ndent; 
lative 
ive tilt 
netries 
ld. In 
o-pairs 
st-field 
of the 
ce the 
dolites 
as the 
en the 
-vision 
hich 3- 
> Very 
tment. 
vo van 
In the 
nbined 
X each 
tfield of 
ntrol 
ints 
Paral 
ion 
t with 
aints. 
s done 
ons (1) 
ntation 
ted as 
ates of 
(c), in 
a. We 
rtions, 
ind for 
(1 
Nx 2 r11 (X - Xo) * 121 (Y - Yo) * 131 (Z — Zo) 
Ny 2 r12 (X - X9) * 122 (Y - Yo) * 132 (Z — Zo) 
D zrj3(X- Xo) * r23 (Y - Yo) * 133 (Z — Zo) 
with: 
XYZ... 5 coordinates of an object point (target), 
Ka Lo Lg perspective center of the camera, 
Yl: ge elements of a 3 x 3 rotation matrix 
modeling the attitude of the cameras. 
Correction terms (additional parameters): 
Q) 
AX z Xp X (1? — 1) aj * x (1* — 1)a2 * (1? + 2 x?)a3 + 2xÿa4 + asX + ay 
Ay =yp+y (12-1) aj +y (14 — 1)ag t 2xy a + (12 + 2y2)a4 — asy 
With: Xp, Yp..... image coordinates of the principal point, 
aj, az .... radial distortion parameters, 
as, a4 .... decentering distortion parameters, 
as, ag .... affine parameters. 
Next, constraints were added to fix the relative 
orientation of the two cameras of any stereo-pair. 
Basically, we have to introduce six parameters to keep the 
relative orientations and scales of all stereo-pairs constant. 
It is of advantage to select the base-vector (bx, by, bz) from 
the left perspective center (Or) to the right one (Og), and the 
three rotations of the right image (Aw, Aq, AK) as relative 
orientation parameters. These values are defined in the left 
camera coordinate system (figure 4). The relative angles 
are small as the two cameras are pointing in almost parallel 
directions, and are mounted orthogonal to the base (= 
normal case stereo-pair). 
  
  
Figure 4: The base vector b and the relative rotation 
matrix AR are defined in the left image coordinate system 
The analytical formulation of this problem is based on 
relative rotation matrices (AR) between the two images of 
any stereo-pair. In the case of fixed stereo-cameras, AR 
must be the same for all image-pairs, which can be 
achieved by keeping the relative rotation angles (Aw, Aq, 
Ax), which form AR, constant. The relative rotation matrix 
1$ computed by (3). 
T A; Ay Aj; 
CCC; 
wi: AR... relative rotation matrix, 
RI ee rotation matrix of left image, 
Rp ie rotation matrix of right image. 
Rr and Rg are defined in the object coordinate system. 
Only the parameters A1, A2, A3, B3, C3 are needed for 
further computations. Their functional relationship with 
the rotation angles is given in (4). 
A3 = sin AQ 
—A2/A1 = tan AK (4) 
—B3/C3 = tan A0 
The base-vector (by, by, bz) can be expressed in the left 
1mage coordinate system by (5). 
by T XR = XL 
B= by SR YR- Yı (5) 
b, ZR —. ZL 
Wh: BI e rotation matrix of the left image, 
(Xi. YL.Z1)......-- left perspective center, 
Xp, YR ZR)... right perspective center. 
Once the base-vectors and the relative rotation matrices 
of two stereo-pairs are available, they can be set equal to 
ensure that the orientation parameters of the corresponding 
images are constrained by the vision system's relative 
orientation. Assuming that b? is the base-vector of stereo- 
pair (i) and ARÓ its relative rotation matrix, and b(? and 
AR) are the corresponding elements of stereo-pair (k), 
they must satisfy equations (6). 
AR® = ARK); (6)