a straight line, and the coordinates for a point with 2D (uy Vm) 
and corresponding to 3D (Xm »Ym »Zm )are known, we can 
determine the translation vectors ¢ using 
XM Xe Xu un 
T= Yum — AR Ye = Ym — AR Von (13) 
ZM Ze Zu mf 
B.4) Distortions rectify 
As a result of several types of imperfections in the design and 
assembly of lenses composing of the camera optical system, we 
think, generally speaking, that main distortion sources come 
from radial and discenteric distortions, i.e., 
ó,(u,v)- ku? tp (3i? tv?) - 2p,uv 
é,(u,v)= K,v(u” +v?) +p, 3u? +v?) +2p,uv (14) 
In order to rectify the distortions, we make full use of the 
geometric properties of straight lines. Provided that any a point 
15 on the line 1-2 is selected, the points 1, 2 and 15 should are 
collinear in ideal case, i.e., the area of triangle constructed by the 
points 1, 2 and 15 is zero (Fig. 4.). In fact, owing to existing 
distortion, the three points are impossible to be strictly collinear. 
So, the collinear property of three points can be used to rectify 
the distortion, i.e., 
ui t Öö(uj, vi) vit ó(up,vi) 
1 
uy +6(uy,vy)  vpitó(uj,v;) 1=0 (15) 
Mis 0(uis,vis) vi+8(uis,vis) 1 
It is easy to obtain the distortion parameters Kj, pp, Ps. 
values through solving (15). 
C) Experiments and Analysis 
Test 1: The first set of data is a simulated cube, whose 
coordinates of 3-D and corresponding to 2-D are designed by 
CAD, whose image is generated by back-projection (Fig. 5). The 
size of image is 200X200 pixels, and the sample distance is 
504m. The natural landmarks are edges of the cube, which 
strictly meet the conditions of our algorithm described above. 
The experimental results with four calibration methods are 
shown in Tab. I. In selecting distortion parameters, we only 
considered radial distortion. 
Test 2: A real image is chosen as experimental data, whose 
size is 512X512 pixels with A ss (Fig. 6). In order to 
compare the precision to be reached by using our approach with 
others. Edges with one pixel level accuracy are considered as 
natural landmarks. To other algorithms, we select corners, which 
are really an intersection point of two or more than straight lines, 
as distinct points. The experimental results with four calibration 
approaches are shown in Tab. 2. 
Test 3: Another type of CCD camera (type: COSMICAR) was 
tested. A scene, which consists of many industrial parts, was 
grabbed when the camera was located with depth distance about 
880mm long. The size of image is 512X 512 pixels with 856 
gray level. As illustrated Fig. 7. The experimental results with 
four calibration approaches are shown in Tab. 3. 
  
  
  
  
  
  
  
Table | 
Rotate Angle( ©) Position Par.(mm) 
0 A T tl T2 3 
Yakimov. |-0.664 [0.904 10.663 [234.711 [-311.76 [282.911 
Tsai -0.663 10.904 10.663 |234.208 |-311.89 [282.345 
DLT -0.664 |0.911 0.667 |234.617 |-311.85 [282.773 
Our Method|-0.666 |0.909  |0.669 |234.437 |-311.90 [282.941 
  
  
  
Intrinsic Par. (pixel 
(pire) Distor. Par( 10% 
Uo Vo f kl 
Yakimov. {103.2 [100.8 1320.3 |- 
Tsai 101.8 [101.7 320.6 15.6577 
DLT 1036 1102.5 0320.1 [1.051 
  
  
  
  
  
  
  
  
  
  
  
  
[Our Method[102.9 [100.4 [320.7 [4.8956 (*) | 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
Table 2 
Rotate Angle( °) Position Par. (mm) 
0 A T tl T2 t3 
Yakimov. [-0.193 [15.451 10.838 -308.56 1276.51 |-1275.51 
Tsai -0.188 [15.613 10.843 -308.31 [1276.03 |-1275.55 
DLT -0.180 |15.226 [0.894 -308.48 1276.21 |-1275.25 
Our Method|-0.185 |15.268 10.876 -308.32 [1276.04 |-1275.53 
Intrinsic Par. (pixel) Dis. Par (1 07$, 
Uo Vo f kl 
Yakimov. [255.91 |251.05 |1279.44 |- 
Tsai 256.07 |243.11 |1273.45 |19.655 
DLT 255.58 |248.41 |1281.04 124.813 
Our Method |255.94 (241.79 |1277.81 |58.766(*) 
Table 3 
Rotate Angle( ? ) Position Par. (mm) 
O A T tl T2 a 
Yakimov. [0.32031 |0.671134 |-0.81435 |620.1516 |516.1162 |825.9164 
Tsai 0.31988 [0.675613 |-0.82147 |621.0311 |516.0393 |825.5547 
DLT 0.32180 [0.669961 |-0.81994 [620.9882 |516.1237 |825.5553 
Our Method |0.31895 [0.668682 |-0.81769 [620.9824 [516.0457 |825.1373 
  
  
  
  
Intrinsic Par. (pixel 
ntrinsic Par. (pixel) Dis Par (107$, 
  
  
  
  
Uo Vo f kl 
Yakimov. [260.012 |251.733 [1614.44 |- 
Tsai 257.973 |253.622 [1625.45 [149.158 
DLT 251.584 |258.017 [1609.04 [114.136 
  
Our Method [259.947 [251.588 [1617.81 |258.563(*) 
(* The value of distortion parameter is a mean value of several lines) 
  
  
  
  
  
  
  
D) Remarks and Conclusions 
An approach for calibrating camera is proposed here. We 
result in following conclusions from experimental results: 
(1) From the result of simulation, the calibration parameters in 
our approach are close to others (Table 1). 
(2) From the result with the real data, the solved calibration 
parameters using our approach are close to Tsai method, and a 
little far Yakimovskyy method (Table 2, Table 3). 
In a word, We adopt the straight lines instead of the points as 
our calibration landmarks, and utilize the geometric information 
of the straight lines to accomplish the camera calibration. The 
advantages can be summarized as follows: 
(a) Without known the equations of straight lines and 
coordinates of any control points, the interior and rotation 
parameters can be determined. 
(b) The computational process is linear, without any iteration 
and initiated values. 
(c) Orthogonal constrains about three axes are considered: 
(d) Distortions are rectified using straight line geometry. 
        
a 
Fig. 5. Artificial image. Fig. 6. Real Image. Fig. 7. Real image. 
4.2 GEMS-based Object Reconstruction Using LP 
A) Representation of object in GEMS 
SIVE is also required to locate and reconstruct 3D industrial 
objects. We developed a GEMS-based object reconstruction 
using line photogrammetry. So-called GEMS is a CAD system 
developed by CAD research group of department of computer 
science and technology at Tsinghua university (Sun 1989, 1990, 
Ren 1991). In GEMS, object representation is to combine CSG 
and Boundary represent (B-rep). CSG represents each complex 
object by geometric transformations (shift, rotation, and scale). 
Boolean set operators including two-tuple operators 
(intersection, merger, and difference), and mono-tuple operators 
204 
(rotation, SC? 
representation 
represent vari 
operator (Fig. 
This repre 
distinct. But i 
has a good efl 
object. In B- 
every surface 
representation 
vertex. B-pre 
topology info 
The geon 
characters of 
surface, suc 
x = R cos(6 
topologic ini 
relation of the 
the object yie 
  
ED - 
Vi 
Fig. 9. Ol 
Our recon 
each industri 
primitives, a 
curves, planz 
parametric ec 
In the foll 
location of 
accomplishe« 
are establishe 
B) Mathemai 
In GEMS, 
transformatic 
scene (Fig. 
feature are e;