in image 
)e repre- 
] points, 
l'he spa- 
nd point 
1 ??) se- 
cies. For 
Ling Wu, 
estima- 
are set 
ch). The 
‘imation 
on via 
1 control 
possibly 
e image, 
matches 
it corre- 
ors. The 
| in that 
rrespon- 
orrectly. 
npletely 
ge infor- 
1dencies 
IAMPEL 
mon fit 
e orien- 
'rpreted 
rom the 
imator, 
n whith 
(7) 
(8) 
1ations 
(9) 
inction 
ique to 
| adap- 
n itea- 
ights is 
al mat- 
action 
calcu- 
w;* - w- f(t*-!) (10) 
leads to an iterative LS procedure weighting down those ob- 
servations with large test statistics t. 
As weight function a weighted combination of the following 
two functions are used 
A(t) 3 I and = ezp(—0.5(t/cy^) 
1 + (t/c)? 
(11) 
  
  
  
with the critical value c = 3. 
The first function guarantiees a global convergency. The se- 
cond one cancles the effect of large outliers onto the result 
[cf. HAMPEL 86 / FÓRSTNER W. 89]. The influence of the 
second function at the first iteration is zero and continously 
increasing with each iteration. Therefore the inconsistencies 
depending on the problem of partial matches are cleaned in 
the first few iterations while the effect of the wrong correspo- 
dencies between image edge and model edge onto the result 
is increasingly reduced with each iteration. 
The observation equation for the LS-fitting of the 3-D con- 
trol point models into the image are given by the well known 
equations of a spatial resection here applied to homologous 
straight line segments. For each homologous line a set of fol- 
lowing 4 equations is added to the normal equations resulting 
from equating the start point and end point of an image edge 
p! to the start and end point of the model edge p" using the 
projective equations. 
Ps : es p, Y" 
(soya w« 
The weight matrix is given by eq. 3. 
The start and end point for each homologous line are treated 
seperatily to allow all the partial matches shown in Fig. 2, 
especially when calculating the weights using eq. 3 respec- 
tively eq 5. The measurement checking the start point-, the 
end point- and line-hypotheses are test statistics with respect 
to the local coordinate system (u, v). To calculate these sta- 
tistics the residual vectors e,, e, and the covarinace matrix 
of the residuals has to be transformed into the local coordi- 
nate system using the same rotation matrix R4 shown in eq. 
6. Thus the test statistics t are calculated as 
1; E eu I 0, 
te = Cue] Tu. : (13) 
i = (Cu eye) Dr ( - )re 
The residuals in the local (u,v) system are analogous to lon- 
gitudinal and lateral error of the fit of image and model edge. 
Thus these test statistics are the proper measurements for 
checking the start point-, the end point- and line-hypotheses. 
The test statistic are normally and (square root) Fisher dis- 
tributed. We have 
ncNO, &NO,) AVE (1) 
The test for the point or line-hypotheses consists of compa- 
ring the test statistics with critical value e.g. c — 3. If the 
test statistcs exceeds the critical value, for numerical reasons 
the weight of the observation is set to zero so having no in- 
fluence on the result. Otherwise it is used to calculate the 
respective weights in the manner previously described. 
The test statistics ¢, and t, are used to weight down the 
wrong the start and (or) end point corespondencies. From 
the statistical point of view this means: Keep the line infor- 
mation while rejecting the hypothese that the start and (or) 
end point of the image edge matches the start and (or) end 
point of the model edge. 
The test statistic tj, which measures the lateral error is used 
to weight down or eliminate wrong egde correspondencies. 
Thus after 6 to 8 iterations the above descriped outliers are 
eliminated or weighted down so heavily that their influence 
onto the result can be neglegted. 
4  Selfdiagnosis 
It is important for each automatic system that it is able to 
make a selfdecision for the acceptance of the result. The ori- 
entation procedure presented in this paper, is part of an au- 
tomatic process which is planned to run in night-time or on 
weekends without any human manipulation. Therefore the 
system must be able to decide wether the determined orien- 
tation parameters are correct or better had to be rejected. 
Thus an objective quality control measure is necessary. Gross 
errors can hide behind small residuals or excellent fitting of 
data and model, therefore they do not necessarily produce 
large variances in the estimated parameters. Therefore a ad- 
ditional sensitivity analysis for selfdiagnosis is used. 
The concept of sensitivity analysis developed by Baarda 
[BAARDA W. 67 ,68] is based on the measures for the in- 
ternal and external reliability. The elementary theory has 
been expanded and specified for our purpose [cf. FÓRSTNER 
W. 83, 92 ]. The sensitivity analysis is used to investigate 
the influence of a single control point model onto the esti- 
mated orientation parameters, taking the geometry of the 
design (control point arrangement) into account. A single 
control point model is represented by several image to mo- 
del edge matches. Therefore the sensitivity analysis is applied 
to groups of observations, namely all the edges belonging to 
one control point model. 
The following measures, calculated for each control point mo- 
del in the aerial image, are used for evaluating the quality of 
the orientation: 
The Fisher test statistic 
Ty-1 
e 31.1 ei 
TP = A (15) 
: riot? 
i09 
depends on the geometry of the design and the size of an 
undetected gross error in the observation group x;. This test 
statistic is Fisher distributed. A gross error in the observation 
group x; could be detected whith a significance level a by 
checking 
T? > F(a,n;,n — ny) (16) 
with 
n; (Number of image edges from model i)-4 
n —n; = (Number of the remainding image edges) - 4 
a = significance level e.g 95% 
595