(8) 
(9) 
(10) 
(11) 
(13) 
Vg = vector of right camera exterior 
orientation observation residuals; 
V - vector of object coordinate observation 
residuals; 
B, = matrix of left camera exterior 
orientation partial derivatives; 
Bj, = matrix of right camera exterior 
orientation partíal derivatives; 
B - matrix of point coordinate partial 
derivatives, 
A, = vector of left camera exterior 
orientation parameter corrections, 
À pn = vector of right camera exterior 
orientation parameter corrections, 
A = vector of point coordinate corrections; 
e = discrepancy vector of plate observations; 
C, = discrepancy vector of left camera 
exterior orientation parameters; 
C p = discrepancy vector of right camera 
exterior orientation parameters, 
C = discrepancy vector of object coordinates. 
The normal equation matrix structure for this model is 
shown in figure 3. 
  
  
  
  
  
Figure 3 Normal equation structure of the camera 
parameter portion of the modified collinearity equation 
bundle adjustment of stereopairs. 
Each left camera of a stereopair is represented by a 6x6 
symmetric sub-matrix generated by the conventional 
collinearity equations. The relationship between the left 
and right cameras is represented by one 6x6 symmetric 
sub-matrix and two 6sx6 sub-matrix generated by the 
modified collinearity equations for the right cameras. The 
number of camera parameters is reduced to 6s +6 
compared to the conventional model however this model's 
degrees of freedom is the same as the conventional model. 
Object space orientation and position of the right cameras 
of each stereopair can be computed from the following 
relationships by back substitution after the adjustment is 
completed from: 
29 
= RpAXE 
RRB AXE (14) 
i = Fi, 
Xp = vector of right camera coordinates; 
ip = rotation matrix of right camera in 
object coordinates, 
AXF = vector of objectiright camera 
coordinate differences in right 
camera coordinates; 
Ry, = rotation matrix of right camera in 
left camera coordinates 
Il 
2 5 
X - B/X, (15) 
= AUX, 
X, = vector of left camera coordinates 
X = vector of object coordinates 
A, 
= rotation matrix of left camera in 
object coordinates 
TEST DATA 
Four stereopairs were taken of a simple object. All points 
on the object were coordinated by theodolite intersection 
to a precision of +0.1mm in each of the three coordinate 
axes. Two stereopairs had horizontal camera bases and 
two vertical. A total of 19 points were observed on all 
images with an ADAM Technology MPS-2 Micro 
Photogrammetric System. 
The photographs were taken by two Canon AE-1 Program 
cameras mounted on a bar with a nominal base of 0.200m. 
The object filled the image frame at a distance of 
approximately 1 metre giving a base:height of 1:5. The 
MPS-2 and both cameras were calibrated by the ADAM 
software prior to the observations being made. 
Approximations of the camera parameters for each 
stereopair were obtained from the ADAM software. Refined 
approximations using all stereopairs were obtained from a 
simple bundle adjustment using unweighted plate 
observations. A self-calibrating bundle adjustment (Fraser 
1982) showed that gross image correction parameters Af 
(camera principal distance correction) and K, (first radial 
lens distortion coefficient) were statistically not significant in 
the plate observations. 
This data was processed by three adjustment models: 
a. a CONVENTIONAL bundle adjustment; 
b. the CONSTRAINED bundle adjustment and; 
c. the MODIFIED collinearity equation model. 
Termination of each adjustment was controlled by either 
the change in the reference variance or magnitude of 
parameter corrections reaching specified limits. Tables 1 
and 2 give the results of the three adjustments with the 
object points treated as control (Table 1) and as unknown 
(Table 2). The precision of the quoted results reflects the