1
1
x
1
2
3
4
xx f
−=−⋅
=
,
0
x
rx
1
w
r y
2
w
rz
3
w
t
ax
w
a y
w
a z
w
a
+
+
+
+
+
+
3
3
3
z
9
10
11
12
rx
1
w
r y
2
w
rz
3
w
t
+
+
+
w
w
w
ax
+
ay
+
az
+
a
im
2
2
2
y
yy f
−=−⋅
=
5
6
7
8
,
(29)
0
y
1
w
2
w
3
w
w
w
w
rx
+
r y
+
rz
+
t
ax
+
a y
+
a z
+
a
3
3
3
z
9
10
11
12
i
j
r
(
,
w
c
where
i j
=
) is the element of
13
R
at the
i
th
row and
j
th
column and
()
T
T
t
t
t
Rt
ww
.
=−
x
y
z
c
c
()
T
Therefore, as
RRI
, we know immediately that:
w
w
=
c
c
, and
, and
a
a
+
a
a
+
a
a
=
0
a
a
+
a
a
+
a
a
=
0
1
5
2
6
3
7
1
9
2
10
3
11
a
a
+
a
a
+
a
a
=
0
.
(30)
1
5
2
6
3
7
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