International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B2. Istanbul 2004 
o 
  
  
  
Rel. Cluster Modeled Relation Side Cond? — v;i,ldmin:dmaz] — Vmax|dmin:dmoax] Original Relation 
C1 disjoint no 14 [0,0] i4. [0, 0] disjoint 
(er disjoint yes jo [doin , daz] Yo [win : d nau] = 
C1 vo [0, 0] 1/4 [0, 0] touch 
C vj [0, 0] V4 [0, 0] weak overlap 
C» E = v [0, 0] v4. 0, 0] strong overlap 
C2 contains no/yes v. [0, 0] V— [0, 0] contains 
C2 contains no/yes v [0, 0] ivo [0, 0] covers 
C» contains no/yes [0,0] io[0. 0] equal 
Co - = vio [0, 0] v4. [0, 0] covered by 
Co = = V4 [0, 0] V4. [0, 0] contained by 
  
Table 1: Equivalence between {C, 
of the distance function, this problem will be addressed 
later on. 
It is interesting, to what extent (probability) two segments 
maintain the modeled topologic relation, considering their 
certainties. The probability P(w|7/;) that a distance Ÿ be- 
longs to a certain class /; is derived from an equipartition 
which depends on the sum of all certainties, assigned to the 
objects (A in sections 4.1 and 4.2) and the relation cluster, 
refer to Fig. 2. As all possible values for Ÿ have to be 
considered, the boundaries 95,4, possible 8nd Pas possible 
have to be chosen carefully. Up to now this simple statis- 
tical model with equipartitioned variables is used, but an 
extension to arbitrary density functions is feasible. For ex- 
ample the value A reflecting the certainties of the objects 
also contains the precision in the position of the extracted 
object opo (ref. to section 4.1), but inserted as a 95% con- 
fidence value. Another density function will result from a 
' convolution of the certainties and the precisions (i.e. a con- 
volution of a equipartitioned density function with a Gaus- 
sian). This combination has not been realized here because 
in practice the impact of opo compared to the equiparti- 
tioned certainties is relatively low. 
À 
P(0}w;) 
— tho Vy 
Up 
hin poss. dmin dmin dmin dmax dmax dmax Ümax.poss. 
ZA PUN -À +A 
Figure 2: Density Functions for Distance Classes 
From the figure the a-priori probabilities P(1/;) can be also 
derived, i.e. the probability for each class compared to the 
whole range of possible values. Together with P()|w;) 
conditional probabilities P(1^;|0) can be derived: 
P(9|v;) P(v,) 
220, ev, P(OIU;) Pts) 
  
PG) 
By multiplying the respective P(1/;|0,,;,) and P(v;|0,,a;) 
according to the equivalences given in Tab. 1 the probabil- 
ity p; that a given pair of segments maintains the modeled 
topologic relation can be achieved. This probability value 
can be used for a Hint which contains the hypothesis that 
the current segments coincide with the model. The focal 
sets and the assigned probabilities for a Hint 7/7. concern- 
ing the topologic relation are shown in Tab. 2. Here two 
more parameters are involved in the confidence measure: 
806 
; 
/ . / 
Umins Ÿ 
max } and topologic relations 
QAI. P 
(21 Dt * Qcov * Pcon 
Or 1 — p(why) 
  
~7 
Wry 
  
  
f y 
Uma 
Table 2: Hint 3£7. 
Peon 18 the confidence assigned to the extracted object. The 
evidence given by an object is the more credible the more 
confident it is. The parameter q.,, expresses to what de- 
gree the segment of the extracted object covers the ATKIS 
segment which is to be assessed. This factor is important in 
order to limit the impact of a Hint given from an object to 
the proportion it influences the ATKIS segment. The focal 
set ©7 represents ignorance. 
The final Hint #7 regarding the topologic relation is also 
influenced by the width of the two objects in case the side 
condition identical width is given for the relation contains 
(if it is not required then # = #H;). The difference of 
widths must be zero, but the certainty of the widths mea- 
sure must also be considered. Therefore 2f, is introduced 
which supports 7' depending on the difference of width and 
the given certainties. From the combination of 2/7. and Hıy 
applying Dempster’s Rule follows Hr. 
43.2 Hints Regarding Geometric Relations Similar 
to the judgment of topologic relations a measure is needed 
describing to what extent the modeled geometric relation 
is maintained by two objects. 
The calculation of the Hint #; concerning the question 
whether two segments are parallel is done in the following 
manner. If the direction of those two segments is given by 
t 4 and to the angle enclosed is a = |ta —to|. In the given 
problem it is sufficient to define o on [0, 5]. Parallelism 
means that a. may not exceed a certain value. This fixed 
threshold is o; and set to ap = 15 . 
The probability p(œ < &p) depends on the precision given 
for the objects (it is presumed that quality measures as- 
signed to the single segments are the same as assigned to 
the object). Further it is assumed that the certainty of the 
segments has no significant effect on the computation of 
the direction, because a certainty in the given context is un- 
derstood as an unknown displacement of the whole object, 
and such a displacement has no impact on the direction. 
As a standard deviation for the position of an ATKIS seg- 
ment is not given in the model the standard deviation c 
just depends on c,o, the precision given for the extracted 
segment: 
O pO 
Lo 
  
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