ii). The relationship between the projective 
coordinates of object points and image points: 
If we substitute the coordinates of 5 control 
points and an unknown point and their 
correspondence image points' projective 
coordinates into the image forming equation, 
we can get a relationship: 
w 6 (u 5 - v 5 ) XY + v 6 (w 5 - u 5 )XZ + 
u 5 (v 6 - w 6 )XT+u 6 (v 5 - w 5 )YZ + 
v 5 (w 6 - u 6 )YT + w 5 (u 6 - v 6 )ZT = 0 
If we define the following marks: 
il= W 6 (u 5 -v 5 ), 
i 2 = v 6( w s -u 5 ), 
i 3 = u 5 (v 6 -w 6 ), 
14 = u,,( v 5 -w 5 ), 
1 5 = v 5 ( w 6 -u 6 ), 
i6=w 5 (u 6 -v 6 ); 
I, = XY, I 2 =XZ, 
I 4 =YZ, i 5 =yt, 
then we get: 
Ml "*■ M2 M 3 M4 I5I5 **■ ^6 = 0 
This equation builds another relationship 
between object points' projective coordinates 
and corresponding image points' projective 
coordinates besides image forming equation. 
From it we can get the unknown object points' 
3D projective coordinates. Moreover, it brings 
the linear solutions with multiple images. 
3.3 Explanation 
By analyzing the procedure and the geometric 
constraints, we find that the way based on two 
images meets the coplanarity condition and 
the additional relationship between image 
points and object points; the way based on 
I 3 = XT, 
I 6 =ZTo 
three images meets the condition of 
I 1 l6 == I 2 l5 = I 3 I 4 ; the way based on four images 
just meets the condition of I,I 6 =I 2 l5 or I 1 I 6 =I 3 I 4 . 
Because the constraints in the way of two 
images are more strict, even though the 
solution procedure is relatively complex, it is 
more robust to error. On the contrary, the 
constraints in the way based on four images 
are relaxious, the solution procedure is linear 
but it is sensitive to noise. 
4. CONCLUSION 
The method of object positioning in projective 
space is a fast, direct solution, which is fit to 
uncalibrated camera. It is a potential method 
in real-time photogrammetry, computer vision 
and other application, such as, mobile 
mapping system. 
In this method, multiple images can be fully 
taken into use. The three ways, which based 
on two, three and four images respectively, 
have different solution conditions and meet 
different geometric constraints. The way 
based on 2 images, which needs more 
homologous points than other two ways, 
meets the most strict geometric constraints so 
that it is the most robust way among them; the 
way based on 3 images, which needs certain 
approximate values about object points, is less 
robust than the way of 2 images; the way 
based on 4 images, which is the most direct 
and simple solution, abandon some geometric 
constraints in solution procedure so that is the 
worst in robustness. 
5. Acknowledgement 
This paper is supported by the 3S Integration 
Theory and Key Techniques (49631050), one 
of the projects of National Natural Science 
Foundation, which is gratefully 
acknowledged.