Objects with Fuzzy Spatial Extent

SpatialExtent Objectswith Fuzzy Tao Chengand Martien Molenaar geometry,but it is rathera problem of obiectdefinition and thematic vagueness.For example,in interpretation of remote The determination of the spatia) extent of geo-obiects is sensingimages,uncertainty exists in the thematic aspect generally approached through their boundaries or, more precisely, thrcugh the positions of their boundary points. The expressedby the likelihood of pixels belongingto thematic analysis of the geometric uncertainty of the objects is therefore classes.Imagesegmentscan then be formed from adjacentpixels falling within the sameclass.If thesesegmentsrepresent often based on accuracymodels for the coordinatesof these spatial obiects,then the uncertainty of the extractedgeometry points. In many survey disciplines objects are mapped, of theseobjectsis mainly due to the fact that the value of the however, that are not crisp wel) defined. In that case, the Iikelihood function variesper pixel. Therefore,existing solugeometric uncertainty is not only a matter of coordinate for handling the uncertainty of objectshave not been tions accuracy, but also a problem of object definition and thematic found satisfactory. vagueness.The spatial uncertainty of such objectscannot be The syntacticapproachfor handling spatial objectinforhandled by a geometricapproach alone, such as the epsilon (1994;1996;1998)makesit band method. This paper investigatesthe reasonsfor the fuzzy mation aspresentedin Molenaar possible distinguish three types of statementswith respectto to spatialextent of objectsand proposesan approach to map the the existenceofspatial obiects: spatial extent of objects and their uncertainties when obiects . an existentialstatementassertingthat therearespatialand theare extracted ftom field observation dato. The relationship of maticconditionsthat imply that an objectexists; uncertaintiesbetweenthematic aspectsand geometricaspects . an extensional statementidentifyingthe geometricelements is investigated. A practical example of a coastal geomorphothe spatialextentof the object;and describing logy study is discussed to illustrate the approach. . a geometricstatement identifyingthe actualshape,size,and positionof the obiectin a metricsense.

Abstmct

lntroduction

Spatial objectsthat arerepresentedin a conventionalcls are generallyconsideredto be crisp with determinedboundaries. For example,the land parcelsin cadastralsystemsare differentiated and identified by sharpboundaries.The basic assumption is that the classificationof landscapeunits is crisp and spatialobjectswithin theseclassescan be clearly determined. The secondassumptionis that obiectsare internally homogeneousand can be differentiatedby crisp boundaries.Under the first assumptionthe threshold valuesor criteria for classification are sharply defined.Classesdo not overlap;thus, each objectwill be assignedto only one class.Under the second assumption,the spatial extent of eachobiectcan be defined unambiguouslyand it will not contain unidentified inclusions of areasnot belongingto the obiect.The determinationof the spatial extent of geo-objectsis then generallyapproached through their boundariesor, more precisely,through the positions of their boundary points. The analysisof the geometric uncertaintyof the obiectsis thereforeoften basedon accuracy morlelsfoithe coordinatesof thesepoints. The epsilon band methodis well known in this context (Dunn et 01.,1s90). Theseassumptions,however,are not valid when the spatial extentsofobiects areto be extractedfrom field datathat changegraduallyand continuously over spaceso that no crisp boundariescan be identified, The boundary betweena grassland and a woodland may be gradualthrough a transition zone ratherthan shaply being defined.In such cases,the geometric uncertaintyof objectscannotbe expressedthrough the position accuracyofboundary points. It is then not only a problem of

InternationalInstitute for AerospaceSurvey and Earth Sciences(ITC),P.O.Box 6, 7500AA Enschede,The Netherlands ({cheng,molenaar}@itc.nl). Tao Chengis currently with the Joint Laboratoryfor Geoinformation Science,The ChineseUniversity of Hong Kong, Shatin, NT, Hong Kong. SENSING & REMOTE ENGINEERING PHOTOGRAMMETRIC

Thesethreetypes of statementsareintimately related.The extensionaland geometricstatementsimply the existential statemenUthus, if an objectdoesnot exist, it cannothavea spatial extent and geometry.The existential statementoften relatesto the uncertainty of thematic information, though that is not explicit in the other two statements.The geometricstatement also implies the extensionalstatement,and often the actualgeometryof the objectis derived from the extensional description.Object detectionthrough imageinterpretationis, in fact. an example of the formulation of extensionalstatements.Thesethiee types of statementscan all have a degreeof uncertainty and, although thesestatementsare related,they give us different perspectivesthat may help us to understand the different aspectsof uncertainty in relation to the description of spatialobjects. In this paperwe will concentrateon the uncertaintyrelated to the extensionalstatement.The remainder of the paper is organizedas follows. The next sectionintroducesa syntactic schemato representthe extensionaluncertaintyin terms of the fuzzy spatial extent of objects.The approachto extractobiects from field datais proposedin the third section.It first explains the reasonsfor indeterminateboundaries.Then it investigates the conversionof uncertaintiesfrom thematic aspectsto geometric aspectsduring the identification of the fuzzy spatial extent of objects.Finally, the forming of conditional boundariesand their syntacticrepresentationis discussed.The proposedapproachis illustrated by a casein the fourth section. The test area,datacollection, datapreparation,and modeling

PhotogrammetricEngineering& RemoteSensing v o l . 6 5 , N o . 7 , f u l y 1 9 9 9 ,p p . 7 9 7 - 8 O 1 . /O I ss/ 6507-7 97$3.O0 ooss-r'11.2 O 1999 American Society for Photogrammetry and RemoteSensing

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resultsarepresented.The last sectionof the papersummarizes the major findings and further research.

Representation of FuzySpatial Extent

Molenaar(199a;1998)proposeda syntacticrepresentationof objects,which can repr-esentthe veclor data an'draster data in a unified way. Let's assumethat the geometryof a spatial database (shortly,a map M) hasthe topologic structureof a planar graph. Eachedgewill always have one face at its left-hand sideind one qn its righthand side. The relationship can be expressed by following functions: Edge e; has a facef at its left-hand side = Lelei,f.l : 1, otherwise : 0. Edge ei has a face/, at its right-hand side + Ri[e1,f"] : 1, otherwise : 0. With this function, we can defineB[e;,f l: Le[ei,f"] + Rilei,f"l. If the value of this function equals0, e;is not relatedtof ; if the value equals2, then the edgehas a faceon both sidesand is thus inside the two faces;if the value equals1, then the edgehas a fac_e on only one side,so that it must be part of the bound-ary. The boundarvofa face f" is then defined as

Baf": lNaf",Eaf"l : 1] representsall the edgesthat have whereEdfo: {e;lB[e,,7,1 the face/" on only one side and Llaf : {njlnj . e,le, e ETf,l representsthe nodesofthe edgesofEOf". The relationship between a face and an area obiect can be

represented esentedas asPart[f., Part[f",O"]. O,]. If it takes takes a avalue value of 1, it implies implies tha that the facebelongs belonesto the object. obiect.If it ittakes valueof nf 0, o itimplies itlmnlieq takesa value that the facedoesn'tbelongto the object. Therefore,the relationship betweenedgeof an objectcan be expressedas follows: Lele1,O"l : Lelei,f.] " Partlf", O"l, Ri[ei, O,] : Ri[et,f,] * Partffo, O"). The boundary ofan objectis then defined as B[e;, O.] :

Lelei, Oo] + Ri[ei, O"].

(1)

. ^ _ If the value is equalto 0, the edgeis not relatedto the object; if the value is-equ_al to 2, then the edgeis inside the object;if it has a value of t, the edgeis the boundary of the objects,i.e., BaOo: lelB[e,O"] : tl. If the objecti s htzzy in the sensethat its spatial extent is uncertain,the relationship of the fact and the obiectis uncertain,which canbe exoressed as Face(O"): {f"lPartlf,, O"] > 0].

(2)

The boundary of a finzy objectcan be defined in the same manneras in Equation1. But the value of the function will vary from 0 to 2. Ifthe value is equal to 0, the edgeis not relatedto the object;ifthe value is biggerthan Oand lessthan 2, the edge is an indeterminateboundary of the object;if the value is eqrial to 2,the edgeis inside the obiect.Therefore,the indeterminate boundarvof an obiectis definedas 798

luly 1999

BAO.: IelO < B[e, O,] < 2)

(3)*

- Basedupon certain criteria, the conditional spatialextent of objectscan be identified, so the indeterminateboundarv becomesa conditional boundary.This syntacticrepresentaiion ofthe relationship betweenedges,boundaries,facis, objects are can be applied for both vector and rasterstructures.The cells of a rasterarethen consideredas faceswith a rectanqular shape(Molenaar,1994;Molenaar,1998).

Extraction ofFuuy0bjects Procedure of Extraction of Objects fiomField0bseryation Data When natural phenomenahave a field character,they can often only be sampledsparselyat a limited number of points, which can then be interpolatedto generatea full raster.In the case describedin the fourth section,this will be an elevationraster. Three types of objectswill be extractedfrom theseheight data through a segmentationof the elevationraster:the foreshore, the beach,and the foreduneareas.Eachtvpe of obiectwill be related to a height interval. A six-stepprocedurewill be followed to identify objectsfrom sampledfield data: (f ) Samplilg datavaluesat specificsamplepoints. (2) Interpolation,alsocalled"regionalization," of the observed data to generate a complete elevation raster covering the observedarea. (3) Classification of all grid cells into pre-defined classes.Bach grid cell is assignedto a class interval that can be related to one of the natural units. Segmentation of the classified raster into areas.Each contieuous set of grid cells belonging to one class will form a.r ar*ea that represents the spatial extent of a particular natural unit. Merging areasthat are smaller than a pre-defined lower threshold for mapling units with an adjacent area. Traditional merging methods, such as "window filtering," "nibbling,', "dropping the longestsharedboundary," and "maximum area merging" (Ma and Zhao, 1995) can be used to remove these small areas. ( 6 ) Identification of objectsrepresentedby the areas,i.e., identification of the actual objectswhose spatial extentsare lepresented by the final segments (after merging).

. Figure 1 illustratesth_eprocedureby meansof a crisp example. The stepsrepresentedin this figure startwith the inlerpolated grid cells aiter Step 2. In Step 5 the grid cells are classified into three elevation classes:"H" (high), ranging from 15 20; "M" (medium), rangingfrom rolhr6ugh 14; and !!1o9Sh 'lt " (low), rangingfrom 5 through g. The segmentationof Step 4 identifies three areas,In Step 5, Area 3 hai beenmergedinto Area 2. Finally, two objects-"A" and "8"-are identified in Step6. ldentification ofSpatialExtentandBoundary of Fuzy0bjects As pres,ented in the previous subsection,classificationand segmentation are essentialfor extractingobjectsfrom field observation data.The classificationofgrid cells is uncertain for two o The height values of the grid cells do have a limited accuracy due to the measurementand interpolation process;and e The height classesrelated to the three obiect types are fuzzy, as will be explained in the fourth section. The combination of these two kinds of uncertainties has been discussed in Cheng et aI. (1992). The uncertain classification can be expressed in terms of the membership function

*The concept of a crisp boundary of an object is only valid when the face belongs to the object (Partlf, Ol : 1) and the edge has the face on either the left or right side. Therefore,Equation 3 is valid only for htzzv obiects.

PHOTOGRAMMETRIC ENGINEERING & REMOTESENSING

Interpolation

t2)

Classification

(3)

t

lTlrfLTLEl ET:t-r-TI ILt UMIVltNrI

lLlruMrulllll tt,tuMMt.tvll I Segmentation

+

assumedto be spatially exclusive,then eachgrid cell belongsto at most one class,and if the setof identified objectsforms a spatial partition ofthe mappedarea,then eachgrid cell belongsto exactlyone object.In oiher applications,fuzzy spatial overlaps amongobjectsare permitted, i'e', the spatial extentof obiects through crisp boundariesthat also carry ."ntroi be-expressed but the oblectshave fuzzy transition inlormation, adiacencv zonesthat may overlap (Burrough,L996;Usery,1996),In the the pixels might belongto multiple objects. transition zon-es, The fuzzy topologic ielationships of spatial objectsare discussedin Dil-kmeilerand De Hoop (1996)and Zhan (1997)'In our case,the landicape units form spatial partitions. So each erid cell should belongto exactly one classand thereforeto one 6biect,which can be d'eterminedby criteria such as we will deiine here. Let NMP;;, Crl : f - MFlPii,C1lrepresentno-membership, i.e.,the certaintythat Padoesnot belongto classC1,and let XIIIIp,,,Colexpressthe membershipthat P;;belongsexclusively to ii ind not to any other classesQ for any I + k. XMPii, Ckl can be derived by applying minimum operations as XMPii, Ckl: MIN(MFIPii,Cpl,MINpp(NMPtj, CID)' (4)

(4)

MHIffiIMHilil lmIHtE

IffiruIil

W

BecauseP;ican only belongto one class,only one classis reouired for wliich the function XMI has a maximum value for P,,.If there are more classeswith the samemaximum values, tlien additional evidenceis required in order to anive at a selectionof a unique class.It can be representedas if XMPii, Cel : MAX6,(XMPii, Ci)

I

/5\

(1: 1, ..., N), then let D[P1i,C*] : 1: otherwise, D[Pii, C*] : o.

(5)

For example,in our casea grid cell hasthe membershipvector

MFIP,Cl (6)

:{:'?}

where G is a foreshoreclass,C, is abeachclass,and C3is a foredune class.Therefore,

N M P ,C I : Figure1. Procedure for (L:5-9; objectidentification M: 10-14; H: 15-20).

andXMp, cl =

[i,s,?]:{:,i} o.s))'l [o.zl (Mrr{(o.2,MrN(0.3, o.e))f: 10.?I lurNlo.z,urN{to.B,

Lo.1J LM/N(O.1,M1N(0.8,0.3)U value that variesper cell and is lessthan one.To identify the spatialextent,*" h"lt" to first assignthe grid cells into classes thu.t clusterthe cells of the sameclassesinto areaswhich ""rd representthe spatialextentsof fu1zyobjects'Here we will disthe effectbf fuzzy classificationon identification of the cu^ss spatialextentof objecls.We will also investigatehow existential uncertainty is ionverted into extensionaland geometric uncertainties. A membership vector lMFlPti, Cl, MI\Pi1, C4' ' ' ' ,, MI!,,, Crollr(0 = MFlPti,Z*l < 1) will be createdfor eachgrid cell Pa ^nu iurry classificition.Herc, MFIP,Cl representsthe memto classC, and [..rfrip ni".tion value ofgrid cell P;Tbelonging N is th-etotal number of the classtypes. For eachclassc*, areascanbe identified withMF(Pii, Ci > Threshold*.Theseaieascan then be interpretedas the fuzzy extent of spatial obiectsbelongingto Cp Ifthe classesare SENSING & REMOTE ENGINEERING PHOTOGRAMMETRIC

Because XMP, c2l = MAX

[:,?]

= 0.7, thereforcD[P, C"] : t.

This meansthat this cell is assignedto classCz(thebeacharea) with a certainty of 0.7. After assigningthe cells to classes,an areaSoof classtype Cl will be forfied 6y the fgllowing two conditions (Molenaar, 1996): e So,D[Pii,Ci: for all pixels P11

1, and

if. Pi1e So and ADIACENTIP41,Pi = 1 and D[P,;, Cr] : t, then P,i e So. luly 1999

(6)

_l

ADJACENTIPu,Piil exp_resses the adjacencyrelationship , between cells Pp1and P4,and it has a v-alueor eiiher Oor 1. p,; will only be assignedto S if " DIPij, Crl : r. " The certaintythiat this assignmentis correct" dependson the certainty that ihe pixel hasbeenassignedconectly to C1.Therefore,ihe relationship betweenP;;and So, PartlP1i,Sol,can be written as PartlP11, S,l : MIN(DIPii, C*], XMpri, Cpl).

(7)

representsthe spatial extent ofthe object O,, _ Because-So the relationshipbetween PilandOocanbe defined (Ct u"g, ", 1e99) PartlPil, O,l:

PartlPil, SJ : MIIt(D[pij, Cr], XMpii, C*)).

(B)

EquationB expressesthe relationship betweenthe uncertainty of a p.ixelbelongingto the spatialextentof an objectand the uncertainty.ofa pixel belongingto classes,i.e., the ielation_ snrp betweenthe geometricuncertainty and the thematic uncertainty.This meansthat the uncertiinty is transferredfrom thematic aspects,togeometricaspectsof obiectsduring spatial c.tustering,i.e., the existential uncertainty is converted to ext en si on al tn certainty. Becausethe spatialextent is fuzzy,there is no crisp bound. ary betweenobjects.After identificatibn of the spatialextentof objects,i.e., assigningpixels to areas,boundari6sare formed. We call them conditional boundariesto distineuish them from crisp boundaries.In our case,the conditiona"lboundarv betweentwo objectsis the transit boundary between two classes. According to the Equation1, the boundary of an objectcon_ . sistsof edgesthat have the obiecton one side.'Tochec(if the edgehas an objecton one side, the relationship betweenthe edgeand the facesbelongingto the obiectshould be checked. Therefore,to identify the conditionaiboundary of afitzzy object,we have to first identifythe facesthat belong to thL object and then find the edgesthat have fa"". o., o.riy one side. The faceof two fuzzy objectsshould satisfy Face{O") : CelI(O,) : lP1)Part[pii,Oo] > partlpii, 06l

(el Face(O) : CelI(Oi : lPillPart[p11, Oa] > partfpli, Then the transitionboundary consistsof edgesthat have simultaneouslythecells of Ooon the left side and the cells of 06 on the right sjde,or the cells of O, on the right side and the ceils of 06 on the left side.Therefore,the edges6fthe boundary should satisfy(Cheng,1999) E",b: leilBle,,f"l : t and B[e;,fA : t and f e Face(O") and /6 e Then the transition boundary is

Ba(O,,Oi :

4,rl and N,,6 : {n;ln;e

0

ftr.*or-ur, (u;a132b2+fl2)

Figure2. Fuzzyclassification function.

foreduneis the first row of the dunesinland from the dune foot. According to thedefinitions given by geomorphologistsfor the situation oJAmeland, the^heightvaiGs of the closriredepth, low water line, and dune footire suggestedto be about -'o.O*, - 1.1T, and 2m, respectively. Thesedefinitions appearto be vaguebecausethe expertsdo not agreeexactly on i6esevalues. l,neretorerwe adopteda trapezodialmembershipfunction as illustrated in Figure 2 and define the transition zonebetween the classesrelatedto theselandscapeunits as in Table1 (Cheng et al.,19sz). . . Height observationshav-ebeen made by laserscanningof the beachand dune areaand by echosoundineat the foresliore. Thesedatahavebeeninterpolited to form u frril h"ight,aster of the test area..Expe-riments show that the uncertaintylf the inter_ polated heightsofthe rastercan be expressedby thLir standard d e v i a t i o n( o : 0 . 1 b m )( H u i s i n ge t o 1 .i,g S O ) . As shown in Plates1,{, 18, and 1C, eacherid cell has a membershipvector containing a value for eac"hof the three c,lasses. After identifying the riost likely classtype for eachcell, the mapped areahas been segmentedby clusteringthe cells belongin^gto the sameclass.fhe u.u"s oi diffe."rrt c"las""srlpru_ sentthe fuzzy spatial extent of the objects,which are shown'in Plate 1D. The transit boundariesareihown in plate 1E.

e; e Eo,6l.

Ameland is a barrier island in the north of The Netherlands, wheregeomorphologic processes occuralongthe coast.partic_ ularly the erosionand accumulationof sediments.Thes'epro_ cessescanbe monitored throygh the observationof changls of Iandscapeunits such as foreshore,beach,and foredune.ihe foreshoreis the areaabovethe closuredepth and beneaththe low water line, the beachis the areaabov^e the low water line and beneaththe dune foot (Reineckand Singh, 1980),and the luly 1999

(Uptt