The International Archives of the Photogrammetry. Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B4. Beijing 2008 
characterizes each topological relation by a different set of 
empty or non-empty values. This concept is shown in equation 
2 for the contain relation. 
. . . /x° n y° 
contain{x,y) := I 
Sx n y 0 Sx n Sy 
In the 9-IM, the exterior x c of a region is considered 
supplementary. Thus the topological relation is represented by a 
9-intersection-matrix. But for the set of fl R relations “(...) 9- 
intersection do not discriminate any further than the 4- 
intersection, they just make the terms larger” (Hernández, 1994, 
p. 61). Additionally two different resolutions for representing 
relations in the IM exist. The so-called high resolution includes 
eight relations and the medium one represents the needed set of 
H R relations. 
In contrast to the IM, the RCC is based on the single axiom 
connected which implies that the regions x and y share a 
common point. Conditions of this axiom allow as well the 
definition of topological relations in two resolutions. On the one 
hand, the RCC9 characterizes nine possible relationships of two 
regions x and y and on the other hand the RCC5 explicitly the 
five of Ü R . The concept of representation is shown in equation 3 
again for the contain relation (PP). 
For further considerations, the semantic of the iI R relations has 
to be defined with respect to the disaster management domain. 
The semantic defined by an operator serves as a reference 
because it has to be similar to human interpretation and their 
respective mental image. It turned out that humans do not 
distinguish between the interior and the boundary of a region. In 
Figure 1 for example the operational area 1 and 2 are disjoint 
and not overlapping according to human consideration. 
The concept of human interpretation results in a constrained set 
of possible topological relations between regions. These 
topological constraints define the semantic of the H R relations 
unambiguously by the equations 6 to 10. Therein x denotes the 
regular closed region x = Sx A x° composed by the regions 
interior Sx and the regions boundary x°. 
disjoint^, y) := Sx n Sy = 0 
inter sect (x,y) ■= Sx n Sy 0,x £ y,y £ x 
contain(x,y) := y c x 
inside(x,y) ■■= x c y 
equal(jx,y) ■— x = y 
PP{x,y) := P(x,y) A -iP(y,x) (3) 
where P{x,y) := Vz[C(z,x) -» C(z,y)] (4) 
C(x,y) ■■= VxC(x,x), Vx,y[C(x,y) -* C(y,x)] (5) 
An additional requirement of the regions respectively the 
relations between them, is that they have to be reflexive (eq. 11) 
and symmetric (eq. 12). 
In both concepts the type of representation as well as the 
characteristic of the topologic relations is different. For instance 
a general disadvantage of the RCC is the restriction to regular 
closed regions for the reasoning (pointless geometry). In 
contrast, relations represented by the IM are also valid for the 
geometric primitives line and point. However, this feature of the 
IM is not important for the application, because relevant objects 
are represented by regular closed regions, due to the 
applications range of scale. 
VxR(x, x) 
Vx,y[/?(x, y) 
R(y,x)] 
(11) 
(12) 
A comparison of the domain topological constraints and their 
respective characteristic (cf. equations 6 to 10 and , Figure 4) to 
the topological constraints of the IM* as well as the RCC5 
showed that the semantic of the RCC5 is congruent with the 
semantic required for the domain. This aspect can be seen in 
figure 3 and 4. 
^^-intersection 
disjoint(x.y) 
overlap(x.y) 
contain(x.y) inside(x.y) equal(x.y) 
^RCC5 
proper 
part’(x.y) 
equal(x.y) 
Figure 3. Characteristic of basis relations £2 in the RCC5 and 
the 4-intersection calculus (medium resolution) 
A further aspect of both methods is the different semantic of 
topological relations. In contrast to the IM the RCC5 does not 
distinguish between the boundary and interiors of regions in its 
Dl(x.y) 
intersect 
IT(x,y) 
contain 
CO(x,y) 
Figure 4. Characteristic of the set of H R relations 
Nevertheless it is also possible to define the topological 
relationships of the IM according to the semantic of the fl R 
relations. In contrast to the RCC the determination of the 
intersections in the IM is semantically correct, because the 
* Topological constraints of the IM for the critical relations 
disjoint(x,y) := x n y = 0 and overlap(x,y) == x n y =£ 
0, x £ y, y £ x. 
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