2.2. Relative Orientation 
In our stereo vision system, two cameras are 
mounted on a stationary platform. This means 
that the relative relationship between the two 
cameras is constant. There are two ways to 
determine the relative orientation: 
a) Stereo camera calibration with relative 
orientation constraints. This is done by 
introducing six constraints in the camera 
calibration procedure to keep the 
relative orientation and the scale of all 
stereo pairs constant. 
b) An individual relative orientation based 
on the co-planarity condition. 
2.2.1 Relative Orientation Constraints 
The relative orientation constraints are 
introduced to the camera calibration 
procedure[2.1]. The analytical formulation of 
this problem is based on the relative rotation 
matrix AR and the relative base vector B 
between left and right images of any stereo 
pair. The relative rotation matrix is computed 
as: 
Ary Ary Ans) 
AR=RIR EA," Ary] A (3) 
Ary, Ary, Arg 
where : 
RL Rotation matrix of left image 
RR Rotation matrix of right image 
There are only three independent variables in the 
relative rotation matrix AR. We use three 
rotation angles (A@w,Ap,AK). A constant AR 
matrix is achieved by keeping the relative 
rotation angles (AW, Aq, AK) constant. Only the 
elements Arj,, Arj;, Arj4, Àrj, Ar44 are needed 
for further computation. Their functional 
relationship with the rotation angles is given by : 
Ar,; = sin Ag 
These three angles are equaled for every image 
pair. Assuming that image pair (i) has a relative 
rotation matrix AR”, and image pair (k) has the 
corresponding matrix AR, they must satisfy 
the following equations : 
Ar = Ar? 
Ar Ar = Ar Ar (5) 
Ar arf) = ar)ar(? 
The relative base vector B is the base vector 
defined in the left image coordinate system: 
b, Xg £4 X, 
Be J: (2p loyraug (6) 
With 
RL Rotation matrix of left image 
(X,,Y,,Z,) Position of left image 
(X,,Y,, Z4) Position of right image 
The base vectors are also equaled for every 
image pair : 
p? = pO 
X X 
b= pio (7) 
p? = po 
Z Zz 
After linearization, equations (5) and (7) are 
used as constraints in the bundle adjustment to 
improve the accuracy of the camera parameters 
(He, Novak, 1993). 
2.2.2 Co-planarity Condition 
The co-planarity condition means that the two 
conjugate image points and the two perspective 
centers are in one plane. It is defined by : 
b, b, b, 
u v wlzO (8) 
WERE Ww 
where (u,v,w) and (u’,v’,w’) are the three- 
dimensional image coordinates of the left and 
right images. We selected the base vector (1, 
by, bz) and three rotations (A90, AQ, AK) of 
the right image in the left image coordinate 
system as the relative orientation parameters. 
After linearization of this equation, the relative 
orientation can be computed easily. No control 
points are necessary for this computation. For 
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