Metric operations on fuzzy spatial objects in databases.
Uncertainty management for geometric data is currently an important problem for (extensible) databases in general and for spatial databases, image databases, and GIS in particular. In these systems, spatial data are traditionally kept as determinate and sharply bounded objects; the aspect of spatial vagueness is not and cannot be treated by these systems. However, in many geometric and geographical database applications there is a need to model spatial phenomena rather through vague concepts due to indeterminate and blurred boundaries. Following previous work, we first describe a data model for fuzzy spatial objects including data types for fuzzy regions and fuzzy lines. We then, in particular, study the important class of metric operations on these objects.
INTRODUCTION
Representing, storing, quering, and manipulating spatial information
is important for many non-standard database applications. But
so far, spatial data modeling has implicitly assumed that the extent
and hence the borders of spatial objects are precisely determined
("boundary syndrome"). Special data types called spatial
data types (see [7] for a survey) have been designed for modeling
these data in databases. We will denote this kind of entities as crisp
spatial objects.
In practice, however, there is no apparent reason for the whole
boundary of a region to be determined. On the contrary, the feature
of spatial vagueness is inherent to many geographic data [3]. Many
geographical application examples illustrate that the boundaries of
spatial objects (like geological, soil, and vegetation units) can be
partially or totally indeterminate and blurred; e.g., human concepts
like "the Indian Ocean" or "Southern England" are implicitly
vague. In this paper we focus on a special kind of spatial vagueness
called fuzziness. Fuzziness captures the property of many spatial
objects in reality which do not have sharp boundaries or whose
boundaries cannot be precisely determined. Examples are natu-
ral, social, or cultural phenomena like land features with continuously
changing properties (such as population density, soil quality,
vegetation, pollution, temperature, air pressure), oceans, deserts,
English speaking areas, or mountains and valleys. The transition
between a valley and a mountain usually cannot be exactly ascertained
so that the two spatial objects "valley" and "mountain" cannot
be precisely separated and defined in a crisp way. We will designate
this kind of entities as fuzzy spatial objects.
The goal of this paper is to deal with the important class of metric
operations on fuzzy spatial objects. Examples are the area operation
on fuzzy regions or the length operation on fuzzy lines. It turns
out that their definition is not as trivial as the definition of their crisp
counterparts. The underlying formal object model follows the au-
thor's previous work in [8] and offers fuzzy spatial data types like
fuzzy regions and fuzzy lines in two-dimensional Euclidean space.
Our concept to integrate fuzzy spatial data types into databases
is to design them as abstract data types whose values can be embedded
as complex entities into databases [9] and whose definition
is independent of a particular DBMS data model. They can,
e.g., be employed as attribute types in a relation. The future design
of an SQL-like fuzzy spatial query language will profit from the
abstract data type approach since it makes the integration of data
types, predicates, and operations into SQL easier.
The metric operations described in this paper are part of a so-called
abstract model for fuzzy spatial objects. This model focuses
on the nature of the problem and on its realistic description and
solution with mathematical notations; it employs infinite sets and
does not worry about finite representations of objects as they are
needed in computers. This is done by the so-called discrete model
which transforms the infinite representations into finite ones and
which realizes the abstract operations as algorithms on these finite
representations. It is thus closer to implementation.
Section 2 discusses related work. Section 3 introduces fuzzy
spatial objects. It gives some basic concepts of fuzzy set theory
and then informally presents the design of fuzzy regions and fuzzy
lines. Section 4 describes and formalizes metric operations on
fuzzy spatial objects and identifies the two classes of real-valued
and fuzzy-valued metric operations. Section 5 draws some conclusions
and discusses future work.
2. RELATED WORK
Mainly two kinds of spatial vagueness can be identified: uncertainty
is traditionally equated with randomness and chance occurrence
and relates either to a lack of knowledge about the position
and shape of an object with an existing, real boundary (positional
uncertainty) or to the inability of measuring such an object precisely
(measurement uncertainty). Fuzziness, in which we are only
interested in this paper, is an intrinsic feature of an object itself and
describes the vagueness of an object which certainly has an extent
but which inherently cannot or does not have a precisely definable
boundary (e.g., between a mountain and a valley). This kind of
vagueness results from the imprecision of the meaning of a con-
cept. Models based on fuzzy sets have, e.g., been proposed in [1,
2, 4, 8].
3. FUZZY SPATIAL OBJECTS
In this section we very briefly and informally present the basic elements
of an abstract model for fuzzy spatial objects as it has been
formalized in [8]. The model is based on fuzzy set theory (and
fuzzy topology) whose main concepts are introduced first, as far as
they are needed in this paper. Afterwards the design of spatial data
types for fuzzy regions and fuzzy lines is shortly discussed.
3.1 Crisp Versus Fuzzy Sets
Fuzzy set theory [10] is an extension and generalization of Boolean
set theory. It replaces the crisp boundary of a classical set by a
gradual transition zone and permits partial and multiple set mem-
bership. Let X be a classical (crisp) set of objects. Membership in a
classical subset A of X can then be described by the characteristic
function f0;1g such that for all x 2 X holds c A
and only if x 2 A and c A This function can be
generalized such that all elements of X are mapped to the real interval
[0,1] indicating the degree of membership of these elements
in the set in question. We call -
the membership function
of -
A, and the set -
A called a fuzzy set
in X . A [strict] a-cut of a fuzzy set -
A for a specified value a is the
crisp set A a [A
a
1g. The
strict a-cut for a = 0 is called support of -
. A
fuzzy set is convex if and only if each of its a-cuts is a convex set.
A fuzzy set -
A is said to be connected if its pertaining collection of
a-cuts is connected, i.e., for all points P;Q of -
A, there exists a path
lying completely within -
A such that -
A (R) - min(-
A (P);-
A (Q))
holds for any point R on the path.
3.2 Fuzzy Regions
We first describe some desired properties of fuzzy regions and also
discuss some differences in comparison with crisp regions. After
that, we informally outline a data type for fuzzy regions.
3.2.1 Generalization of Crisp to Fuzzy Regions
A very general model defines a crisp region as a regular closed set
[5, 6, 7] in the Euclidean space IR 2 . This model is closed under (ap-
propriately defined) geometric union, intersection, and difference.
Similar to the generalization of crisp sets to fuzzy sets, we strive
for a generalization of crisp regions to fuzzy regions on the basis of
the point set paradigm and fuzzy concepts.
Crisp regions are characterized by sharply determined boundaries
enclosing and grouping areas with equal properties or attributes
and separating different regions with different properties
from each other; hence qualitative concepts play a central role. For
fuzzy regions, besides the qualitative aspect, also the quantitative
aspect becomes important, and boundaries in most cases disappear
(between a valley and a mountain there is no boundary!). The distribution
of attribute values within a region and transitions between
different regions may be smooth or continuous. This important feature
just characterizes fuzzy regions.
A classification of fuzzy regions from an application point of
view together with application examples is given in [8].
3.2.2 Definition of Fuzzy Regions
We now briefly give an informal description of a data type for fuzzy
regions. A detailed formal definition can be found in [8]. A value
of type fregion for fuzzy regions is a regular open fuzzy set whose
membership function -
predominantly continuous.
F is defined as open due to its vaguenessand its lack of boundaries.
The property of regularity avoids possible "geometric anomalies"
(e.g., isolated or dangling line or point features, missing lines and
points) of fuzzy regions. The property of -
F to be predominantly
continuous models the intrinsic smoothness of fuzzy regions where
a finite number of exceptions ("continuity gaps") are allowed.
3.2.3 Fuzzy Regions As Collection of a-Level Regions
A "semantically richer" characterisation of fuzzy regions describes
them as collections of crisp a-level regions [8]. Given a fuzzy region
F , we represent a region F a for an a 2 [0; 1] as the regular
crisp set of points whose membership values in -
F are greater than
or equal to a. F a can have holes. The a-level regions of -
F are
nested, i.e., if we select membership values
a n ?a a n+1 .
3.3 Fuzzy Lines
In this section we informally describe a data type for fuzzy lines
whose detailed formal definition can be found in [8]. We start
with a simple fuzzy line -
l which is defined as a continuous curve
with smooth transitions of membership grades between neighboring
points of -
l , i.e., the membership function of -
l is continuous too.
The end points of -
l may coincide so that loops are allowed. Self-intersections
and equality of an interior with an end point, however,
are prohibited. If -
l is closed, the first end point must be the leftmost
point to ensure uniqueness of representation.
Let S be the set of fuzzy simple lines. An S-complex T is a finite
subset of S such that the following conditions are fulfilled. First, the
elements of T do not intersect or overlap within their interior. Sec-
ond, they may not be touched within their interior by an endpoint of
another element. Third, isolated fuzzy simple lines are disallowed
(connectivity property). Fourth, each endpoint of an element of T
must belong to exactly one or more than two incident elements of
T to support the requirement of maximal elements and hence to
achieve minimality of representation. Fifth, the membership values
of more than two elements of T with a common end point must
have the same membership value; otherwise we get a contradiction
saying that a point of an S-complex has more than one membership
value. All conditions together define an S-complex as a connected
planar fuzzy graph with a unique representation.
A value of the data type fline for fuzzy lines is then given as a
finite set of disjoint S-complexes.
4. METRIC OPERATIONS ON FUZZY
A very important class of operations on spatial objects are metric
operations usually getting one or two spatial objects as arguments
and yielding a numerical result. They compute metric (i.e., mea-
surable) properties such as area and perimeter and are commonly
used in the analysis of spatial phenomena. While their definitions
are well-known, clear, and relatively easy for crisp spatial objects,
it is not always obvious how to measure metric properties of fuzzy
spatial objects and hence how to define corresponding operations.
A central issue is whether the result of such a metric operation
is a crisp number or rather a "fuzzy" number. From an application
point of view both kinds of numerical results are acceptable and
even desirable. A resulting single crisp number can be interpreted
as an appropriately aggregated or weighted real value over all membership
values of a fuzzy spatial object. An arising fuzzy number
satisfies the expectation that, if the operands of a metric operation
are fuzzy, then the numerical result should be fuzzy too. Therefore,
we will consider the crisp (Section 4.1) and the fuzzy (Section 4.2)
variant of several metric operations. Both variants have in common
that they operate on fuzzy spatial objects and that they are reduced
to the ordinary definitions in the crisp case.
4.1 Crisp-Valued Numerical Operations
In this section we view operations on fuzzy spatial objects that yield
crisp numbers. We first present a special view on membership functions
which simplifies an understanding of the metric operations
discussed afterwards.
4.1.1 Membership Functions Considered as Functions
of Two Variables
Essentially, a membership function for a spatial object s associates
with each point
which p belongs to s. A slightly modified view considers - as a
function of the two variables x and y. This view has the benefit that
we can visualize how - "works" in terms of its graph. The graph
of - is the graph of the equation z = -(x;y) and comprises all three-dimensional
points
If s is a fuzzy region, the graph of the corresponding spatial membership
function of two variables is a collection of disjoint surfaces
(one for each fuzzy face) that lie above their domain s in the Euclidean
plane. Each surface determines a solid or volume bounded
above by the function graph and bounded below by a fuzzy face
of s. If s is a fuzzy line, we obtain a collection of disjoint three-dimensional
networks each consisting of a set of three-dimensional
curves. As an example, Figure 1 shows the three-dimensional view
of the membership function of a fuzzy region
(showing the expansion of air pollution caused by a power
station, for instance). Three-dimensional visualizations of membership
functions of fuzzy spatial objects lead to an easier understanding
of the metric operations discussed in the following.

Figure

1: 3D representation of the membership function of a
fuzzy region.
4.1.2 Metric Operations on Fuzzy Regions
Metric operations on fuzzy regions usually yield a real value as
a result and can be summarized as a collection of functions
real with different function names for g and, of course,
different semantics. Let -
fregion where the - f i are
the fuzzy faces of -
F .
The definition of an area operator applied to -
F requires that -
F is
integrable. This condition is always fulfilled since our definition of
a fuzzy region requires that -
F is continuous or at least piecewise
continuous. The area of -
F can be defined as the volume under the
membership function -
area( -
RR
F (x;y) dx dy
RR
F (x;y) dx dy
RR
Thus, the integration can be performed either over the entire Euclidean
plane, or equivalently over the support of -
F , which is always
bounded, or equivalently over the supports of the faces of -
F .
Note that -
Crisp holes enclosed by fuzzy faces do not cause problems during
the integration process; they simply do not contribute to the double
integral. Evidently, if -
G, we have area( -
G). In
particular, we obtain
area( -
RR
F (x;y) dx dy
RR
A special case arises if -
F is piecewise constant and -
F consists
of a finite collection fF a of crisp a-level regions. Then
the area of -
F is computed as the weighted sum of the areas of all
a-level regions F a :
area( -
RR
(x;y)2F a i
a
Next, we determine the height and the width of a fuzzy region. For
computing the height (width) of a fuzzy region -
F , all maximum
membership values along the y-axis (x-axis) and parallel to the x-axis
(y-axis) are aggregated and (their square roots are) added up,
F is projected onto the y-axis (x-axis), and the maximum membership
value is determined for each y-value (x-value).
R
F (x;y) dy
and similarly we obtain
R
F (x;y) dx
Both integrals are finite since -
F has bounded support. Let -
crisp. For the height operator we obtain max x2IR -2
1 for a fixed y 0 2 IR if the intersection of the horizontal line
with F is non-empty. Otherwise, the expression yields
Hence,
R
F (x;y) dy is the measure of the
set of y 0 's such that F intersects In the case that F consists
of several connected components, each component of F gives rise
to a y-interval. Then height(F) is the union of these intervals. If F
is connected, we only obtain one interval, and height(F) is just its
length. Analogous thoughts hold for the width operator.
So far, we have not explained the meaning and the necessity of
the exponent 1
2 as part of the integrands. The introduction of this
exponent is essential since it ensures our expectation that the area
of a fuzzy region is equal or less than its height times its width, i.e.,
LEMMA 1. area( -
PROOF. We can show this as follows:
area( -
RR
F (x;y) dx dy
R
F (x;y) dy \Delta R
F (x;y) dx
F (x;y) dy \Delta R
F (x;y) dx
Without the exponent 1
2 we would have area( -
F) which is a completely different result and which does not
correspond to our intuition. Hence, metric operations on fuzzy spatial
objects do not only dependon their geometric extent but also on
the nature of their membership values so that a kind of "compen-
sation" is necessary. In accordance with the definitions for height
and width, the exponent 1
will also appear in the definitions of the
following operators.
If -
F consists of a finite collection fF a of crisp a-level
regions, we obtain
R
dy
R
The diameter of a spatial object is defined as the largest distance
between any of its points. We will give here two definitions and
distinguish between the outer diameter and the inner diameter. For
the computation of the outer diameter we may leave the fuzzy re-
gion; for the computation of the inner diameter we have to remain
within its interior. The outer diameter of -
F is defined as
R
F (u;v) du
where u and v are any pair of orthogonal directions and where the
maximum is evaluated over all possible directions u (we can also
imagine that -
F is smoothly rotated within the Cartesian coordinate
system). If -
is crisp, the value u yielding the maximum is the
direction of the line along which the projection of -
F has the largest
size. Obviously, height( -
F) and width( -
F), and we can further conclude that area( -
The inner diameter operator only relates to connected fuzzy re-
gions. Let P and Q be any two points of -
F , and let p PQ be a path
from P to Q that lies completely in -
F . Such a path must exist since
F is assumed to be connected. The inner diameter is defined as
min
RR
F (x;y) dx dy
where the maximum is computed over all points P and Q of the
Euclidean plane and where the minimum is determined over all
paths between P and Q such that -
F (P);-
F (Q)) holds
for any point R on p PQ . Since -
F is connected, such a path always
exists.
If -
path p PQ from P to
Q will not yield the maximum. Otherwise, if both P and Q are in F ,
must lie completely in F so that min pPQ
RR
F (x;y) dx dy=
length(p PQ ). In this case, the meaning of innerDiameter(F) amounts
to its standard definition as the greatest possible distance between
any two points in F where only paths lying completely in F are
allowed.
The relationship between inner and outer diameter is different
for crisp and fuzzy regions. If -
is crisp, we can derive two
propositions:
LEMMA 2. If F is a connected crisp region, then
outerDiameter(F) - innerDiameter(F).
PROOF. Select a line (which is not necessarily unique) upon
which the projection of F onto the u-axis is largest. Since F is
connected, this projection is an interval, and its length is given by
outerDiameter(F). Let us assume that P and Q are those points of
F that coincide with the end points of this interval. Then the shortest
path p in F between P and Q is at least as long as the straight
line segment s joining P and Q. Segment s is at least as long as the
interval since the interval is a projection of s.
F
s

Figure

2: Example of the relationship between inner (p) and
outer diameter of a connected crisp region.
An example of this relationship is that s crosses the exterior of
F . Then path p is longer than s (Figure 2).
LEMMA 3. If F is a convex crisp region, then
PROOF. Since F is convex, F is connected so that
outerDiameter(F) - innerDiameter(F) holds according to
Lemma 2. Let P and Q be the end points of the shortest path
yielding the maximum in the definition of innerDiameter(F).
Since F is convex, the shortest path in F between P and Q is the
straight line segment joining P and Q. The projection of F on this
line segment thus has length at least innerDiameter(F) so that
outerDiameter(F) - innerDiameter(F).
For convex fuzzy regions the situation is different in the sense
that the outer diameter can even be greater than the inner diameter.
An illustration is given in Figure 3. Let c;d 2 IR ?0 . We consider a
fuzzy region -
F having the membership function
a
and thus consisting of two circular, concentric a-level regions F a 1
and F a 2 . Moreover, we assume that a 1 is much larger than a 2 and
that d is only slightly larger than c. If R and S are two points on
the boundary of F a 1 which are located on opposite sides so that
their straight connection passes the center of F a 1 , then for F a 1 the
inner and the outer diameter are equal according to Lemma 3, and
we obtain innerDiameter(F a 1
and Q are two points on the boundary of F a 2 , they have the largest
distance if they are located on opposite sides so that their straight
connection passes the center of F a 2 . This is just the outer diameter
of -
F , and we have outerDiameter( -
c). But the
inner diameter of -
F can be smaller if we consider a shortest path
This path avoids F a 1 with
its high membership value a 1 , and we can compute the value for
a 1 so that innerDiameter( -
F) holds. We must
require that length(p PQ ) \Delta a c). This is the case
if a 1 ?
a 2 (length(p PQ )\Gamma2(d \Gammac))
2c .
The observation that the outer diameter of a fuzzy region -
F can be
larger than its inner diameter is only valid if -
F is convex:
LEMMA 4. If -
F is a convex fuzzy region, then
F).
PROOF. If -
F is a convex fuzzy region, we know due to the definition
of connectedness that -
F (P);-
F (Q)) holds
for any point R on the straight line segment PQ between any two
R
d
c
F a 1
F a 2

Figure

3: Example of the relationship between inner and outer
diameter of a convex fuzzy region.
points P and Q. Therefore, we obtain RR
F (x;y) dx dy -
RR
F (x;y) dx dy for the shortest path p PQ between P and
Q. A projection of -
F onto the line segment PQ in any direction
u yields R
F (u;v) du - RR
F (x;y) dx dy. Fi-
nally, we obtain
RR
F (u;v) du
RR
F (x;y) dx dy
RR
F (x;y) dx dy
In all other cases where -
F is a more general fuzzy region, no
general statement can be made about the relationship between inner
and outer diameter.
Based on the concept of outer diameter we can specify two other
operators which characterize the shape of a fuzzy region. They rate
the opposite geometric properties "elongatedness"and "roundness"
and are defined in terms of the proportion of the minor outer diameter
to the major outer diameter. The first operator is given as
min
R
F (u;v) du
The geometric property "roundness" can be regarded as the complement
of "elongatedness":
The next operator of interest computes the perimeter of a fuzzy
region -
F . Assuming that the membership function -
F is continuous
and that -
-x and -
-y denote the partial derivatives of - with respect
to x and y, respectively, we can define the perimeter of -
F as
RR
-x -2
F
-y -2
F
dx dy
RR
-y - 1-
f
dx dy
One can prove that, if -
G, we obtain perimeter( -
G). If the membership function -
F is piecewise constant
so that -
F consists of a finite collection fF a of crisp
a-level regions, the perimeter of -
F is defined as
where calculates the length of the kth arc along which
the discontinuity between the a-level regions with membership degrees
a i and a j occurs and where this length is weighted by the
absolute difference of a i and a j .
4.1.3 Metric Operations on Fuzzy Lines
For fuzzy lines we consider operations g with the signature
l n g be a fuzzy line. We start with
the operation length measuring the size of -
L and first determine the
length of a simple fuzzy line -
l. We know that the membership function
of -
l is given as -
l just yields the support of -
l, i.e., supp( -
l ([0;1]). Hence
RR
l)
F
-y - 1-
F
dx dy
Consequently, we obtain
Another operation is strength which follows the principle that "a
line is as strong as its weakest link". It computes the minimum
membership value of a fuzzy line and is thus defined as
4.2 Fuzzy-Valued Metric Operations
Another interpretation of metric operations on fuzzy spatial objects
is that they yield a fuzzy numerical value as a result. This accords
with the fuzzy character of the operand objects. We first explain
the concept of fuzzy numbers needed for a description of the metric
operations afterwards.
4.2.1 Fuzzy Numbers
The concept of a fuzzy number arises from the fact that many quantifiable
phenomena do not lend themselves to a characterisation in
terms of absolutely precise numbers. For instance, frequently our
watches are somewhat inaccurate, so we might say that the time
is now "around five o'clock''. Or we might estimate the age of an
elder man at "nearly seventy-five years". Hence, a fuzzy number
is described in terms of a central value and a linguistic modifier
like nearly, around, or approximately. Intuitively, a concept captured
by such a linguistic expression is fuzzy, because it includes
some number values on either side of its central value. Whereas
the central value is fully compatible with this concept, the numbers
around the central value are compatible with it to lesser degrees.
Such a concept can be captured by a fuzzy number defined on IR.
Its membership function should assign the degree of 1 to the central
value and lower degrees to other numbers reflecting their proximity
to the central value according to some rule. The membership function
should thus decrease from 1 to 0 on both sides of the central
value.
Formally, a fuzzy number -
A is a convex normalized fuzzy set of
the real line IR such that (i) 9!x
A is called the
central of -
A is piecewise continuous. The membership
function of -
A can also be expressed in a more explicit form. Let
where a is a piecewise continuous function increasing
to 1 at point b, and g is a piecewise continuous function decreasing
from 1 at point b. We introduce freal as the type of all fuzzy (real)
numbers.
For the representation of the fuzzy-valued result of metric operations
we introduce two restricted kinds of fuzzy numbers. The
first kind contains numbers that are characterized by the property
that function f is lacking, i.e., these numbers only have a right-sided
membership function. The second kind comprises numbers
that are characterized by the property that function g is lacking, i.e.,
these numbers only have a left-sided membership function.
4.2.2 Metric Operations on Fuzzy Regions
For fuzzy regions we now consider fuzzy-valued operations g with
the signature first confine ourselves to a
subset of operations g 2 farea, perimeter, height, width, diame-
terg, because their result can be expressed in a generic way by a
restricted fuzzy number with a right-sided membership function,
as we will see.
To determine the result of g we switch to the
view of -
F as a collection of crisp a-level regions fF a
where n is possibly infinite. Since the regions F a i are crisp, we can
apply the corresponding known crisp operations g c to them. The
relationship between g c and the membership values a i is given in
the following lemma:
LEMMA 5. a ? b , g c
PROOF. From the definition of a fuzzy region as a collection
of a-level regions we know that a ? b , F a ' F b . Since g c is a
monotonically increasing function, we obtain F a ' F b , g c
We now define g( -
F) as the following fuzzy number:
This fuzzy number has a right-sided membership function, because
for the smallest a-level region F a 1 the membership value is a
and for all other a-level regions F a increasing i and thus
increasing g c the membership value a i decreases from 1 to 0.
In particular, the support of g( -
F) does not contain any smaller values
than g c
F is finite, we obtain a stepwise constant and
hence piecewise continuous membership function for g( -
F). Other-
wise, if -
F) is continuous, L -
F is infinite, and we get a continuous
membership function for g( -
F).
Intuitively, this result documents the vagueness of the operator
F), because with the increase of g c the certainty and
knowledge about its correctness decreases. We can confirm with
the membership value 1 that g c is the value of g( -
F). Thus,
c is the lower bound (or the core). But the value could be
higher, and if so, we can only confirm it with a lower membership
value. The membership value here indicates the degree of imprecision
of the operation g.
Let now g 2 felongatedness;roundnessg. These two operations
cannot be treated as fuzzy numbers with right-sided membership
functions since they are not monotonically increasing, i.e., in general
F a ' F b
How to measure these two operations as fuzzy-valued numbers is
currently an open issue. It is even doubtful whether they can be
represented as general fuzzy numbers at all.
4.2.3 Metric Operations on Fuzzy Lines
Each operation g 2 flength;strengthg on fuzzy lines has the signature
L be a fuzzy line, and let fL a
be a collection of crisp a-cuts where n is possibly infinite. For
both operations we pursue a similar strategy as for the operations
on fuzzy regions. The main difference in the definition of both operations
is that length is an increasing function whereas strength
is a decreasing function. Hence, analogously to Lemma 5 we can
conclude that
a
But since L a ' L b , strength c (L a ) - strength c (L b ), we obtain
a
Nevertheless, we can define the value of g( -
L) in the same manner
for both operations as the fuzzy number
For length this leads to a fuzzy number with a right-sided membership
function, and for strength we obtain a fuzzy number with a
left-sided membership function.
5. CONCLUSIONS AND FUTURE WORK
As part of an abstract model for fuzzy spatial objects in the Euclidean
space, we have defined metric operations that can have either
real-valued or fuzzy-valued results. All operations considered
have been unary functions. For future work we will also have to
consider binary metric operations, and here we will have, in par-
ticular, to deal with fuzzy distance and fuzzy direction operations.
Having achieved a formal and rather complete data model for fuzzy
spatial objects, we can transform it to a discrete model where we
have to think about finite representations for the objects and algorithms
for the operations. These are efforts that should later lead to
an efficient implementation.
6.



--R

Fuzzy Set Theoretic Approaches for Handling Imprecision in Spatial Analysis.
Natural Objects with Indeterminate Boundaries
Geographic Objects with Indeterminate Boundaries.
Qualitative Spatial Reasoning: A Semi-Quantitative Approach Using Fuzzy Logic
Vague Regions.

Spatial Data Types for Database Systems - Finite Resolution Geometry for Geographic Information Systems
Uncertainty Management for Spatial Data in Databases: Fuzzy Spatial Data Types.
Inclusion of New Types in Relational Database Systems.
Fuzzy Sets.
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fuzzy sets;fuzzy-valued metric operation;fuzzy line;real-valued metric operation;fuzzy region;fuzzy spatial data type