nly 
ion 
(19) 
the 
ter 
ga 
(20) 
ind 
hat 
'Or- 
[he 
ing 
an 
21) 
2a) 
2b) 
or 
ect 
ent 
23) 
kq = Ma M's wi; 
Al = Q(Bk); 
Aui cu EPAIS 
927 r-u+s’ 
t t -1 
Q-P; M-BQB; N- ALM A,; w - A, x; 
This is a nonlinear problem, why the process has to be 
iterated until Axi*! is less than a certain selected limit. 
For each iteration, then new parameters are computed 
by 
XH = Si + Axi+l (24) 
4. TESTS 
The main discussion concerns the critical geometric 
situation where a straight line under observation lies 
in the same plane as the projection centres of the 
image pair. The situation makes it impossible to cal- 
culate the line parameters. This is also known as the 
epipolar case. 
In the test we do not use the angle in the image be- 
tween the line and the epipolar plane. Instead the 
angle between the direction vector d and the difference 
vector b of the two image projection centres in object 
space is used as an approximation. In the following, 
this angle is called EPI. Several image combinations 
have been chosen to give a wide variation of the angle 
EPI, i.e. between 0 and 100 gon. The result from tests 
where more than two images have been used is not 
dependent of the epipolar case. Also the influence 
from the geometric constraints on the resulting line 
parameters, i.e. vertical and horizontal, are checked. 
Test results are shown in column diagrams, where 
black is used to represent results where no geometric 
constraints have been used. White represents results 
with geometric constraints. The origo of the ground 
coordinate system has been moved into the 
approximative centre of each building. This is done to 
achieve a better numerical stability. Two buildings, A 
and B, are used in the tests. Building A is projected in 
five and building B in four images. 
4.1 Separate lines 
All lines of the two test buildings were calculated 
using a) measurements from all available images 
which gives a result independent of the angle EPI and 
b) measurements using image pairs. 
It is assumed that the changes of the angle EPI is one of 
the most important factors influencing the precision 
and reliability of the calculated 3D lines in case b. 
Therefore all estimated a posteriori standard devia- 
tions of the adjusted line parameters are presented in 
relation to the variations of the EPI-angle. 
The standard deviations of the angles 8, ¢,y do not 
have the same units and dimensions, as the metric 
distance r why the results are presented separately. 
673 
Figure 3 show the mean of the standard deviations of 
the three angles 6, ©, y for test case a described above 
where each line of building A and B is shown 
separately. One result from figure 3 is that geometric 
constraints leads to a reduction of the standard 
deviations. The standard deviations of the angles of 
the horizontal lines, which are line 1 to 6 for building 
A and 1 to 4 for building B, are not as much decreased 
as the standard deviations of the vertical lines. On the 
other hand the standard deviations for the metric 
parameter r does not give a much better result when 
geometric constraints are introduced as shown in 
figure 4. These results are independent of the angle 
EPI. 
Figures 5 and 6 show the results from the test case b. 
The mean of the standard deviation of the angle 6, Q, y 
is plotted as a function of angle EPI. This is done to see 
if it is possible to derive a logarithmic function which 
can describe the variation of the calculated standard 
deviations as a function of angle EPI. 
An examination of the approximate logarithmic 
curves in figure 5 shows, that the introduction of the 
horizontal constraint decrease the mean of the stan- 
dard deviations of the angles with approximately 
power of one, especially for building B. Without geo- 
metric constraint the mean values increase significant, 
if the angle EPI becomes less than about 25 gon. Figure 
5 shows that. The introduction of the horizontal con- 
straint do not result in different standard deviations 
for parameter r. 
4.2 Combination of straight lines 
The same measurements and results from the 
adjustment process are used as in the previous section. 
Results from using stereo photogrammetry and mono- 
scopical line photogrammetry will be compared with 
each other and with the results achieved using a geo- 
detic measuring method. 
The subject of the investigation are horizontal dis- 
tances between and height differences of the corner 
points of the two test buildings. The coordinates of the 
corner points are directly measured using stereo 
photogrammetry. In line photogrammetry the corner 
points are defined as the point of gravity of those 
vectors, which are the shortest connection between the 
calculated 3D straight lines. 
The geodetic system, which is used as reference system, 
is based on a local coordinate system. Thus the high 
precision of this method is prevented. The two photo- 
grammetric methods are based on the official ground 
coordinate system. Therefore horizontal distances and 
the differences of the corner point heights to the mean 
height of each building are compared, instead of the 3D 
point coordinates. 
Horizontal distances The distance s between two 
corner points i, j are calculated as 
  
E 5 mS 
5 = A 6,-X02« (,- Y) (25)