196 
ISPRS, Vol.34, Part 2W2, “Dynamic and Multi-Dimensional GIS", Bangkok, May 23-25, 2001 
Figure3-1 parallelogram l,j,k,l and volume of a tetrahedron 
The volumes of the 4 tetrahedrons(Vc-ijk,Vc-ilk,vc-ijl,Vc-jkl) can 
be computed from the 3 vectors defined by the image 
coordinates, the focal length and the distances from the 
projection center to the 3 corner points. Volume of the 
tetrahedron (Vc-ijk) 
didjdk I , j, k jj 
4№l 
where 
(20) 
Xi = i = (Xi,yi-f) 
|/j, |y|, |A:| express the size of the vectori.j.k. 
di.dj.dk express the distance between c and l.j.k. 
Because of the parallelity, a diagonal of the parallelogram always 
splits its area in 2 equal parts. And thus, the areas of the 4 
triangles^*,Aiik.Ajki,Aim) in the plane of the parallelogram are all 
equal. As the same holds for the distance h of the projection 
center to the plane of the parallelogram, ilt follows that the 
volumes of the 4 tetrahedrons are identical. 
dc0[i,j,k\ jkk^j,kj\ jldct[l,i, j\ 
~m~ = isr HIT TT 
The distance ratios Cij can be written as follows 
[kj,j] 
[jM 
(21) 
Cik = 
4 model coordinates and 3D coordinates 
Model coordinates (X m i) can be written as a function of one 
distance di that defines the scale of the model 
tiXl\ 
(22) 
x image coordinates. 
X the available coordinates in the object system 
X° the coordinates of the projection center. 
5 experimental result 
One simulated test and one real test have been conducted to 
check the feasibility of the new method. 
simulated test The first set of data are a simulated cube, whose 
coordinates of 3D and corresponding to 2-D are designed by 
back-projection. Three couples of parallel straight lines in cube 
are parallel to X.Y.Z axes in object coordinate system. The 
simulated orientation parameters of the camera are listed in table 
5-1.The simulated control points are listed in table 5-2. The 
results are listed in table 5-3 and use OpenGL function in visual 
C++ to display result as figure 5-1. 
Table 5-1 the simulated orientation parameters 
Interior 
orientation 
parameters 
xO(pixel) 
yO(pixel) 
F(pixel) 
-274.50 
-236.98 
2604. 76 
Exterior 
orientation 
parameters 
<P 
w 
K 
-0.6910 
0.5642 
0.9172 
JtM 
Xs(mm) 
Ys(mm) 
Zs(mm) 
57910.90 
-47492.94 
54088.57 
Table 5-2 the simulated control points 
number 
X(m 
m) 
Y(mm) 
Z(mm) 
u(pixel) 
v(pixel) 
1 
0 
0 
0 
-492.26 
-40.85 
2 
100 
0 
0 
-490.25 
-41.89 
3 
100 
0 
100 
-490.57 
-39.60 
4 
0 
0 
100 
-492.54 
-38.56 
5 
0 
100 
0 
-490.19 
-39.51 
6 
0 
100 
100 
-490.51 
-37.22 
7 
100 
100 
100 
-488.54 
-38.25 
(23) 
X m j = XiCtjXj 
From the first step, interior and exterior orientation parameters 
are known. So, object coordinates can be computed in following 
form 
ÁRx¡ = X,-X° 
XRCijxj = Xj - X° 
(24) 
where X scale factor 
R rotation matrix 
Cij distance ratio projection center-points i and j from step 2.