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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