This paper introduced a model based on contour lines to represent a vague region modeled as a fuzzy set. The model allows the user to adjust the accuracy of the membership function to the needs of the application and can enforce the continuity of this function whenever desirable

De Caluwe, Rita

De Tré, Guy

Hallez, Axel

Verstraete, Jörg

English

Ghent University Academic Bibliography

Contourline Based Modeling of Vague RegionsA. Hallez, R. De Caluwe, J. Verstraete, G. De TreGhent University, Dept. of Teleommuniations and Information ProessingSint-Pietersnieuwstraat 41, B-9000 Ghent, BelgiumAxel.Hallezrug.a.beAbstratThis paper introdues a model basedon ontour lines to represent a vagueregion modeled as a fuzzy set. Themodel allows the user to adjust theauray of the membership fun-tion to the needs of the appliationand an enfore the ontinuity of thisfuntion whenever desirable.keywords: Vague Regions, SpatialData.1 IntrodutionReently there has been a growing interestin the modeling of vagueness in spatial re-gions. Several authors have proposed a rep-resentation of spatial regions with non-rispboundaries by introduing broad boundaries([1℄,[2℄,[4℄). In those papers a vague region isharaterized by an inner and an outer bor-der as shown in gure 1. The outer borderdelimits the set of points that belong to theregion to a ertain extent, the inner borderdelimits the set of points that belong to thebroad boundarykernelinner border outer borderFigure 1: Sample of a vague region.region for ertain. The subregion of points ly-ing between these borders is alled the broadboundary.The papers ited above mainly disuss thetopologial properties of regions that are rep-resented by an inner and an outer border, ig-noring the available information about pointsin the broad boundary.In [6℄ a method to assign a membership de-gree to the points lying in the broad bound-ary has been presented. In this method themembership degree of a point is based on thedistanes from the point to the inner and tothe outer border. This is a very straightfor-ward method, but the membership funtion isompletely determined by both borders only,so there is no possibility to rene the mem-bership funtion in the broad boundary if de-sired.This paper presents an extension of thismethod, that allows for a more aurate mod-eling of the membership funtion in the broadboundary by introduing ontourlines. Infat, these ontourlines provide a very exi-ble means of dening a membership funtionwith arbitrary auray.Setion 2 explains how ontourlines are de-ned. Setion 3 and setion 4 explain how theontourlines are used to alulate the mem-bership degrees of individual points of a vagueregion.2 Contourlines in the broadboundaryA region is a subset of a two dimensional Eu-lidian spae U . Vague regions are a gener-alization of regions. In the model explainedbelow, vague regions will be represented byfuzzy sets. The outer border of a vague re-gion X is the boundary1of the support of X(supp(X) = X0)2, the inner border is theboundary of the kernel of X (ker(X) = X1).The onept of using a border to dene -leveluts an be applied to membership degreesother than 0 and 1, using information aboutpoints in the broad boundary. It is possible todene additional borders, eah of these bor-ders orresponding to a spei membershipdegree m 2℄0; 1[ suh that it is the boundaryof Xm.The set of membership degrees for whih aborder is dened to model a vague region X,will be denoted MX [0; 1℄.To be useful in the ontext of the presentedmodel (as will be explained further), MXmust be a nite set that ontains at least twoelements, inluding 0. It is not required to in-lude 1 inMX, whih makes it possible to usenon-normalized fuzzy sets to represent vagueregions. Sine X0is undened, the member-ship degree 0 has to be treated as a speialase.The border orresponding to a given member-ship degreem does not neessarily orrespondto one single line. In the ase that Xmhas a"hole" in it or is a disonneted set, the or-responding border will orrespond to two ormore losed lines.A losed line that is part of a border will bealled a ontour line3. The set of all ontourlines that are dened to model a vague region1A point p is alled a boundary point of a (risp)set S i every non-empty environment of p intersetswith both S and the omplement of S. The set ofall boundary points of S is alled the boundary of S,denoted by Bd(S).2X::= fxjX(x) >= g for  2℄0; 1℄,X::= fxjX(x) > g for  2 [0; 1[, see [3℄.3In [5℄ (p.66) ontour lines are dened as follows:"Contour lines are a representation of isolines for asample set of elevation values".X will be denoted CX. A ontour line or-responding to the membership degree m willbe alled an m-ontour line. The subset ofCXof all ontour lines that orrespond to themembership degree m will be denoted CXm.In order for the presented model to be appli-able on a vague region, the following restri-tions must be satised: The vague region must be bound. (Itis not possible to dene the membershipfuntion of an unbound region in thismodel, as at least the 0-ontour lines an-not be dened). The support of the vague region must bea onneted set. If neessary, a vagueregion with a disonneted support anbe modeled as the union of two or morevague regions with a onneted support. The -uts for membership degrees otherthan zero an be disonneted sets, butthey have to be the union of a limitednumber of onneted subsets (to exluderegions with an innite number of on-tour lines).Figure 2 is an example of a vague region.This region has one ontour line for the sup-port (0-ontourline) and one 0.5-ontour line.The kernel onsists of two disonneted parts,hene its boundary is made up of two ontourlines.Given the denition of ontour lines, it is pos-sible that several ontour lines, orrespondingto dierent membership degrees, (partially)oinide. As a most striking example, the on-tourlines of a risp region dened with this00.511Figure 2: Sample of a fuzzy region with a dis-onneted kernel (marked grey).model will be the same for all membershipdegrees, as the outer border (0-ontour lines)and the inner border (1-ontour lines) willompletely oinide.Given a fuzzy region dened by a set of on-tour lines, the omputation of the member-ship degree of a point lying in the broadboundary onsists of two parts. Firstly, theontour lines that will be used for the ompu-tation of the membership degree of the point,alled the relevant ontour lines, have to bedetermined. Seondly, the membership de-gree of the point has to be omputed fromthe membership degrees assoiated with theserelevant ontour lines.3 Determining the relevantontour linesThe ontour lines of a region divide the broadboundary in adjoining non-overlapping subre-gions, so that eah subregion is bound by asubset of the ontour lines and eah ontourline is part of the boundary of at least onesubregion.This notion of subregions is used to determinethe relevant ontour lines of a point lying inthe broad boundary of a vague region. If apoint p is not loated on a ontour line, itis possible to nd the unique subregion thatontains p. The ontour lines that border thissubregion will be used to alulate the mem-bership degree of p (setion 4).3.1 A ontour line treeA tree of ontour lines will be used to deter-mine the relevant ontour lines.A ontour line divides the plane into twoparts. One part is the inside4of the ontourline and the other part the outside of the on-tour line. The points of the ontour line itselfbelong to the inside. When a point p belongsto the inside of a ontour line C, it is said thatC enloses p. If all points of a seond ontourline C0are enlosed by C, then C enloses C0.4the region dened by the inner normal of the on-tourline.Consider a vague regionX. The following ruledenes a tree of all ontour lines in CX:8A;B 2 CX:A is an anestor of B, A enloses B:If there are no ontour lines that totally oin-ide, this rule denes the tree unambiguously.If two ontour lines ompletely oinide, oneontour line enloses the other and vie versa,whih auses ambiguity. In this ase, a nat-ural order for these ontour lines has to befound. If the ontour lines orrespond to holesin their orresponding -level uts, the on-tour line with the higher membership degree isdesignated as the anestor. In all other ases,the ontour line with the lower membershipdegree is designated as the anestor.As it is assumed that the support of a vagueregion is onneted, CXontains exatly one0-ontour line that enloses all other ontourlines. As a result, this ontour line will be theroot of the tree.A ontour line A 2 CXwill be a hild of on-tour line B 2 CXif B enloses A and:9C 2 CX: (B enloses C) ^ (C enloses A)Figure 3(a) shows a more elaborate exampleof a vague region. Figure 3(b) shows the or-responding ontour line tree.If the ontour line tree is known, the searh forthe relevant ontourlines for a point p is ar-ried out by applying the following algorithm: Step 1: Start with the root of the treeas the urrent node. If the orrespond-ing ontour line enloses p, arry out thestep 2 of the algorithm, otherwise the al-gorithm ends as the region does not on-tain p (whiwh will be assigned the mem-bership degree 0). Step 2: Repeat this step until the urrentnode does longer hange:For eah hild-node of the urrent node:if the orresponding ontour line enlosesp, make this hild the urrent node.a)B - 0.33C - 0.66D - 1E - 0.33G - 0.66F - 0.66A - 0b)AB EF GCDFigure 3: (a) An example of a vague regionwith labeled ontourlines. (b) the orrespond-ing ontourline-tree.The relevant ontour lines are the lines orre-sponding to the urrent node, obtained at theend of the algorithm, together with its hild-nodes if any. The proess always ends beausethe set of speied ontour lines must be nitein this model.The next setion shows how the ontour linesand their assoiated membership degrees areused to alulate a membership degree for p.4 Interpolation methodThe distane d between a point p and a on-tour line C is dened as:d(C; p) = d(p;C) = minp02Cd(p0; p)It an be proved that this funtion is ontin-uous in every point p 2 U for a given C.If a point lies on a ontour line, this point isassigned the membership degree orrespond-ing to this ontour line. If the point lies onmore than one (oiniding) ontour lines, it isassigned the highest membership degree or-responding to these ontour lines.The alulation of the membership degree of apoint p not lying on a ontour line in a vagueregion X is based on the interpolation of themembership degrees of the relevant ontourlines. Based on the number of relevant on-tour lines for p, three general ases an bedistinguished.4.1 One relevant ontour lineIn this ase the only relevant ontour line doesnot enlose any other ontour line (e.g. on-tour line F in gure 3) p is assigned the mem-bership degree assoiated with the relevantontour line.As a result the membership funtion of X willbe ontinuous for all points inside the relevantontour line.4.2 Two relevant ontour linesIn this ase (e.g. points lying between Band C in gure 3) the membership degree ofp is dened as the weighted average of themembership degrees assoiated with both rel-evant ontour lines. The weights are alu-lated from the distanes from p to the ontourlines:X(p) =1d(p;C1)X(C1) +1d(p;C2)X(C2)1d(p;C1)+1d(p;C2)(1)=d(p;C2)X(C1) + d(p;C1)X(C2)d(p;C1) + d(p;C2)(2)where C1and C2are the relevant ontourlines, X(C1) and X(C2) the assoiated mem-bership degrees.Taking into aount that d(C1; p) and d(C2; p)are ontinuous funtions in all points p 2 U , itan be proved that (1) is a ontinuous fun-tion for all points p 2 Un(C1SC2). More-over, the limit of the funtion in points of C1isX(C1) and the limit of the funtion in pointsof C2is X(C2).4.3 More than two relevant ontourlinesThis is the most ompliated ase. (e.g. thease whih holds for points lying between theontour lines A, B and E in gure 3.) Thereare several options for alulating X(p). De-pending on the option that is used, the result-ing membership funtion will have dierentproperties.One option is to hoose two elements offCiji = 1; :::; ng, thus reduing this ase tothe previous ase. It is obvious that two on-tour lines will be hosen based on the distaneof the ontour lines to the p. However, simplyseleting the two ontour lines that lie los-est to p will not result in a good membershipfuntion. First of all the membership degreeof points that are equidistant to two or moreontour lines is not always well dened in thisase. Of ourse, this ould be solved by hoos-ing one of both equidistant ontour lines (forinstane the ontourline that results in thehighest membership degree), but this way ofdening the membership funtion would on-it with the provided ontour lines! For in-stane, if there are three relevant ontour linesC1, C2and C3with X(C1) = X(C2) = m1,X(C3) = m2and m1> m2, the points forwhih C1and C2are used to alulate themembership degree, will be assigned member-ship degree m1. However, the lower member-ship degree of C3indiates that the subregionbetween these three ontour lines does not be-long to Xm1. By denition, points that donot belong to Xm1have a membership degreelower than m1. To get rid of this problem therestrition is introdued that the two ontourlines that are used to alulate the member-ship degree of p, have to orrespond to twodierent membership degrees whenever possi-ble (e.g. if not all ontour lines orrespond tothe same membership degree). With this re-strition, the resulting membership funtionwill not only be onsistent with the denitionof the ontour lines, but it will also be on-tinuous and the limit of the funtion in pointslying on a (relevant) ontour line will be themembership degree that orresponds to thisontour line. These last two properties makethe hoosen approah useful.Another option is to generalize (1), so thatX(p) beomes the weighted average of allmembership degrees assoiated with all rel-evant ontour lines:X(p) =Pni=11d(p;Ci)X(Ci)Pni=11d(p;Ci)(3)with fCiji = 1; :::; ng the set of relevant on-tour lines for p.It an be proved that the funtion (3) is on-tinuous in all points p 2 UnS1inCi. Thelimit of funtion (3) taken in a point p 2 Ciis X(Ci).4.4 Continuity of the membershipfuntionIn the previous subsetions, it has beomelear that with the presented method themembership funtion of a vague region ismade up of many dierent funtions. In fat,every subregion as mentioned in setion 3 hasits own membership funtion. It has beenmentioned that all of these funtions are on-tinuous in the part of the region where theyare used as the membership funtion.As the points lying on a ontour line are as-signed the orresponding membership degreeand the limit of the membership funtions,used in the adjaent subregions, in a pointlying on the ontour line equals to the samemembership degree, the membership funtionwill also be ontinuous on the ontour linesitself as long as the ontour lines are not o-inident. Where two or more ontour linesoinide, the membership funtion will be dis-ontinuous.As a result, the membership funtion beingontinuous or not is not a side eet of thealulations, but is determined by the relativeposition of the ontourlines only. This meansthat the user of this model has full ontrolover the ontinuity of the membership fun-tion, whih an be highly desirable.5 ConlusionA very general method for modeling vague re-gions has been introdued. With this methodit is possible to model a very wide range ofvague regions and to modify the preision ofthe membership funtion by inreasing or de-reasing the number of membership degreesfor whih ontour lines are dened. It is alsopossible to ontrol the ontinuity of the mem-bershipfuntion by dening the ontour linesto be oinident or not.Referenes[1℄ E. Clementini, P. Di Felie; An algebraimodel for spatial objets with undeter-mined boundaries; 1994; GISDATA Spe-ialist Meeting - revised version.[2℄ A.G. Cohn, N.M. Gotts; Spatial regionswith undetermined boundaries; 1994; Pro-eedings of the Seond ACMWorkshop onAdvanes in Geographi Information Sys-tems; pp. 52-59.[3℄ D. Dubois, H. Prade; Fundamentals ofFuzzy Sets; 2000; Kluwer Aademi Pub-lishers.[4℄ N.M. Gotts, A.G. Cohn; A mereologi-al approah to representing spatial vague-ness; 1995; Working Papers, Ninth Inter-nationalWorkshop on Qualitative Reason-ing; pp. 246-255.[5℄ P. Rigaux, M. Sholl, A. Voisard; SpatialDatabases with Appliation to GIS ; 2002;Aademi Press.[6℄ J. Verstraete, B. Van Der Cruysen, R. DeCaluwe; Assigning Membership Degrees toPoints of Fuzzy Boundaries; 2000;Proeedings of the 19th InternationalConferene of the North Amerian FuzzyInformation Proessing Soiaty; pp. 444-447.

Contourline based modelling of vague regions

https://biblio.ugent.be/publication/1946761/file/1946794.pdf

Contourline based modelling of vague regions

Abstract

Similar works

Full text

Available Versions

Ghent University Academic Bibliography