[ Pobierz całość w formacie PDF ]
that for
algorithm are reasonably accurate.
26
4.3.2 Simulations: II
In this section we will compare the results of the geometric algebra algorithm with the closed-form
solution described in [Weng et al. 1989]. We will choose parameters for the motion and structure as
described in [Weng et al. 1989] so that the results here may also be compared to their results for the
closed-form algorithm against other linear algorithms. The advantages of such algorithms is their
speed { the geometric algebra algorithm is iterative and is therefore slower. However, we hope to
show that the results obtained with the GA algorithm are signi cantly more accurate. Indeed, for
real noisy data and approximate calibration the geometric algebra is able to obtain good solutions
for cases in which the closed-form algorithms fail completely.
In the following simulations 12 object points are used and these are generated randomly from a
uniform distribution in a cube of side 10 units whose centre is 11 units from the centre of projection
of the camera. The focal length will be taken as 1 and the image size as 2 units. Unlike the
simulations in the previous section, the noise is now solely due to quantization e ects in the image
plane and we consider two noise scenarios, one given by an image resolution of 256 256 and the
other by a resolution of 128 128. Such a decrease in resolution will considerably increase the
quantization noise. The rotation of the camera is 8 about an axis in the (;0:2 1 0:2) direction
and 21 di erent translations, t, are used with these given by
t =3 cos(i ; 1) + ; 3sin(i ; 1) 3 (98)
1 2
for i = 1 to 21, and = 90=20 = 4:5 . Thus, when these translation vectors are projected
onto the x ; z plane they describe a sequence of equally spaced vectors between the x axis and
the ;z axis. This situation is designed to be similar to that described in [Weng et al. 1989]. The
translation vectors were chosen in this way so as to illustrate the greater potential for error in the
motion estimation when the translation is parallel to the image plane ([Weng et al. 1989] explain
why this is via explicit reference to their algorithm). We would not expect the performance of the
GA algorithm to be signi cantly a ected by the translation direction.
For each of the 21 values of the translation vector the estimated values of t, n and are found by
averaging over 100 trials (in each trial the 12 points are generated at random within the allowed
region). The relative error is calculated by taking the norm of the di erence between the mean
value and true value divided by the norm of the true value. In the case of t the trial values are
normalized before forming the mean. To form the mean values of n and , we form the product n
for each trial (having rst normalized n) and then form the average of n over the total number of
trials. The mean values of n and separately are then extracted from this mean value of n . Recall
that above we identi ed n as being a sign-invariant quantity it also has 3 degrees of freedom (dof).
A general rotation has 3 dof and in forming an `average' rotation it therefore seems sensible to nd
the mean value of a quantity with a su cient set of independent quantities. This would therefore
suggest that one should average over the quantity n rather than over and n individually or over
the components of the rotation matrices from each trial.
Figures 5 and 6 show the relative error for t, n and for the two di erent resolutions. In addition,
for comparison with [Weng et al. 1989], we also plot the relative error in the estimated rotation
matrix R in this case the `mean' value of R is that rotation matrix formed from the mean value
of n and the relative error is formed from the norm of the di erence of the estimated R and
the true R divided by the norm of the true R { where the norm of a matrix A is given by
27
0.04 8.5
0.02 8
0 7.5
0 5 10 15 20 0 5 10 15 20
0.03 0.01
0.008
0.02
0.006
0.004
0.01
0.002
0 0
0 5 10 15 20 0 5 10 15 20
Figure 5: Relative errors in a) the rotation axis b) the rotation angle c) the translation direction
and d) the rotation matrix, plotted against the 21 di erent translation vectors used. The results
are the average of 100 samples at each point and use a resolution of 256 256. The solid and
dashed lines respectively show the results of the GA algorithm and the linear algorithm of Weng
et al.
qP
jjAjj = aij2. In each gure the relative errors are plotted against the 21 di erent translation
ij
[ Pobierz całość w formacie PDF ]