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
estimated
ned by a
s in this
jeters are
> required
tical tests
ways give
15 bundle
very weak
e given in
nstant and
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e.g. 14 or
parameter
ock on the
be 14x14,
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adjust the
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= Jacobian
linearized
sparse and
ts on each
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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