Graphics Reference
In-Depth Information
and
(
)
d
=
bmmbmmbmm bmm
¢
¢-
¢
¢
¢-
¢
≤
≤-
≤
≤
≤-
≤
.
1
34
14
2
34
24
1
34
14
2
34
24
This gives four equations in three unknowns. Such an overdetermined system does
not have a solution in general; however, if we can ensure that the matrix A has rank
three, then there is a least squares approximation solution
-
1
(
)
+
T
T
(
)
=
aa a
dd
A
=
A AA
.
(4.19)
123
using the generalized matrix inverse A
+
(see Theorem 1.11.6 in [AgoM05]).
Next, consider the camera calibration problem. Mathematically, the problem is to
compute M if equation (4.13) holds for known points
a
i
and
b
i
, i = 1, 2,..., k. This
time around, we cannot normalize the
a
i
and shall assume that
a
i
= (a
i1
,a
i2
,a
i3
,a
i4
) and
b
i
= (c
i
b
i1
,c
i
b
i2
,c
i
). It is convenient to rewrite equations (4.16) in the form
mam
∑- ∑
b
a
=
0
(4.20a)
(4.20b)
1
i
3
i
1
i
mam
∑- ∑
b
a
=
0
.
2
i
3
i
2
i
We leave it as an exercise to show that equations (4.20) can be written in matrix form
as
n0
A =
,
where
T
T
T
Ê
ˆ
a
a
L
a
0
0
L
0
1
2
k
Á
Á
˜
˜
T
T
T
A
=
0
0
L
0
a
a
L
a
1
2
k
T
T
T
T
T
T
-
b
a
-
b
a
L
-
b
a
-
b
a
-
b
a
L
-
b
a
Ë
¯
11
1
21
1
n
1
1
12
1
22
1
nn
2
and
n
=
(
)
mmmmmmmmmmmm
11
43
.
21
31
41
12
22
32
42
13
23
33
This overdetermined homogeneous system in twelve unknowns m
ij
will again have a
least squares approximation solution that can be found with the aid of a generalized
inverse provided that
n
is not zero.
4.13
Robotics and Animation
This section is mainly intended as an example of frames and transformations and how
the former can greatly facilitate the study of the latter, but it also enables us to give
a brief introduction to the subject of the kinematics of robot arms. Even though we
can only cover some very simple aspects of robotics here, we cover enough so that
the reader will learn something about what is involved in animating figures.
