Area, Perimeter and Shape of Fuzzy Geographical Entities Cidália Costa Fonte 1 and Weldon A. Lodwick 2 1 Secção de Engenharia Geográfica, Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade de Coimbra, Apartado 3008, 3001 - 454 Coimbra - Portugal, e-mail: [email protected]. 2 Mathematics Department, University of Colorado at Denver, Campus Box 170 – P.O.Box 173364, Denver, Colorado USA 80217-3364, e-mail: [email protected]Abstract This paper focuses on crisp and fuzzy operators to compute the area and perimeter of fuzzy geographical entities. The limitations of the crisp area and perimeter op- erators developed by Rosenfeld (1984) are discussed, as well as the advantages of the fuzzy area operator developed by Fonte and Lodwick (2004). A new fuzzy pe- rimeter operator generating a fuzzy number is proposed. The advantage of using operators generating fuzzy numbers is then illustrated by computing the shape of a FGE, through its compactness, using the extension principle and the fuzzy area and perimeter. 1 Introduction In object based Geographical Information Systems (GIS) the geographical information is represented by geographical entities. These entities are characterized by an attribute and a spatial location. The spatial location of a geographical entity may be represented either in a vector data structure or considering a tessellation of the geographical space formed by elemen- tary regions to which an attribute is assigned. In the second case, the geo- graphical entities are formed aggregating contiguous regions to which the attribute characterizing the geographical entity was assigned. The elemen- tary regions forming the tessellation may be, for example, cells in a raster data structure, Voronoi polygons or Delaunay triangles. In this paper, for simplicity, we restrict ourselves to the second type of geographical entities,
Transcript
Area, Perimeter and Shape of Fuzzy Geographical Entities
Cidália Costa Fonte1 and Weldon A. Lodwick2
1 Secção de Engenharia Geográfica, Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade de Coimbra, Apartado 3008, 3001 - 454 Coimbra - Portugal, e-mail: [email protected]. 2 Mathematics Department, University of Colorado at Denver, Campus Box 170 – P.O.Box 173364, Denver, Colorado USA 80217-3364, e-mail: [email protected]
Abstract
This paper focuses on crisp and fuzzy operators to compute the area and perimeter of fuzzy geographical entities. The limitations of the crisp area and perimeter op-erators developed by Rosenfeld (1984) are discussed, as well as the advantages of the fuzzy area operator developed by Fonte and Lodwick (2004). A new fuzzy pe-rimeter operator generating a fuzzy number is proposed. The advantage of using operators generating fuzzy numbers is then illustrated by computing the shape of a FGE, through its compactness, using the extension principle and the fuzzy area and perimeter.
1 Introduction
In object based Geographical Information Systems (GIS) the geographical information is represented by geographical entities. These entities are characterized by an attribute and a spatial location. The spatial location of a geographical entity may be represented either in a vector data structure or considering a tessellation of the geographical space formed by elemen-tary regions to which an attribute is assigned. In the second case, the geo-graphical entities are formed aggregating contiguous regions to which the attribute characterizing the geographical entity was assigned. The elemen-tary regions forming the tessellation may be, for example, cells in a raster data structure, Voronoi polygons or Delaunay triangles. In this paper, for simplicity, we restrict ourselves to the second type of geographical entities,
formed by elementary regions ri ( )1,...,i = n . However, the operators pre-sented herein can be extended to the continuous space.
Since there may be uncertainty in the construction of the geographical entities, there may be uncertainty in their spatial location (Fonte and Lod-wick 2003). In these cases, the spatial extent of the geographical entities may be represented by a fuzzy set, generating a Fuzzy Geographical En-tity.
A Fuzzy Geographical Entity (FGE) EAt, characterized by the attribute ‘At’, is a geographical entity whose position in the geographical space is defined by the fuzzy set
{ }: belongs to the geographical entity characterized by attribute ' 'At i iE r r At= , with membership function [ ]( ) 0,1
AtE irµ ∈ defined for every elementary
region in the space of interest. The membership value one represents full membership. The membership value zero represents no membership, and the values in between correspond to membership grades to E
ir
At, de-creasing from one to zero.
Many GIS applications require the computation of geometric properties of the geographical entities, such as their area, perimeter or shape. There-fore, it is necessary to develop operators capable of computing these geo-metric properties of Fuzzy Geographical Entities (FGEs).
In section 2 some area operators are presented, such as the Rosenfeld area operator and the fuzzy area operator. Section 3 is dedicated to the pe-rimeter operators. A crisp operator due to Rosenfeld (1984) is presented, and a new fuzzy operator, generating a fuzzy number is proposed. The evaluation of the shape of a FGE is analysed in section 4 based on the compactness operator, where the fuzziness in the area and perimeter values are propagated to the compactness using the extension principle.
Throughout this paper it is assumed that the reader is familiar with the basic ideas of fuzzy set theory (Klir and Yuan 1995). The presented con-cepts are applicable to FGEs represented by normal fuzzy sets, called nor-mal fuzzy geographical entities.
2 Area of a Fuzzy Geographical Entity
Even though the computation of the area of geographical entities is a ba-sic operator of any GIS, little attention has been given to the computation of the area of FGEs. Katinsky (1994) stated that an ambiguous determina-tion of the area of a FGE required its defuzzification, while Erwig and Schneider (1997) stated that the area of a FGE was an interval, but since
Area, Perimeter and Shape of Fuzzy Geographical Entities 317
new operators had to be developed to process intervals only used the infe-rior and upper limit of the interval.
Since a FGE is represented by a fuzzy set, the operator proposed by Rosenfeld (1984) can be applied to compute the area of a FGE. The method proposed by Rosenfeld to compute the area of a FGE considers that the contribution of the area of each elementary region to the total area is proportional to the membership function value assigned to it. That is, for a fuzzy set E, in the discrete case,
( ) ( ) ( )E i ix y
AR E r A rµ=∑∑ (2.1)
This operator is appropriate when the concept of FGE arises from mixed pixels and the membership function values correspond to the percentage of the pixel area occupied by the attribute characterizing the FGE. However, if the membership function values represent the degree of uncertainty of whether the pixels can be classified as belonging to a geographical entity characterized by a certain attribute, the spatial extent of the geographical entity is not known and its corresponding area cannot be known either. For this situation the value of the Rosenfeld area operator is just a crisp ap-proximate value of the entity’s area, and no information regarding other possible values is given. Similarly, if the membership function values translate a degree of belonging based on similarity to the attribute, the Rosenfeld area operator has also a limited applicability (Fonte and Lod-wick 2004).
To overcome the limitations of the Rosenfeld area operator a new opera-tor with a fuzzy output, called fuzzy area operator (AF), was developed (Fonte and Lodwick 2004).
A FGE is characterized by a fuzzy set. A fuzzy set can be represented in a unique way by a family of level cuts, for . Since an alpha level
cut is a crisp set , its area, denoted here by
, is the sum of the areas of the regions belonging to the level
cut. That is, for a FGE E:
[ ]0,1α ∈
( ){ }:i E iE r rα µ α= ≥
( )Area Eα
[ ]
( ) ( )0: 0,1
.E
E
Area
Area Area E zαα α
+→
= =a
Since the level cuts of a fuzzy set are nested, that is E Eβ γβ γ< ⇒ ⊇ , the function AreaE is decreasing because
( ) ( ) ( ) ( )E EE E Area E Area E Area Areaβ γ β γ β γ⊇ ⇒ ≥ ⇒ ≥ .
318 Cidália C. Fonte and Weldon A. Lodwick
Let us consider a set of values ( )1,...,i iα = n belonging to ( ]0,1 and a set of values , belonging to( 1,...,iz i n= ) 0
+ , such that ( )i Ez Are ia α= , where 10 1i iα α + ≤< < . Definition: The fuzzy area of a FGE E, is the fuzzy set
( ) ( ) ( )( ){ }, AF EAF E z zµ= where
( ) [ ]
( ) ( )
( )
( )
( ) ( ))
0
11
: 0,1
max :
:
0 1
i E i
AF E
i iz Area
kk k k iAF E
k k
E E
when i z z
z zz z when i z zz z
when z Area Area
α
µ
α
µ α α α
+
=
++
∃ = −
= − + ∃ =−
∉
a
a
, 0
(2.2)
and and ( )maxi
k iz zz z
≤=
( )( )min
k E ik iz Area α
α α=
= .
The fuzzy area operator generates a fuzzy number (Fonte and Lodwick 2004). It’s support is the set of all values the area can take and each alpha level cut is the set of all values the area can take if level cuts of the FGE corresponding to values larger or equal to alpha are considered.
Table 1 shows the areas of the level cuts of the FGE represented in Fig. 1 considering that each cell has a unitary area. The resulting fuzzy area is represented in Fig. 2.
0 0 0 0 0 0 0 0
0 0 0 0.5 0 0 0 0
0 0 0.6 0.9 0.8 0.7 0.3 0
0 0.3 0.8 1 1 0.8 0.4 0
0 0 0.4 1 1 0.9 0.5 0
0 0 0.2 0.6 0.7 0.9 0.4 0
0 0 0.1 0.3 0.2 0.3 0 0
0 0 0 0 0 0 0 0
Fig. 1. Fuzzy geographical entity E.
0 5 10 15 20 25 300
0.2
0.4
0.6
0.8
1
Area z
Deg
ree
of m
embe
rshi
p
0 5 10 15 20 25 300
0.2
0.4
0.6
0.8
1
Area z
Deg
ree
of m
embe
rshi
p
Fig. 2. Fuzzy area of the fuzzy geographi-cal entity E.
Area, Perimeter and Shape of Fuzzy Geographical Entities 319
Several properties of both the Rosenfeld area operator and the fuzzy area operator are analysed by Fonte and Lodwick (2004).
3 Perimeter of a Fuzzy Geographical Entity
Rosenfeld (1984) proposed an operator to compute the perimeter of a fuzzy set formed by a finite set of contiguous and homogenous regions. This operator, herein designated by Rosenfeld perimeter (PR), may be ap-plied to FGE represented by a tessellation. The Rosenfeld perimeter is given by:
( ) ( ) ( ) ( ), 1 1
ijnn
E i E j ijki j ki j
PR E r r l aµ µ= =
<
= −∑∑ (3.1)
where n is the number of elementary regions forming the FGE, nij is the number of arcs separating regions ri and rj and l(aijk) is the length of k arc aijk of contact between regions ri and rj.
This operator considers that each arc separating contiguous elementary regions has a degree of belonging to the perimeter equal to the difference of the membership grades associated to the elementary regions. Therefore, as for the Rosenfeld area operator, the Rosenfeld perimeter operator gener-ates an approximate real value for the perimeter of the FGE, giving no other information about the other values it can take, nor about its variabil-ity with the grades of membership to the FGE. To overcome these limita-tions of the Rosenfeld perimeter a fuzzy perimeter operator is proposed.
320 Cidália C. Fonte and Weldon A. Lodwick
A level cut of a FGE E is a classical set. Let ( )P Eα be the perimeter of
the alpha level cut of the fuzzy set representing the FGE. Let us now con-sider for each FGE E a function PE such that:
[ ]( ) ( )
0: 0,1
.E
E
P
P P Eαα α
+→
p= =a
The perimeter values ip obtained for the FGE E represented in Fig. 1 corresponding to the alpha levels iα are shown in Table 2.
Notice that while for the area ( ) ( )E E Area E Area Eβ γ β γβ γ< ⇒ ⊇ ⇒ ≥ ,
since the level cuts of a fuzzy set are nested, that does not happen for the perimeter. It can vary in any way with the several level cuts of the FGE. That is,
E Eβ γβ γ< ⇒ ⊇ ⇒ ( ) ( )P E P Eβ γ≥ .
Table 2 and Fig. 3 show the values of the perimeter of several level cuts of the FGE represented in Fig. 1. Notice that the perimeter of alpha levels 0.1 and 0.2 are smaller than the perimeter values of alpha level 0.3 (which has a larger area, as can be seen in Table 1). Therefore, the fuzzy perimeter cannot be built in the same way the fuzzy area was, because otherwise, in some cases, the output might not be a fuzzy set.
Let us denote by , 1,...,ip i n= , a set of values of 0+ such that
(:i i E ip P )α α∃ = , where 1i i0 1α α +< < ≤ .
Area, Perimeter and Shape of Fuzzy Geographical Entities 321
Definition: The fuzzy perimeter of a FGE E, is the fuzzy set
( ) ( ) ( )( ){ }, PF EPF E z zµ= where
( ) [ ]
( ) ( )0: 0,1PD E
PD Ep p
µ
µ
+ →
a
and
( ) ( ) ( )
( ) [ ]
[ ]
11 1
1
1, :1
max if min ,max
0 min ,max
k kk k k k
k k
kk k k ip p
k kp p p p p pPD E p p
i i
p pip p p
p pp
if p p p
α α αµ +
+ +
+
++≤ ≤ ∨ ≤ ≤
∧ ≠
−− + ∈ − =
∉
(3.2)
The continuous line in Fig. 3 shows the fuzzy perimeter of the FGE of Fig. 1.
The fuzzy perimeter operator satisfies the following properties (proofs can be found in Appendix A) Property 1: For all FGE E, the support of ( )PF E is a subset of . 0
+
Property 2: If E is a normal FGE then ( )PF E is a fuzzy number.
5 10 15 20 25 300
0.2
0.4
0.6
0.8
1
Perimeter p
PF(E)
pi
Mem
bers
hip
func
tion
5 10 15 20 25 300
0.2
0.4
0.6
0.8
1
Perimeter p
PF(E)
pi
Mem
bers
hip
func
tion
Fig. 3. Fuzzy perimeter of the geographical entity E.
Such as with the fuzzy area operator, the support of the fuzzy perimeter shows the set of values the perimeter of the FGE can take. Such as for the fuzzy area, each alpha level cut of the fuzzy perimeter is the set of all val-ues the perimeter can take considering level cuts of the FGE corresponding to values larger or equal to alpha.
322 Cidália C. Fonte and Weldon A. Lodwick
4 Shape of a Fuzzy Geographical Entity
The shape of a FGE may be evaluated in several ways. One of them is the compactness, given by
( ) ( )( )
2 Area EC E
P Eπ
= (4.1)
where Area(E) is the area of entity E and P(E) represents its perimeter (Forman 1997). The compactness takes values between zero and one. A value of one is obtained for a circle, the value 0.88 for a square and values gradually smaller for more contorted shapes.
As the fuzzy operators for the area and perimeter generate fuzzy num-bers, a fuzzy shape can be obtained applying the extension principle to the results of these fuzzy operators (see Appendix B). That is,
( )( )
( )2 AF E
CF EPF Eπ
= (4.2)
As the area and perimeter of a geometric figure are not independent variables, the cartesian product of the fuzzy area and perimeter values should not be considered, but a relation between these values. The result of applying the extension principle to the cartesian product generates the fuzzy set represented in Fig. 4. Notice that the support of this fuzzy set is even larger than the set [0,1]. That is, we obtain impossible values for the compactness. This happens because we are considering that it is possible to have the area of one level cut and the perimeter of another. That is, we are combining areas and perimeters of different geometric figures, which makes no geometrical sense.
A relation should then be defined between the values of the fuzzy area and perimeter. For each area value corresponds a certain membership grade. So, that area value can only be related to the perimeter value having the same membership grade. Considering the fuzzy area and perimeter of FGE represented in Fig. 1, the relation presented in Table 3 is obtained.
Applying the extension principle considering this relation (see Appen-dix B) the fuzzy set of Fig. 5 is obtained. Its support represents the possi-ble variation of the compactness of the FGE and the level cuts of the fuzzy compactness the set of all values the compactness can take considering al-pha level cuts of the FGE corresponding to values larger or equal to alpha.
Area, Perimeter and Shape of Fuzzy Geographical Entities 323
0 0.5 1 1.5 2 2.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Compactness
Mem
bers
hip
func
tion
0 0.5 1 1.5 2 2.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Compactness
Mem
bers
hip
func
tion
Fig. 4. Compactness of a FGE ob-tained through the application of the extension principle to the cartesian product of the fuzzy area and perime-ter of a FGE
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Compactness
Mem
bers
hip
func
tion
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Compactness
Mem
bers
hip
func
tion
Fig. 5. Compactness of a FGE obtained through the application of the extension principle to the relation in Table 3 de-fined between the fuzzy area and perime-ter of the FGE represented in Fig. 1.
Table 3. Relation between the area and perimeter values of a FGE.
Relation between the area zi and the perimeter pi
Degree of membership αi Area zi Perimeter pi Compactness
The inclusion of FGEs in a GIS requires operators capable of computing their geometric properties, such as area, perimeter and shape.
The computation of crisp areas and perimeters of fuzzy sets, and there-fore of FGEs, can be done using respectively the area and perimeter opera-tors developed by Rosenfeld (1984). The crisp values generated by these operators are not satisfactory for many semantics, as they just provide an
324 Cidália C. Fonte and Weldon A. Lodwick
approximate value, giving no information about the area or perimeter variation with the grades of membership, nor about the set of the other possible values. The limitations of the crisp operators motivated the devel-opment of the fuzzy area and perimeter operators. When applied to normal FGEs, both operators generate fuzzy numbers. Their support is the set of all values the area or perimeter can take, and the degrees of belonging to the fuzzy area and fuzzy perimeter represent the possibility of occurrence associated to each area or perimeter value. Fuzzy operators incorporate therefore much more information than the crisp ones.
The shape of geographical entities may be evaluated computing their compactness, which is a function of area and perimeter. As the output of both the fuzzy area and the fuzzy perimeter operators are fuzzy numbers, the compactness of a FGE can be computed using the extension principle, generating a fuzzy value for the compactness. Therefore, fuzzy area and perimeter operators enable not only the delivery of more information to the user, but also the propagation of uncertainty to other quantities. The com-putation of the compactness of FGE is an example of this propagation.
Appendix A
Proof of property 1: The support of ( )PF E is the set [ ]min , maxi ip p . Since the perimeter of a geometric figure is always a positive number,
, for all . Consequently, the support of 0ip ≥ 1, ,i = K n ( )PF E is a sub-
set of .■ 0+
Proof of property 2: A fuzzy number is a fuzzy interval with a bounded support whose core has one and only one value of (Dubois et al., 2000) and a fuzzy set A is an upper semi continuous fuzzy interval if and only if:
R
1. The core of A is a closed interval, represented by [ ],a a , or an interval of the form ( ],a−∞ , [ ),a +∞ or ( ),−∞ +∞ ;
2. The restriction of ( )xµ to ],a−∞ (when appropriate), denoted by , is a right-continuous non-decreasing function;
A (A−
The restriction of to [ (when appropriate), denoted by , is a left-continuous non-increasing function.
( )A xµ ),a +∞
A+
If E is a normal FGE then its core is not an empty set. According to the definition of fuzzy perimeter, the perimeter p of the core of E has a degree
Area, Perimeter and Shape of Fuzzy Geographical Entities 325
of belonging to the fuzzy perimeter equal to one, and consequently is a normal fuzzy set. Since in the definition of fuzzy perimeter
the only value p that has a degree of belonging to the fuzzy perimeter is the core’s perimeter, it can be stated that there is only one value
( )PF E
p∈ such that ( ) ( ) 1E p =PFµ . Denoting this value by pc, the core of ( )EPF is the
closed set [ ],n np p . According to the definition of ( ) ( )PF E pµ , the restriction of ( ) ( )PF E pµ
to [ , n ]p−∞ is an increasing function continuous to the right and the re-striction of ( ) ( )PF E pµ to [ ],np +∞ is a decreasing function continuous to
the left. Then, ( ) ( )PF E pµ is an upper semi continuous fuzzy interval and,
since there is only one value p∈ such that ( ) ( ) 1PF E pµ = , is a
fuzzy number. ■ ( )PF E
Appendix B
Extension principle: Let us consider a function :f X →Y and let ( )XF and be respectively the fuzzy power sets of X and Y. Then, for every set
( )YF
(A X∈F ,)( ) ( )
( ) ( ){ }:
: ,
f X Y
A f A y y f x x A
→
= = ∧ ∈a
F F
and the degree of belonging of each value y Y∈ to ( )f A is given by
( ) ( ) ( )( ) ( )
:sup if :
0 if
Ax y f x
f A
x x y f xy
µµ =
∃ = =
∃ ( ): .x y f x
=
The extension principle is also valid when function f is defined on a car-tesian product, that is, when 1 2 ... nX X X X= × × × and
( ) (1 2
1 2 1 2
: ..., ,..., , ,..., .
n
n n
f X X X Y
)x x x f x x x× × × →
y=a
In this case, if 1 2, ,..., nx x x are non-interactive variables and are fuzzy subsets of respectively
1 2, ,..., nA A A
1 2, ,..., nX X X , the extension principle states that (Dubois et al. 2000)
326 Cidália C. Fonte and Weldon A. Lodwick
( ) ( ) ( ) ( )1 2: nf X X X× × × →LF F F F Y is such that: ( ) ( ) ( ){ }1 2 1 2 1 2 1 2, ,..., : , ,..., , ,..., ...n n n nf A A A y y f x x x x x x A A A= = ∧ ∈ × × ×
y Yand the degree of belonging of each value ∈ to ( )1 2, ,..., nf A A A is given by:
( ) ( ) ( )( )
( ) ( ) ( ) ( )1
1
11
1 1,..., :
,...,,...,
sup min ,..., if ,..., : ,...,
0 if
nn
nn
A A n nx x
y f x xf A A
1 nx x x x y f xy
µ µµ =
∃ = =
∃
x
( ) (1 1,..., : ,..., .n n )x x y f x x
=
The extension principle can also be applied to relations. In this case the extension principle is only applied to the combinations of elements of
1 2, ,..., nX X X that belong to the relation (Dubois et al. 2000).
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