B. One Stereo-Pair 
One Point Added 
A. One Stereo-Pair 
No Points Measured 
  
  
  
  
  
C. One Stereo-Pair with 
One Point Measured. 
One Stereo-Pair Added 
Figure 3: Changes in Matrix Structure as Points and Stereo-Pairs are Added 
At this point the datum is established by fixing 
seven parameters. Typically, these are the 
coordinates of the perspective centers and one camera 
rotation angle. When additional stereo-pairs or new 
points are added to the system, memory for the upper 
triangular matrix, its inverse, and the solution 
vector is allocated accordingly, and approximate 
values for new parameters are computed. To avoid 
singularities, a minimum of five points must be 
measured for each additional  stereo-pair. The 
determination of approximate values is addressed 
later. 
the observations and 
residuals appear on the screen. The solution 
vector, corresponding precision values and the 
variance factor are visible to the operator upon 
request. A simultaneous solution can be performed 
anytime at the operators convenience. 
When points are measured, 
Variable Size Parameter Vector 
Two types of parameters can be added to the system; 
the twelve orientation parameters of a new stereo- 
pair or the three coordinate parameters of an 
additional point. For both types of additions, the 
matrix storing D and R increases in size. To better 
illustrate how this matrix structure changes when 
stereo-pairs or points are added, Figure 3 can be 
used. 
The shaded portions in Figure 3 represent additions 
to the matrix structure. This is also indicated in 
the bold type in the caption under each matrix. In 
Figure 3A memory has been allocated for the initial 
stereo-pair. This is a 13 x 13 matrix where columns 
1-12 are devoted to the orientation parameters of the 
left and right cameras. Column 13 contains the right 
hand side, and element (13,13) maintains the residual 
sum of squares. 
In Figure 3B we still have one stereo-pair but one 
new point has been measured. This adds three rows 
and three columns to the original matrix structure to 
account for the three dimensional coordinates of the 
point. The matrix is now 16 x 16. 
In Figure 3C, one stereo-pair is added to the matrix 
structure in Figure 3B. This adds 12 rows and 12 
columns for the orientation elements of the second 
stereo-pair to the matrix structure. The matrix has 
grown to 28 x 28. 
Adding a new point or stereo-pair to the matrix 
structure is not simply a matter of allocating the 
proper amount of memory. The triangular matrices 
used in this program are stored as one dimensional 
vectors. When additional memory is allocated to an 
existing matrix, it is simply tacked onto the end of 
that vector, leaving all previous elements in their 
original memory locations. As can be seen in Figures 
3B and 3C, the correct position for the newly 
allocated memory actually lies within the vector, not 
at the end. Therefore, all original elements must 
be shifted to maintain their correct location. 
Approximate Values 
The choice of approximate values can be a problem 
with sequential estimation in a non-linear system. 
Sequentially updating the parameters is not a viable 
solution unless the entire system is re-linearized. 
Otherwise, if the initial values were not good, the 
parameter vector would "drift off" [Gruen, 1985]. 
One answer to this problem is simply to start with 
good enough approximate values. This however, may 
not always be possible. The more practical solution 
is to perform a simultaneous solution periodically. 
We follow this approach in our implementation. 
For the first stereo-pair in a strip, the true 
coordinates of the perspective centers are determined 
by relating the position of the cameras to the known 
coordinates of the GPS antenna and utilizing the 
information from the inertial system for orientation. 
This information is available for each stereo-pair as 
long as satellite lock is maintained and is stored 
in a feature file. For stereo-pairs during the 
loss-of-lock period, approximate values for are 
determined from the distance and direction traveled 
from the previous stereo-pair. This information is 
available from the inertial system or can be 
estimated from the speed of the vehicle. Approximate 
values for the coordinates of new points are obtained 
by intersecting light rays from the two perspective 
centers of the camera-pair. 
Control 
For a strip of stereo-pairs obtained by the mapping 
van, there are no control points along the road. The 
datum is defined by the known GPS positions of the 
van before and after a loss-of-lock period, and from 
the orientation information at these positions, 
which is available from the inertial system.