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al. are given. It is worth noting at this point that the relative errors in R formed by the method
just described will not be equal to the relative errors formed by averaging the individual rotation
matrices which have 9 dof. We believe that one should therefore form an average R from an average
n . It is not clear which method of averaging is used in [Weng et al. 1989].
We can see from Figures 5 and 6 that the errors in the linear algorithm of Weng et al. are greater
for translation directions parallel to the image plane (corresponding to small values on the x axis),
whereas the GA algorithm shows no such tendencies, giving relative errors which are roughly of
equal magnitude over the range of directions used this is in agreement with our expectations.
Overall we see that the GA algorithm gives more accurate predictions than the linear algorithm for
all parameters but more particularly for the vectors n and t { in many cases there is almost a factor
10 improvement on average. The obvious advantage of the linear algorithm is its simplicity and
its speed. If we start the GA algorithm with an initial guess for the rotor as the identity (i.e. no
rotation) then the algorithm is robust, in the sense that it will converge to the given solutions but
will take perhaps 100 iterations to do so. However, in generating the results shown in Figures 5 and
6 the procedure was to evaluate R for the linear algorithm and to use this as the initial estimate
of the rotation in the GA algorithm. This had the e ect of decreasing the number of iterations
28
theta (degrees)
relative error in n
relative error in t
relative error in R
9.5
9
0.1
8.5
8
0.05
7.5
0 7
0 5 10 15 20 0 5 10 15 20
0.025
0.1
0.02
0.08
0.015
0.06
0.01
0.04
0.005
0.02
0 0
0 5 10 15 20 0 5 10 15 20
Figure 6: 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 128 128. The solid and
dashed lines respectively show the results of the GA algorithm and the linear algorithm of Weng
et al.
.
29
theta (degrees)
relative error in n
relative error in t
relative error in R
Figure 7: Two views of Lego house from a single camera undergoing translation and rotation.
dramatically. Indeed, the current algorithm can be seen as an addition to the linear algorithm of
Weng et al. to re ne the estimates if greater accuracy is required. We note also from the gures
that the improvements in accuracy given by the GA algorithm increase as the resolution decreases,
i.e. as the noise increases. The algorithm presented here therefore seems to be a better tool if there
is a high likelihood that the data is very noisy.
4.4 Real Data
In this section we will give some results of applying the algorithm to a pair of real images. The
camera was approximately calibrated before the experiment and was then used to observe a `LEGO'
house from two di erent viewpoints. The two views are shown in gure 7. The motion of the camera
and the distances to various points on the house were measured. From the two images a set of
12 corresponding features (mostly corners) were found and these were used as input to both the
algorithm of Weng et al. and to the GA algorithm with an identity rotor as starting point. The axes
in both camera positions were such that the z-axis passed through the optical centre perpendicular
to the image plane and towards the scene, the x-axis was vertically upwards and the y-axis was
then such that the directions formed a right-handed orthogonal set. Relative to the frame in view
1 the true values of the rotation and translation between the views was:
n = [1 0 0]
30
= ;27
t = [0 ;72:6 25:1]: (99)
The translation values are given in centimetres writing t as a unit vector gives t =[0 ;0:945 0:327].
Giving the 12 point matches to the algorithm of Weng et al. produced the following results:
n = [0:456 ;0:290 0:842]
= 94:7
t = [;0:801 0:598 ;0:037]: (100)
In this particular case it is clear that the closed form solution above gives something very far from
the truth. Now, if we give the same 12 points to the GA algorithm with an initial identity rotor
(i.e. no rotation), after 4500 iterations the results are
n = [;0:999 0:006 ;0:039]
= 28:33
t = [;0:004 ;0:941 0:339]: (101)
It is clear that we are able to obtain a very accurate solution given a fairly large number of iterations
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