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Uncertainty of Spatial Metric
Relations in GIs
Xiaoyong
hen', Takeshi ~oihara'and Mitsuru ~ a s u '
Abstract.---In this paper, after an introduction to the notations of metric
topology, a novel theory of spatial metric relations between sets is developed
in which the relations are defined in terms of the intersections of the
boundaries, interiors and exteriors of two dynamically generated sets. Then,
the presented theory is extended to quantitatively deriving the spatial metric
relations between sets in consideration of conceptual and positional
uncertainties based on the fuzzy set theory. Finally, some potential
applications of presented theories and the ideas for spatial and temporal
reasoning in GIs are also suggested.
INTRODUCTION
One of the most fbndamental requirements for GIs is to modeling and
communicating error in spatial databases. With increased research into error
modeling in GIs over the past few years, there has a considerable body of models
and techniques available for measurement spatial database error from researching to
real applications [Goodchild and Gopal, 1989; Vergin, 19941. Spatial relationships
(such as distance, direction, ordering, and topology) between objects, as very useful
tools for spatial queries, constraints checking and sorting in GIs, are strongly
influenced by the uncertainties of original spatial data. These practical needs in GIs
have led to the investigation of formal and sound methods for driving spatial
relations and their variations with uncertainties [Chen and et .al., 1995, 1996;
Egenhofer and Franzosa, 1991;Frank, 1992; Kainz, et.al., 1993; Mark, et. al., 1995;
Peuquet and Zhang, 19871. However, how to derive spatial relations between sets
(non-point-like) in consideration of conceptual and positional uncertainties based on
an mathematically well-defined algebra framework is still an open problem up to
now. The lack of this comprehensive theory has been a major impediment for
solving many sophisticated problems in GIs, such as formally deriving complex
spatial relations among spatial objects with multiple representations, and generation
of the related standards for transferring spatial relations.
This paper focuses on the development of the theory and associated models of
spatial metric relations between sets in consideration of data uncertainties. At first,
based on the metric topology, a novel theory of spatial metric relations between sets
is developed in which the relations are defined in terms of the intersections of the
boundaries, interiors and exteriors of two dynamically generated sets. Then, the
AAS Research Institute, Asia Air Survey Co., LTD.,&IO, Tarnura-Cho, Atsugi-Shi, Kanagatva 243, JAPAN
545
presented theory is extended to quantitatively deriving the spatial metric relations
between sets in consideration of conceptual and positional uncertainties based on
the fbzzy set theory. Finally, some potential applications of presented theories and
the ideas for spatial and temporal reasoning in GIs area are also suggested.
FUNDAMENTAL DEFINITIONS
Metric Spaces
A metric space is a pair consisting of a set E and a mapping ( p ,,p2) +d ( p ,,p2) of
ExE into R+,having the properties: (1). p 1 = p 2 0d ( p ,,p2)=0; (2). d ( p ,,p2)=d(p2,p,)
(symmetry); (3). d ( p ,,p2)<d(p,,p, )+d ( p ,,p2)(triangle inequality). The function d is
called a metric and d ( p ,, p 2 ) is called the distance between the points p, and p2 .
Distance between points pi (xi,,.xi, ,,x) in R~ is described in terms of the Minkowski
d, -metric:
Conventional Euclidean distance is defined by the d2-metric. Similarly, the
Manhattan distance defined by the dl -metric, and the maximum distance defined by
the dm-metric.
Metric Topological Spaces
Topological spaces
A topological space is a pair consisting of a set E and a collection Aof subsets of
E called the open sets, satisfjmg the three following properties: (1). every union
(finite or otherwise) of open sets is open; (2). every finite intersection of open sets is
open; (3). the set E and the empty set 0 are open.
Metric topology
A metric d on a set E includes a topology on E, called metric topology defined by
d . This topology is such that UCE is an open set if, for each p,& , there is an E > O
such that the d-ball of radius E around p, is containedu . A d-ball is the set of points
whose distance from p, in the metric d is less than E , i.e. { p 2d p ( p l,p2)<&). Notice
that the metric topological spaces are Hausdorffand separable.
The Hausdorff Metric
The Hausdorff metric is defined on the space K where each point is a non empty
compact set of^'. If K, and K, denote two non empty compact set in R' (or
equivalently two points in K ), and B ( E ) is the closed ball with a radius E , then the
quantity:
0
Q
.. .
.:..:.
..............
...
........
............
.....
:,...,...
..........
.. . .
.:<:<A;$!:
::.............:.
.
..
.
@
18 8 il
disjoint
inside
contains
I
[8! il
equal
meet
covers
I
[i I il
coveredBy
overlay
Figure 1. ---The eight topologkal relations between regions in IR2 .
p(Kl,K,)=inf(&:K,
cK2@B(&),K2cK,OB(~))
[21
defines a metric p on K , known as the Hausdorff metric. From equation [2], p is the
radius of the smallest closed ball B such that both K, is contained in the set K20B(&)
generated by the morphological dilation [Serra, 19821 and K, is contained in the
dilated set K, @B(E) . Serra (1982) has proven that the Hausdorff distance p(Kl,K2)
satisfies all properties of distance functions. In particular case, when K, and K2
are reduced to two points, the Hausdorff distance p(Kl,K2) coincides with the
Euclidean distance.
SPATIAL METRIC RELATIONS BETWEEN SETS
Set Intersection Models
9-Intersection
For driving binary topological relations between sets, Egenhofer et al., (1994)
developed the 9-intersection model based on the usual concepts of point-set
topology with open and closed sets, in which the binary topological relations
between two objects, K, and K,, in IR' is based upon the intersection of K, 's
interior ( K; ), boundary ( iK1),and exterior ( K; ) with K, 's interior ( K; ), boundary
(2K2),and exterior ( K ; ). A 3x3 matrix 3,, called the 9-intersection as follows:
By considering the values empty (0) and non-empty (1) in equation [3], one can
distinguish between 2'=5 12 binary topological relations in which only a small subset
can be realized when the objects of concern are embedded inIR2. Egenhofer and
Franzosa (1 99 1) showed that, for two regions with connected boundaries embedded
in IR2 , the %intersectiondistinguishes just 8 different relations, i.e. disjoint, contains,
inside, equal meet, covers, coveredBy, and overlap (figure 1). However, when we
apply the 9-intersection model to describing topological relations between other
types of spatial objects, the situation will be more complicated. According to the
results of Mark et. al. (1995), for two simple lines 33 different spatial relations are
possible, and for a line and a region, 19 are possible.
Dynamic 9-intersection
For integrally deriving different kinds of spatial relations between sets, Chen et
al., (1995, 1996) developed the dynamic 9-intersection model based on the concepts
of the metric topology with open and closed sets and the morphological dilation, in
which the general 9-intersection of equation [3] is extended as follows:
where the Kl and K2 are given two closed sets, the K,BB(&,) means relevant
( E , ) means
morphological dilation by the closed ball B with radius E, , and the 3:,,,)
dynamic %intersection with parameter E , from K to K j . Based on the equation [4],
we can derive dynamic topological relations by using the different parameter E, .
In particular case, when E, =O , the structure element B ( E , )is reduced to the
original point { o }, according to the algebraic properties of morphological dilation
[Serra, 19821, we have K;@B(E,)=K@{o)=K, , then the dynamic 9-intersections
i , j ( ~ t ) defined in equation [4], coincide with the general 9-intersection3, in
equation [3].
Spatial Metric Relations between Sets
Distance Relations between Sets
According to the derived dynamic topological relations by the dynamic 9intersections of equation [4] with different parameter E , , such as dynamic equal,
dynamic covers (or dynamic corveredBy) and dynamic contains (or dynamic
inside), we can easily get the Hausdorff distance p(Kl ,K2) between two closed sets
K, and 6 by calculating the minimum and maximum dilated distances:
p(Kl ,K2)=max{min(s,),m i n ( ,)I;
~ when
1
where " * " means either empty (0) or non-empty (1). The binary distance relations
derived by equation [5] are suitable for different types of spatial objects, such as
point-objects, line-objects and region-objects, as well as combining different types
of spatial objects such as a line and a region, a point and a line, or a point and a
region. Some examples are shown in figure 2.
Figure 2. ---The spatial metric relations between two planar sets A and B.
Directional Relations between Sets
Directional relations between sets can be defined by the Hausdorff metric of
angular bearings, in which the direction fiom one spatial object to another is
identical to that for metric fimction except that the angular bearing is computed for
each ordered pair in the Cartesian product. The angular bearing is measured in the
sense of navigation bearings (i.e. increasing clockwise fiom north).
Similarly, for calculation of the directional relation q(Ki,K, ) between two non
empty compact sets Kl and K2in R2, we can select the angular bearing set R ( a , ) as
instead of the closed ballB(s,)in equation [3], then we can get the Hausdorff
direction q(Ki, K j ) by calculating the minimum and maximum dilated angles as
follows:
[111][011
001 , 001 , [001]
01 1
001 001
[ip[K][iY']
L
contains
001
where " * " means either empty (0) or non-empty (1). The binary directional relations
derived by [6] are also suitable for different types of spatial objects and their
combining types. Some examples are shown in figure 2.
UNCERTAINTY OF SPATIAL METRIC RELATIONS
Conceptual Uncertainty of Spatial Metric Relations
Unlike the spatial metric relations between points, the concepts of distances and
directions between subsets are fuzzy [Zadeh, 19651, since the spatial objects may
contain many subsets, the distances and directions between these subsets are
difficult to be represented just by a single value. If we define the fuzzy memberships
as the covering percentages of generated region areas ( or point numbers and line
lengths ) by the dynamic intersection of sets in R' , we can find that the Hausdorff
metric is just the special case with the fuzzy membership value equal to one. Based
on the changes of the covering areas of regions (or point numbers and line lengths)
from an empty set to a complete set, we can estimate the fuzzy memberships from
zero to one, then we can quantitatively derive the spatial metric relations between
subsets.
For reasons of simplicity the distances and directions between closed subregions discussed in this paper only, related models for estimation of conceptual
fuzzy membership functions are defined as follows:
where A ( * ) means covered area sizes by dynamic intersections with the parameters
n and B for distances and directions separately, the functions Os@,(n)<l and
o<a,(s)slwith the parameters O<<<~(K,,K,) and o s e s p ( K , , ~ , ) k xare called the size
distribution functions [Serra, 19821. An example of conceptual uncertainty of spatial
distance relations between two spatial regions is shown in figure 3.
Positional Uncertainty of Spatial Metric Relations
Generally, the locations of objects in spatial databases are not error-free, they may
contain many kinds of errors, such as the errors of scanning, digitizing, selecting,
projection, overlaying, and et.al. [Goodchild and et.al., 19891. For describing the
uncertainty in the positions of spatial objects (such as lines and areas), we can use
the error model of E -band developed by Chrisman (1982), in which the positional
uncertainty of a spatial object K~ can be represented as K . = K @ ~ ( E ), where E is
) the fuzzy membership function
the buffer distance of error distribution and p ( ~ is
derived by E . According to the equations of [2]-[7], if we use the uncertain spatial
object K as instead of the general spatial object K in the equations of [2]-[7], we
Figure 3. ---The conceptual fuzzy membership fhction between subsets.
Figure 4. ---The positional fuzzy membership fbnction between subsets.
can drive the spatial metric relations between uncertain sets and the conceptual
fuzzy membership functions between uncertain subsets. For quantitatively
estimating the positional fuzzy membership functions of derived spatial metric
relations between uncertain subsets, we can use the measurement of dynamic
covering uncertain areas as follows:
where ki=Ei-4=[KiBP (E)]-Ki means the uncertain area generated by the related E band, A { * ) means the covered size of area or volume when p(*) is a constant or a
general fuzzy member function with the value o ~ J ( * ) separately.
~I
An example of
positional uncertainty of spatial directional relations between two spatial regions is
shown in figure 4.
CONCLUSIONS AND OUTLOOKS
As the natural extension of the general 9-intersection, the dynamic 9-intersection
based on metric topology supplied a general fi-amework for studying metric spatial
relations between sets. The presented integrated theory of spatial relations between
sets makes a new way for formally deriving spatial relations among complex spatial
objects with conceptual and positional uncertainties. Even though the presented
approach is only focus on the applications in GIs field, the related results for
deriving spatial relations between sets and uncertain sets can be also used for many
other fields, such as CAD, computer vision, pattern recognition, robot space
searching and so on. However, only the theoretical models have be presented in this
paper, a wide field of practical application for management and analysis of spatial
data in 2-D and 3-D GIs environments has not been touched. Therefore, the
reported results must be verified and extended in order to be used in different
practical environments.
Two main directions for further research shall be pointed here, one is the
applications of the presented theoretical models and algorithms in 2-D and 3-D GIs
environments for developing the new tools of spatial query and analysis; another one
is the extensions of presented theories and models for formally deriving complex
spatial relations among spatial objects with multiple representations.
ACKNOWLEDGMENTS
The authors wish especially to thank Prof. S.Murai and Mr. H.Yamamoto for their
considerable helps and supports of this research project.
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