Figure 2 Normal equation structure of the camera 
parameter portion of the constrained bundle adjustment of 
stereopairs. 
INVARIANCE MODELLED BY MODIFIED COLLINEARITY 
EQUATIONS 
This approach sees the left camera of each stereopair 
being modelled by conventional collinearity equations and 
the right camera is modelled by modified collinearity 
equations written in terms of the left camera coordinate 
system instead of the object coordinate system. Therefore 
the position and orientation of the right camera is 
determined with respect to the left camera. The result is to 
reduce the number of camera parameters from 12s to 
6s+6 for s stereopairs. 
The set of modified collinearity equations for the right 
camera are developed below. Subscripts define the object 
(L = left camera, R = right camera, P = object point), 
superscripts define the coordinate system (none = object 
space coordinate system, L = left camera coordinate 
system, R = right camera coordinate system), X = vector 
of plate coordinates, s = point scale factor, À = camera 
rotation matrix and X = vector of coordinates. 
Points on the Left image in object space 
coordinates: 
ZL = s B UG-X) (6) 
Points on the Left image in Left camera coordinates: 
Rt = s, Xe (7) 
LP 
from which: 
Xp = R,(X,-X;) 
Points on the Right image in object space 
coordinates: 
XP = SpA (Xp-Xp) (9) 
Points on the Right image in Left camera 
coordinates: 
XP s sU XA (10) 
and so plate coordinates of the right hand camera 
expressed in terms of the left hand camera's 
coordinate system: 
RR = s AHA XS- X -XA (11) 
Expressed in the more conventional form: 
t ICQ - Y) ens (i-Z? 
"i (P-X) «rg (Q- Y) «s (i-Z) 
c (PX (Q- Y) eras (R-Z? 
(PX) «rg(Q- Y) «rss (R-Z) 
c, - principal distance of right camera, 
  
  
Kk 
rly...rs3 = rotation matrix elements of right 
camera in left camera coordinates; 
X^, Y Z' - X,Y,8Z coordinates of right camera 
perspective centre in left camera 
coordinates, 
P = 1y(Xp-X) +r Yp- Y) +115(Zp-2); 
Q - no(Xoe-X)) *roY(Ye- Y.) *fos(Zp-Z1); 
Fl 7 fy (Xo-X)) +a2(Xp- XL) * ras (Z7 Z4); 
M 
lj4.../34 = rotation matrix elements of left 
camera in object coordinates, 
Xp YoZp - object coordinates of imaged point, 
X, Y,Z, - object coordinates of left camera 
perspective centre. 
The resulting set of observation equations is: 
V «BA «C (13) 
V . 
| B, By B X e 
- V, _ = — C, 
Fol hepa Bho ly Lagu À 
- 0 -I 9 5 Cr 
V 0 0 -/ À 
V = vector of plate observation residuals; 
V, = vector of left camera exterior 
orientation observation residuals;