7A-6-3 
solutions of the linear equation are achieved, 
which form a solution plane. On this plane the 
constraint of I,I 6 =I 2 I 5 or I,I 6 =I 3 I 4 is used to 
get a unique group of solution. 
S 3 : Transforming object points' 3D 
projective coordinates into Euclidean 
coordinates: 
To get the Euclidean coordinates of object 
points from their projective coordinates, we 
use five object points, which are the control 
points, to build a transformation matrix. This 
matrix transforms the control points' 
quasihomogeneous from of 3D rectangular 
coordinates into the canonical coordinates of 
3D projective coordinates system. Its inverse 
matrix transforms other object points' 
projective coordinate into their 
quasihomogeneous rectangular coordinates, 
from which we can get the results of the 
positioning. 
3. ROBUSTNESS TEST TO THE THREE 
WAYS 
In our previous work, we used some simulated 
values to examine the method with all the 
three ways and proved it is high accurate in 
theory. In real work, the coordinate of image 
point has its accuracy limits, which has 
discrepancy with the true value. So we use 
some pseudo-stochastic noises to add on every 
point to test the method's robustness. 
3.1 Robustness Test 
Undoubtly, the arrangement of cameras and 
the distribution of object point influence the 
accuracy and robustness of positioning. Here 
we just use well-arranged cameras and well- 
distributed object points to test it. 
We have used tens groups of values to test the 
three ways. We found the way based on 2 
images is the most robust, the way based on 4 
images is the worst robust. In table 1, we list 
one of the results about occasional errors. On 
every image point we introduced normal 
distributed errors, which means value is 1.0 
pixel. 
According to the proportion of the distance 
between image and object, we calculate the 
errors on object points caused by the error of 
image points is about 0.03. We take 0.05 as 
demarcation to mark the coordinate of big 
error with grey color. We found result based 
on 2 images are all within the demarcation; 
one point's coordinates are out of the 
demarcation in the way based on 3 images; 
more than one points' coordinates are out of 
the demarcation in the way baed on 4 images. 
To test the robustness of the three way 
concerned with gross error, we add 5 pixel on 
one of the points, which is not control point. 
We found this gross error does not inflence 
other points' results to all the three ways. One 
of the result is listed in table 2. However, if 
this point is used to calculate the fundamental 
matrix in the way based on 2 images, then the 
results of all the points are influenced. 
3.2 Geometric constraint 
In order to look for the reason about the 
difference of the robustness to the three ways, 
we consider the geometric constraints related 
to image forming in projective space. The 
following are two constraints, which should 
be met in the procedure of solution. 
i). The coplanarity condition between image 
points on two photos: 
To a pair of homologous points on two images 
(u, v, w) T and (u', v', w' ) T , there exist a matrix 
F 3X3 , which meet: