Information Technology Reference
In-Depth Information
The optimization problems in (9.27) can be solved by using various techniques,
such as the simplex algorithm which can handle also nondifferentiable cost func-
tions. The found solutions may be nonoptimal since these functions are nonconvex,
nevertheless it is guaranteed that any found solution is a lower bound of the sought
worst-case error.
Lastly, let us observe that an initialization for the optimization problems in (9.27)
is simply given by (
0
3
,
0
3
), which is admissible for any possible
δ
. Indeed:
p
∗
−
p
∗
∞
w
(
0
3
,
0
3
)=
δ
−
=
δ
≥
0
.
9.4
Examples
In this section we present some examples of the proposed approach. The upper
bounds
s
r
(δ) and
s
t
(δ) in (9.24)-(9.25) and the lower bounds
s
r
(δ) and
s
t
(δ) in
(9.27) have been computed by using Matlab and the toolbox SeDuMi.
9.4.1
Example 1
Let us consider the situation shown in Figure 9.1(a) where a camera is observing
four dice. The chosen object points are the centers of the ten large dots. The screen
size is 640
×
480 pixels, and the intrinsic parameters matrix is
⎛
⎞
500 0 320
0 500 240
001
⎝
⎠
.
A
=
Figure 9.1(b) shows the corresponding camera view. The problem is to estimate the
worst-case location error introduced by image points matching,
i.e.
the errors
s
r
(
δ
)
and
s
t
(
represents the total image error in (9.11).
We compute the upper bounds
s
r
(
δ
) in (9.20) where
δ
) and
s
t
(
δ
δ
) in (9.24)-(9.25) and the lower
bounds
s
r
(
) and
s
t
(
δ
δ
) in (9.27) for some values of
δ
. We find the values shown
in Table 9.1.
Ta b l e 9 . 1
Example 1: upper and lower bounds of the worst-case location errors for the object
points in Figure 9.1
δ
s
r
(δ)
s
r
(δ)
s
t
(δ)
s
t
(δ)
[pixels] [deg]
[deg]
[mm] [mm]
0
.
5
0
.
138 0
.
725 0
.
146 1
.
644
1
.
0
0
.
960 1
.
452 1
.
011 3
.
297
1
.
5
1
.
469 2
.
183 1
.
510 4
.
964
2
.
0
1
.
962 2
.
916 2
.
106 6
.
650
Search WWH ::
Custom Search