Table 2 
  
275.51 
275.55 
275.25 
275.53 
  
Table 3 i 
  
t3 
. [825.9164 
[825.5547 
[825.5553 
825.1373 
  
  
  
  
  
  
  
eral lines) 
d here. We 
sults: 
arameters in 
] calibration 
>thod, and a 
he points as 
information 
bration. The 
t lines and 
ind rotation 
iny iteration 
sidered; 
metry. 
   
eal image. 
D industrial 
onstruction 
"AD system 
Of computer 
1989, 1990, 
mbine CSG 
ch complex 
and scale). 
operators 
le operators 
(rotation, scale transformation and transfer). This kind of 
representation can be illustrated by binary tree, whose leaves 
represent various primitives and whose nodes represent Boolean 
operator (Fig. 8). dius, 9 
This representation for primitives is concise, integrate and 
distinct. But its topologic relation is not clearly described. B-rep 
has a good effect in representing topologic relation of a complex 
object. In B-Pre, every object consists of finite surfaces, and 
every surface is enclosed by finite boundaries. Therefore, this 
representation is divided into volume, face, loops, edge and 
vertex. B-pre can also deposit geometric information and 
topology information in detail. 
The geometric information, which indicates geometric 
characters of the point, line (curve) and face. For example, 
surface, such as sphere surface, is represented into 
xz Rcos(0)sin(p), y = Rsin(0)sin(p), z 7» Rsin(0). The 
topologic information, which describes quantity, type and 
relation of the vertex, edge and face. Its purpose is to guarantee 
the object yielded is unique and legal. 
Cylinder 
   
  
  
C 2IPtop 
LProrward LP backward 
CAS LP bottom 
Ej Edge 4 
DL 
E e cua es 
Vp oMáÁN2 Edge 2 
Fig. 9. Object representation with combination B-pre and CSG. 
Our reconstruction method is based on this assumption that 
each industrial object is constructed by Boolean set operators of 
primitives, and each surface is described by B-pre. These line, 
curves, planar and curved surfaces in 3D space are described by 
parametric equations (Fig. 9). 
In the following analysis it is assumed that the extraction and 
location of features in image space have already been 
accomplished and the correspondences between these features 
are established. 
B) Mathematical model of reconstruction 
In GEMS, an object is constructed by four coordination system 
transformations (Fig. 10). Assuming that T' is a linear feature in 
Scene (Fig. 11), and geometric elements (information) of this 
feature are expressed into 
X = X(5) A 
Y = Y(5) 
Z = Z(5) 
Where $ denotes the parameters to describe primitive including 
position parameters and geometric parameters. We also assume 
that linear features £, £/ in the left and right image plan are 
perspective projection of the spatial linear feature T', and an 
arbitrary point p in the linear feature / is measured (Fig. 11), 
the collinearity equation can be formulated by the ray line pP . 
rg ua a (Xp — X3) +b,(lp = Ys) + (Zp = Zs) (i) 
a,(Xp — Xg) +b, (Vp = Ys) +e, (Zp — Zs) 
_ A7 (Xp — Xs) +b, (¥p = Ys) + C2(Zp — Zs) 
ay (Xp — Xs) + by,(Yp = Ys) + c3(Zp — Zs) 
In addition, the point P still meets the parametric equation 
describing the industrial object. i.e. 
  
  
Y, Ge F (17b) 
X, = XG) 
Y, - Y(3) de 
Z,-Z(3) 
Where a;, bj, c;, i — 1, 2, 3 are components of the rotation 
matrix for left image, X 91$, 2, ^ aie coordinates of the 
perspective center of the image sensor, f denotes a constant of 
the image sensor, Xp: 3p denote the coordinates of the 
point p in image plane, X ,,Y,,Zj denote the coordinates of 
the point P. 
  
  
o 
Fig. 8. CSG modeling. Fig. 10. Object and coordination system. 
The mathematical model of reconstruction using line 
photogrammetry is formed by substituting the parametric 
equation (18) into the collinearity equation (17). We have 
o UU 7 Xs) t S, - Ys) + (Ze, 7 Zs) 
  
  
3p 7 XE 19a 
P as(X psy = Xs) + b;(Yp (3 - Ys) * e(Zp(sy 22 % ) 
y, -%=-f aX psy = Xs) + Balers — Vs) + Ca (Z ms) — Zs) (19b) 
P o a; (X ys - Xs) + b3(Yy 3 = Ys) + C3(Zpcö) = 25) 
If an arbitrary point. p^ (non-correspondence with the point p) 
in feature ¢’ is observed, the scene point P^ corresponds to the 
point P” . The similar equation is formed. 
LAC gy 7 X8) * bre (s, - YD * eZ 5) - 25) 
  
  
tpe aU XD PR TO * er - ZO 
nS ; ay(X ps) - X$) +b3(Yp. 5, -Yg)-* ca (Z pus) — Zs) 
aX pg, - X) * P3 p s, - Y + C3(Z pugs, - 25) 
(20) 
Where aj, bj, c;, i = 1, 2, 3, X5,Y{,Zg, f'+ Xp» Yp 
I 
, 
for right image are similar parameters to left image. p 
corresponds to scene point P". 
  
Fig. 11. Configure of LP reconstruction. Fig. 12. Straight line feature. 
205