it possible
ration tool
| can be de-
iodel range
1989), large
0), medium
eipe, 1990;
aller format
Riechmann,
cult for the
lled on the
lemented in
can get ap-
a on the job
the selected
; problem is
tion. While
non optimal
called con-
c whole ad-
litioned and
'o avoid this
ig technique
ram will al-
Jess the ex-
e protection
> application
bject coordi-
rs four ways
given
ate in the ad-
datum.
articipate in
n the numer-
ith respect to
observations
e datum with
n the quality
e solution to
rientation pa-
ent or defini-
to the object
ement the six
theodolite of
datum. If the
1 to a leveled
ing degree of
distance mea-
> datum by an
results in the
ock geometry
able three di-
> case in close
range applications but not in aerial applications. If
this requirement is fulfilled the free adjustment
: will transform the network build from the observa-
tions only, without any constraint onto the initial
values of selected object points. This results in an
error free datum of very high quality. The extreme-
ly high quality can be proofed by the resulting error
ellipsoids on the object points. They become
smaller than with any other kind of datum defini-
tion.
4.1.4. Algorithmic Aspects
To gain high perfcrmance of a bundle program the
selected numerical algorithms are essential. Typical
for combined adjustment is a not regularly struc-
tured normal equation system, in contrast to the
regular structure of an aerial block adjustment. As
the number of unknowns in bundle adjustments is
usually quite high, the normal equation system is a
sparse matrix. Special algorithms for general sparse
matrices exist (George, Liu, 1981). After various
empirical tests (Hinsken, 1985), the best suitable
algorithm of that class was modified and implem-
ented in CAP. This algorithm requires the minimum
amount of computer memory to store the normal
equation system. The storage technique doesn't
store any zero elements. Before the numerical ad-
justment is performed a logical computation takes
place which figures out, in which position of the
normal equation system non zero elements will ap-
pear during the numerical computations. Besides
the minimum storage requirements and minimum
computation time it ensures also, that round off er-
rors are reduced, because less numerical operations
are performed.
As the non zero structure of the normal equation
system depends on the ordering of the unknown pa-
rameters in the equation system, CAP uses a reor-
dering algorithm to find the best order of unknowns
in the equation system. This is not a trivial problem
in combined adjustments. However the numerical
results are not effected by the order of the un-
knowns. The implemented reordering algorithm is
the so called Banker's algorithm (Snay, 1976). The
aim of the algorithm is to reduce the amount of
computations during the numerical handling of the
normal equation system.
Furthermore the fastest known algorithm to invert a
sparse matrix is implemented in CAP (Hanson,
1978). This special algorithm inverts the matrix
only at those positions where non zero elements
were located by the preprocessing logical algo-
rithm. This allows for full error propagation for un-
knowns and functions of unknowns (e.g.
observations).
4.1.5. Statistical Model
The implemented numerical algorithms were se-
lected in such a way that full statistical error
propagation can be performed. This is an important
feature for industrial applications.
The following statistical values are provided in the
output file:
* standard deviations of object point coordinates
15
+ standard deviations of exterior orientation
parameters (image, model, theodolite)
* Standard deviations of camera parameters
* Standard deviation of adjusted geodetic
observations
* local redundancy of all image, model and
theodolite observations
* correlation matrices of camera parameters
* correlation matrices of correlation between
exterior orientation and camera parameters
* internal reliability of each observation
* Single point test statistic for deformation
analysis
* variance of unit weight
* variance components of model observations
Furthermore statistical information is provided in
separate files:
* covariance matrices of object points
(computation of error ellipsoids)
* covariance matrices of projection centers (see
above)
To help the user analyzing the results the before
mentioned information can be visualized by use of
a special graphical analyzing program.
4.1.6. Blunder Detection
The bundle adjustment is based on a least squares
adjustment. The results of least squares adjustments
are only correct, if the observations are free from
gross errors. If this is not the case, the gross errors
must be eliminated from the adjustment. If those
blunders are not eliminated from the set of observa-
tions they cause so called smearing effects on all
estimated parameters. Therefore a statistical test is
implemented in CAP to detect if the observations
contain blunders. The test is performed on residuals
which are divided by there respective standard
deviation. The computation of the standard devi-
ation of the residuals involves the inverse of the
normal equation matrix. Therefore the geometry of
the block is included in the blunder detection test
(Pope, 1976).
4.1.7. Deformation Analysis
The ability of CAP to perform a free net adjustment
leads to another ability namely to analyze deforma-
tions on object points. Therefore a single point test
statistic is computed. This test value is used to fi-
gure out if individual points have changed their
coordinates significantly. The normal equation in-
verse is included in the computation of the test sta-
tistic therefore the geometry of the block and also
the accuracy is included in the deformation analy-
sis. The deformation analysis is an iterating pro-
cess. The basic aim is to bring the two object
coordinate systems of two different epoches into
the same datum. To figure out which points can be
used to define the datum, the test statistic is used.
The method is used successfully in industry for
regular tool inspections and also on huge deforma-
tions (e.g. crash tests).
