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;