Catadioptric Stereo with Planar Mirrors: Multiple-view Geometry and Camera Localization - Visual Servoing via Advanced Numerical Methods - page 16

Fig. 1.9 Experimental
setup: an eye-in-hand robot
observes a structured 3D
scene directly and reflected
through two planar mirrors

( R c w , t c w )

In Figure 1.8, we ran 200 iterations of the proposed algorithms for increasing
value of the image noise standard deviation. Figure 1.8(a) reports the mirror angle
estimation error | θ − θ |

for the algorithms in Propositions 1.5 and 1.6. The algo-
rithm that uses the virtual epipoles is the most accurate with a maximum mean error
of 1 5 deg. Figure 1.8(b) shows the R c w estimation error computed as the roll-pitch-

yaw angle errors (cf. Proposition 1.7). We experienced a good robustness to noise
of the proposed method, with a maximum mean error around 2 deg. Figure 1.8(c)
finally reports the estimation error
ε( t c w )=

.

( t c w )
℘

( t c w )
℘

℘

− ℘

( t c w ) ( t c w )

for the two algorithms in Proposition 1.8: screw axis (1.21) and Sylvester equation
(1.22). From an inspection of Figure 1.8(c), we observe that the second method is
less sensitive to noise since it does not rely on the direct computation of the epipoles.
A maximum mean error of 5 cm is achieved.

1.5.2 Experiments

In order to test the robustness of proposed algorithms in real scenarios we per-
formed a series of experiments using the setup shown in Figure 1.9. A Lumenera 1
LU071C camera mounted on the end-effector of a 6 axes KUKA 2 KR 3 manipu-
lator, observes a structured scene directly and reflected through two planar mirrors.
The mirrored objects lie in a box of size 0.42 0.3 m 2 .

We exploited the high positioning accuracy of the KR 3 manipulator in order to
have precise ground truth reference measurements of position and orientation of the
camera. The calibration matrix K of the camera is the same as in the simulations 3 .
Figure 1.10(a) shows the mirror angle estimation error

×

| θ − θ |

for θ ∈{ 30 deg ,

45 deg , 60 deg } . The values reported in Figure 1.10(a) are the average of 5

1

Lumenera is a registered trademark of Lumenera Corporation. http://www.lumenera.com

2

KUKA is a registered trademark of KUKA AG. http://www.kuka.com

3

The calibration matrix has been estimated with the camera calibration toolbox [2].