B.1) Straight line feature 
The analytic equation for straight line can be expressed into 
P=C+d-t (21) 
Where P indicates an arbitrary observed point in straight line, 
C 2 (Xc, Ye, Zc) denotes a fixed point, f is a parameter. 
Assuming that the point @ and b in the left and right image 
planes are observed (4 and b are non-correspondence points). 
The observation equation can be obtained. 
left image: 
2a (Xc tat = X5) +hQOc +O- -Y) +g +a-t, —Z5) 
A a E PS Koh C ADR ci) tela ra) 
LA (Xc t0 tj 7 Xs) tbc xa t 7 Y) co (Ze +A- 1 —Z5) 
  
  
  
  
x a(Xc+a-t -X) +50 +A- 1 - Y) ce (Zc att — Z3) 
(22) 
right image: 
—À aj(Xc * «t; - X9) c bj(Yc a: - Ys) c eq(Zc * a0, — Z5) 
as(Xc * &-15 - XS) c b3(Yc c a0 — Ys) + c3(Zc * a-t, — Zs) 
d 2o A Xc £a:5 - X9) e bfc a0 - Y9) t (Zc vat - Z3) 
Yo Jo, e f 
a(Xc * a: t) - XS) -b(Yc at) -Y9) e e(Zc at, —Z5) 
(23) 
Six fixative unknown parameters (Xo. Yo Zoo, B,v) and an 
additional unknown £ or £^ can be determinate by the equations 
22 and 23 for any a point. To the problem, the condition unequal 
N + N’ > 6 is met for two images; unique solution can be 
achieved. 
B.3) Conic Curve feature 
The conic curves indicate ones like circle, hyperbola and 
ellipse, as well as the perspective projection of the regular 
surfaces like sphere and spheroid. Let us take a sphere as an 
example to construct the mathematical model. 
Assuming that the model coordination system (describing 
sphere) is coincided with the spherical coordination system. An 
arbitrary measured point M in the left image is a respective 
projection of the point M in spherical surface. The collinearity 
equations can be formed by ray linemM . Moreover, the 
point M still meets the equation. 
x = Rsin(p)cos(6) 
y R sin(g) sin(0) 
z = Rcos(@) 
Thus, we have (for the left image) 
ay(R sing) cox6) — Xs) +4(R sing) sin(6) — Xs) + (Roos) — Zs) 
az(R sing) cos(6) — Xs) +b3(R sin(g) si(8) — Ys) + cz(Roos(g) — Zs) 
Sn = Ya 7f axR sino) cos(8) — Xs) pino in) = Ys) + (Roose) - Zs) 
a3(R sin(p) c0s(0) — Xs) * b(R sin(o) sin(O) — Y5) + c3(R cos(p) — Zs) 
(25) 
Where R is a radius, ¢,0 are the elevation and azimuth 
respectively. If another point € is observed, and the point E on 
the spherical surface corresponds to the point €, and the 
point M and the point E locate in the same hemisphere, the 
angle (0 is invariable, and the angle © is variable when the 
measured point is changed. If the point h is measured in the 
right image plan, the angles ,9 differ from the point 772, we 
have equation. 
sog, cop AiRsiny) cox0,) — X5) + BR sin.) sin(8,) — Ya) * (cosy) - Z5) 
^ 07^ 7 ayRsiXo,) cos8,) — X5)  b(Rsig,) siX8,) — Y5) + ,(R costp,) - Z;) 
Ws , aX (Rsir(o,,) oo(0,) — Xs) * b(Rsir(o,) si(8,) - Ys) * c; (Roo9,) - Z5) 
"ro as(Rsir(9,) co(,) — XS) * b(Rsi(o,) si(8,) — Y5) + c,(R cop, ) - Zs) 
(26) 
In order to obtain a unique solution by LSM, the unequal 
condition 2(N + N’)>(N + N’+ 3) must be formatted. 
(24) 
X473, 7f 
  
  
206 
Where N and NN ' denote the amount of the observed points in 
the left and right image planes. 
  
Fig. 13. Conic curve feature. Fig. 14. Intersection curve feature. 
B.2) Intersection curve feature 
Another kind of important linear feature is intersection curve, 
Let us take a plane intersecting with a cylinder as an example to 
derive the mathematical model. We also get 
a, (Xy — X5) +b (Yy = Ys) +c (Zy — Zs) (30a) 
a (Xy - XS) eb (Yg -Y,) t e (Zu = Zs) 
ay(Xy — Xs5) + ban — Y5s) + c2(Zn - Zs) (30b) 
  
Xp X, T 
  
Yi =; Bf a4(X, — Xs)+b3(Yıy — Ys) + C3(Zyy - Zg) 
xy = Rcos(0) 
YH = R sin(0) Gn 
Zu = A'Rcos(0) + B’R sin(0) + D’ 
Where: A,B,C are components of the normal vectors of the 
plane; D is a constant; R is a radius of cylinder, Ó indicate a 
parameter, A’ = A/C, B’ = B/C, D’= D/C. In the same way, 
we can obtain equation for the point Ww (non-corresponding to 
point h ) in the right image. 
No matter many points are measured on the intersection linear 
feature in the left or right images, only one unknown 0 is 
added. Thus if the unequal condition (N + N°) > 4 is met, the 
unique solution can be obtained by LSM. 
We here only discussed the mathematical models for straight 
line, intersection curve, and sphere. Others such as spheroid, 
parabola, hyperbola and sweeping can similarly be processed. 
These detail discusses can be located in [Zhou 1994]. 
C) Experiments 
Simulation: The simulated images are generated by back- 
projection. The camera parameters including interior parameters 
and exterior parameters are designed. The industrial objects are 
constructed by combining CSG and B-pre with GEMS data 
structure. The size of all simulated images without any noise 
are 200 x 200 pixels, with a pixel size of 50 um. In simulation, 
we divide experiments into four groups. The first group is to 
reconstruct cube using straight line. The second group is to 
reconstruct regular solid using conic curves. The third group is 
to reconstruct intersected objects by two primitives using 
intersection edges. The fourth group is to reconstruct complex 
objects by comprehensive usage of straight lines, conic curves, 
intersection curves. 
(1) The geometric elements for the cube (length, width, and 
height) are shown in table 4. Figures 15a, 15b are a stereo pair, 
and figure 15c is its 3D reconstruction drawing. 
(2) The geometric elements (radius, height, semi-major axis, 
semi-medium axis, semi-minor axis) for sphere, cylinder, 
spheroid are shown in table 5, table 6, table 7. Figures 16a, 16b, 
17a, 17b, 18a, 18b are the stereo pairs for sphere, cylinder, and 
spheroid. Figure l6c, 17c, 18c are their 3D reconstructions. 
(3) The geometric elements (radius, height, other parameter) 
for intersection of a plane and a cylinder, a sphere and a sphere 
as well as a cylinder and a cylinder are shown in table 8, table 9, 
table 10. Figu 
pairs. Figure: 
drawings. 
(4) The gec 
of the comple 
are a stereo pa 
Length (mm) 
  
T. V. JC. V: 
  
  
1000.0 | 999.9 
  
Table 5 (sphere) 
[Radius (mm) 
  
TV. IC. V. 
  
  
  
800.0 | 799.9: 
  
Table 7 (Sphero 
Semi major axis 
  
TV: CV. 
  
  
  
1200.0 | 1200. 
  
Table 8 (interse 
  
Radius 
  
T. V. T C. V. 
  
800.0 | 800.01 
  
  
  
Table 9 (interse« 
  
  
Big sphere radii 
TM ass] e. V. 
800.0 | 799.9: 
  
  
- Table 10 (inter: 
Big cylinder rad 
  
TV C. \ 
  
  
600.00 — | 600 
  
Table 11 (a com 
Primitive | Par: 
Name Nar 
| len; 
Cube | wid 
hei; 
| Up i 
dov 
hei 
  
  
  
i= 
Frustum 
of cone | trar 
[: 
  
(Hole) | trar 
Cylinder | par 
rot: 
par 
hme 
  
  
(Continue table 
Parameter 
  
Names 
Radius 
Height 
  
  
  
  
Transfer