Full text: XVIIIth Congress (Part B5)

ng to the 
a of least 
p is given 
3.6 
« diagonal 
shown in 
ubmatrix. 
1e camera 
ent matrix 
g N, 6x6 
ce all the 
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15 bundle 
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ts on each 
> non-zero. 
37 
Fig. 3.2 illustrates the sparseness of A. When the number of 
object points is 100, less than 396 of elements in Jacobian 
matrix 4 are non-zero. In this case, inverting the full size 
matrix of (4'WA) or two partitioned matrices which is common 
in the bundle adjustment is not efficient if the full covariance 
matrix is not necessary. The full covariance matrix may be 
valuable in some situations to evaluate the whole system, but in 
many situations (e.g. real-time) the diagonal elements of 
covariance matrix could be adequate to evaluate the accuracy of 
estimated 3-D co-ordinates of the object points. 
The two step separated adjustment makes full use of the special 
properties of the Jacobian matrix A and the principle of iterative 
least squares estimation. An accuracy evaluation of the 
estimated 3-D co-ordinates of object points and camera 
parameters are given approximately by 
Qrop T Gp WAS) 3.8 
Quep = (Ac WAS) " 3.9 
In some industrial applications, for example real-time 
monitoring of moving objects, the camera parameters are 
relatively stable while the 3-D co-ordinates of object points 
may move frequently. In this case, the object points can be 
located with good estimates for camera parameters which can 
also be monitored and if necessary adjusted. In addition real- 
time 3-D co-ordinate measurement for hundreds of targets can 
be achieved using inexpensive computers. Care must be taken 
when the two step separated adjustment is applied in 
photogrammetry in order to get the same results as the 
traditional bundle adjustment. The linearized observation 
equations should be the same for both steps. The objective of 
the minimisation is the sum of squares of the residuals on the 
image plane as is usual in the bundle adjustment but is often not 
the case in many intersection algorithms for example Shmutter 
& Perlmuter (1974). 
4. SIMULATION TESTS 
A simulation network was constructed to test the two step 
separated adjustment method in photogrammetry and compare 
it with the traditional bundle adjustment. Fig. 4.1 illustrates the 
configuration of the simulation test network. The object points 
were randomly distributed in a 400x400x200 mm. box with 
eight control points on the edges which were used to initialise 
the camera parameters. The focal length of the cameras was 25 
mm. The cameras were uniformly located on a circle with a 
distance of 2500 mm. to the centre of the box. The 2-D 
projections of the targets on the image planes were then 
computed. Approximate camera parameters were calculated 
using control points with deliberately added errors. 
Approximate 3-D co-ordinates of the object points were 
computed using the approximate camera parameter. Both the 
approximate camera parameters and the 3-D co-ordinates of the 
object points were then used as the starting values. The results 
of using the simultaneous adjustment and the separated 
adjustment were compared. 
591 
  
  
y 
Fig. 4.1 The simulation test network 
Table 4.1 shows some simulation test results of the bundle 
adjustment and the two step separated adjustment for a four 
camera network. The minimisation of the sum of squares of the 
residuals (v‘Wv) on the image plane is the objective of the least 
squares process. The values of v‘Wv calculated from both 
methods were always same (the small differences in the eighth 
decimal place is caused by the round off of input data), and all 
residuals on the image planes were the same for the two 
methods. A further check was made by comparing the 
difference between 3-D co-ordinates of the object points 
obtained from both methods after a 3-D transformation. The 
results indicated no differences to the level of precision used. It 
can be seen from Table 4.1 that the two step separated 
adjustment is much faster than the bundle adjustment especially 
when the number of targets is very large, since this method 
shows a linear computational expense with the number of 
targets. To measure 1000 targets for this four camera network, 
the two step separate adjustment needs only 103 seconds. It 
should be noted that the two step method iterates more times 
than the bundle adjustment but for real-time applications only 
one iteration may be required. 
v mm v mm 
1 0. 
0.00053658 .00053662 
0. 0. 
200 0.001 0.001 
2 0. 
300 1 0.0014851 32 0.001 
350 3488 0.001 34 0.001 
400 3967 0.00206680 41 0.00206678 
1000 103 0.00506967 
Table 4.1 Number of cameras = 4. (TSSA refers to the two step 
separated adjustment. GAP, the General Adjustment Program 
developed at the City University, is a simultaneous least squares 
estimation program used in survey and/or photogrammetric 
network adjustment - a typical Bundle Adjustment) 
The accuracy of the 3-D co-ordinates of the object points 
estimated by the two methods are the same since their results 
are same. In the two step separated adjustment method, the full 
covariance matrix is not calculated, the accuracy of the 3-D co- 
ordinates of the object points estimated can only be evaluated 
approximately by Eq. 3.8. Table 4.2 shows these approximate 
values and the values calculated from the full covariance matrix 
with a six camera network. It can be seen that the results are 
similar especially when the number of targets increases. So the 
approximately evaluated standard errors appear to be 
acceptable. 
mm 
0.05614 0. 
100 0. 0. 0. | H 0. 
1 0. 0.04693 | : Ä 
0. 0. 0. 0. 6 0 
Table 4.2 Number of cameras = 6, 6, = 0.001 (mm), œ = 90° 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B5. Vienna 1996 
 
	        
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