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left 0.050 0.030 6.80 0.00 0.00 0.00 
| right 0.030 0.050 6.90: 0.10 0.10 0.10 | 
Table 1: Simulated Interior and Relative Orientations of the Vision 
System Cameras 
Practically, the perspective center coordinates and 
one camera rotation angle for the first stereo-pair 
are fixed by using pseudo-observations with high 
weights. This is done by setting the corresponding 
diagonal elements of matrix D to a high weight. 
Points measured in the first stereo-pair then, are 
determined by intersection. Point coordinates and 
orientation parameters for all additional stereo- 
pairs are computed using the full bundle method. 
When lock is regained the GPS position of the van 
will not exactly correspond to the position computed 
by sequential triangulation. Corrections must be 
applied to all perspective centers and orientation 
parameters between the first and last stereo-pair. 
This is accomplished by fixing the orientation 
parameters of the last stereo-pair just as those of 
the first stereo-pair are fixed, and then performing 
a simultaneous adjustment. 
Implementation of a Simultaneous Solution 
To prevent the solution vector from drifting, a 
simultaneous adjustment including a relinearization 
of the design matrix must be performed periodically. 
This is accomplished in the program by actually 
maintaining two versions of the solution vector. One 
version remains constant throughout the updating 
procedure. This version changes only when the entire 
system is re-linearized through a simultaneous 
adjustment. The second version of the solution is 
used for updating. To obtain this solution, a back- 
substitution is performed into the current upper 
triangular matrix to derive parameter corrections. 
These corrections are applied to the constant 
solution vector to obtain the updated solution, 
which is echoed to the screen for the operators 
analysis. 
With regard to on-line triangulation, the time at 
which a simultaneous solution should be performed is 
an important consideration. The most likely time is 
when the operator is involved with tasks unrelated 
to the sequential estimation. The best time for this 
would be as soon as possible after a new stereo-pair 
is introduced because the errors in approximate 
values are largest at this point. In our program, 
the operator has the ability to perform a 
simultaneous solution at his convenience. 
TRIANGULATION METHODS AND RESULTS 
As the calibration for the vision system cameras was 
not available at the time of writing, a simulated 
sequence of 5  stereo-pairs was mathematically 
designed to demonstrate the procedure necessary to 
sequentially triangulate an actual strip obtained by 
the mapping van. 
Several assumptions were necessary in the design of 
this sequence. First, we assume that the van is 
traveling at approximately 55 miles per hour. This 
means that there should be about 24 meters between 
stereo-pairs. To match the stereo-vision system we 
constrain the base length to 1.83 meters and choose 
the interior and relative orientations of the cameras 
to be as close to reality as possible. These 
simulated orientations appear in Table de 
Assumptions are also made about the probable location 
of ground points which would appear in the images. 
By projecting these 3 dimensional coordinates into 
image space we obtain their image coordinates. All 
object coordinates and rotation angles refer to the 
vision coordinate system. 
It must be emphasized that because this strip is 
simulated, the results obtained from the 
triangulation are too optimistic to expect from an 
actual sequence, and that the purpose here is simply 
to demonstrate how the van location can. be 
sequentially tracked through a loss-of-lock period. 
To begin the triangulation process, the datum is 
established by fixing the object coordinates of the 
camera perspective centers of the first stereo-pair 
along with the rotation angle of the left camera 
around the x axis (o). To simplify matters, the 
coordinates of the left perspective center of the 
first stereo-pair are assumed to be at the origin of 
the vision system at (0,0,0) and the right 
perspective center at (1.83,0,0). At this time all 
points are measured in the first stereo-pair. At any 
time during the measurement procedure the current 
point solution and camera exterior orientation are 
available to the operator. In this sequence, a total 
of 22 points were measured. Eleven of these were 
measured in the first stereo-pair and a simultaneous 
adjustment was performed. For subsequent stereo- 
pairs, any tie points appearing previously were 
measured first and then a simultaneous adjustment was 
performed to reduce the errors in the orientation 
parameters of the new stereo-pair. After the 
adjustment, any additional points could be computed 
accurately by simply performing an update. This 
procedure is necessary because new point 
approximations are computed by intersection and rely 
on the orientation parameters of the current stereo- 
pair. Obviously, better orientation values yield 
more accurate approximate values for added points. 
Due to the precision of the simulated strip, the 
exterior orientations of each stereo-pair did not 
change throughout the sequential adjustment. For 
this reason, only the values obtained by the final 
adjustment are presented. These appear in Table 2. 
Included are the coordinates of the camera 
perspective centers for each stereo-pair, the 
rotation angles (w,¢,k), and the standard deviations 
for each. 
In Table 2, it can be seen that as the strip length 
increases, the precision decreases. This occurs 
because the strip is only controlled at the first 
stereo-pair and the geometry worsens as the strip 
becomes longer. 
CONCLUSIONS AND RECOMMENDATIONS 
A mobile mapping workstation has been developed at 
the Center for Mapping at The Ohio State University. 
The workstation integrates a GPS receiver, an 
inertial system, and a stereo-vision system for 
automatic recording of highway data. This article 
focuses on utilizing the stereo-vision system for 
solving the "loss-of-lock" problem which occurs when 
satellite signals cannot reach the GPS antenna due to 
obstructions. 
The computer program developed for this research 
implements a sequential adjustment method known as 
Givens Transformations Without Square Roots to tie 
together a strip of stereo-pairs obtained by the 
vision system during loss-of-lock. This strip is 
formed in an on-line triangulation mode and is 
controlled by the GPS positions of the van before and 
after loss-of-lock. The perspective centers and 
orientations of the cameras are determined 
sequentially through the strip allowing the vehicle 
to be navigated through the loss-of-lock period. 
Givens Transformations provide a method for solving 
least squares adjustments without forming the normal 
equations. Givens Transformations Without Square 
Roots requires significantly less operations than 
conventional Givens Transformations and thus has 
become of interest for problems requiring results in 
real-time, such as on-line triangulation. 
Givens Transformations are advantageous in terms of 
numerical stability, efficiency, and storage 
requirements. Additionally, in an interactive mode, 
observations can be added and deleted and the updated 
solution vector and cofactor matrix can be obtained 
efficiently.