Lemonia Ragia and Stephan Winter 1 CONTRIBUTIONS TO A QUALITY DESCRIPTION OF AREAL OBJECTS IN SPATIAL DATA SETS

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1 D. Frisch, M. Englich & M. Seser, eds, 'IAPRS', Vol. 32/, ISPRS Commission IV Symposium on GIS - Beween Visions and Applicaions, Sugar, Germany. Lemonia Ragia and Sephan Winer 1 CONTRIBUTIONS TO A QUALITY DESCRIPTION OF AREAL OBJECTS IN SPATIAL DATA SETS Lemonia Ragia Insiue of Phoogrammery Universiy of Bonn, Nußallee 15 D Bonn, Germany Ph.: Fax: Lemonia.Ragia@ipb.uni-bonn.de Sephan Winer Deparmen of Geoinformaion TU Vienna, Gusshaussr A-1 Vienna, Ausria Ph.: Fax: winer@geoinfo.uwien.ac.a KEY WORDS: spaial daa qualiy, comparison of spaial eniies, building exracion. ABSTRACT In his paper we presen a qualiy evaluaion of wo-dimensional building acquisiion. We propose mehods for idenificaion and quanificaion of differences beween independenly acquired regions, and we presen a sysemaic classificaion of hose differences. Differences beween acquired ses R j = fr i g j of regions r ij depend on he conex of observaion, on he echnique of observaion, and so on. We disinguish opological and geomerical differences. Topological differences refer o he inerior srucure of a se of regions as well as o he srucure of he boundary of a single region. Geomerical differences refer o he locaion of he boundary of a single region or of a se of regions, independen of heir represenaion and of he srucure of he boundaries. Idenificaion of differences requires a maching of wo daa ses R 1 and R 2 which is done here by weighed opological relaionships. For he idenificaion of opological differences beween wo ses R 1 and R 2 of regions we use he wo region adjacency graphs. For an idenificaion of geomerical differences we use he zone skeleon beween wo mached subses rp1 and r q2 of he given ses. The zone skeleon is labeled wih he local disances of he corresponding boundaries of he subses; especially we invesigae is densiy funcion. An example, based on wo real daa ses of acquired ground plans of buildings, shows he feasibiliy of he approach. 1.1 Moivaion and Idea 1 INTRODUCTION A lo of concurren acquisiion echniques for spaial daabases are available. They need o be evaluaed wih respec o given specificaions, and compared wih respec o a usually grea number of crieria, such as efficiency, accuracy, and compleeness. In his paper we presen a qualiy evaluaion of wo-dimensional building acquisiion. We propose mehods for idenificaion and quanificaion of differences beween independenly acquired regions, and we presen a sysemaic classificaion of hose differences. Two daa ses of he same objec caegory nearly never show inciden eniies (Fig. 1). Differences beween acquired ses R j = fr i g j of regions r ij depend on he conex of observaion, on he echnique of observaion, on error, on numerical problems of discree represenaions, onemporal effecs, and so on. When given a se of observed regions a second se of regions is needed, playing he role of a reference daa se. Fig. 1 shows wo small ses of ground plans of buildings acquired wih wo differen echniques. Neiher he number of buildings nor he pariioning coincides. Addiionally one could imagine differen levels of deail in he daa acquisiion, and here will be differences in he locaion of maching regions. The idenificaion of he various differences obviously is a complex problem as all ypes of errors may inerfere. In his paper we invesigae he differences beween wo independenly acquired daa ses which cover he same wo-dimensional space, wih one daa se considered as he reference. We disinguish opological and geomerical differences. Topological differences refer o he inerior srucure of a se of regions as well as o he srucure of he boundary of a single region. Geomerical differences refer o he locaion of boundaries of single regions or of ses of regions, independen of heir represenaion and of he srucure of heir boundaries. By his way we obain a complee descripion of he differences of wo ses of regions. Idenificaion of differences requires a maching of wo daa ses R 1 and R 2 which is done here based on weighed opological relaionships. For he idenificaion of opological differences beween wo ses of regions, R 1 and R 2, we use he wo region adjacency graphs. For an idenificaion of geomerical differences we use he zone skeleon beween wo mached subses r p1 and r q2 of he given ses. The zone skeleon is labeled wih he local disances of he corresponding boundaries of he subses; especially we invesigae is densiy funcion. Ulimae goal of our work are ools for deermining he qualiy of ses of acquired wo dimensional regions. The use of such a descripion of differences is manifold: Daa capure can be supplied by classificaion of errors onhe-fly. Comparison of wo daa ses yields measures for qualiy assessmen. Having a complee saisics of classified differences may be advanageous when merging wo daa ses. 1.2 Framework and Terms of he Paper In he conex of his paper we confine ourselves o daa ses conaining he same class of areal objecs (regions). We do no consider poins and lines as spaial objecs; objecs in real world have spaial exen, and hey need a represenaion adequae for heir exen. Also we exclude here fields (coverages) as unbounded spaial phenomena (Burrough and Frank, 1996). The objecs shall belong o he same class because hey will be compared only wih regard o heir opology and geomery, bu no in heir aribues or oher properies. Tha presumes also a similar granulariy in boh daa ses; i does no make sense o compare, e.g., buildings wih sree-blocks. 1.3 Srucure of he Paper The paper is organized as follows. Secion 2 summarizes he previous work in he field of qualiy descripions for spaial daa, and in comparing spaial daa ses. In Secion 3 we invesigae differences beween spaial eniies from a opological and a geomerical poin of view, and we discuss he reasons for observable

2 D. Frisch, M. Englich & M. Seser, eds, 'IAPRS', Vol. 32/, ISPRS Commission IV Symposium on GIS - Beween Visions and Applicaions, Sugar, Germany. 2 Lemonia Ragia and Sephan Winer Figure 1: Two real daa ses R 1 and R 2, observed wih differen acquisiion echniques ( cdetemobil 1998). differences. Secion presens he used approaches for characerizing opological and geomerical differences of regions. Wih hese ools a hand, in Secion 5 we invesigae and sysemaically classify geomeric differences. Tha will yield he basic parameers o find operaional qualiy descripions. In Secion 6 we presen an empirical es of he mehod. We conclude in Secion 7 wih an overview on furher invesigaions and developmens in his field. 2 PREVIOUS WORK In his secion we discuss previous spaial daa se comparisons and qualiy descripions. I will become clear ha here is a srong demand for furher research; he presened ideas fi ino his framework. 2.1 Comparison of Spaial Daa Ses Comparison of wo spaial daa ses, referring o he same secion of space, is a fundamenal abiliy for Geographic Informaion Sysems (GIS); i is uilized in handling posiional uncerainy or imprecision (Glemser, 1993), in maching spaial eniies (Knorr e al., 1997, Harvey e al., 1998), in daa fusion (Haala, 199), in change deecion, in generalizaion (Goodchild and Procor, 1997, Tryfona and Egenhofer, 1997), and so on. Comparison yields a descripion based on common and disinc feaures (Tversky, 1977); he ype of descripion number and meaning of parameers depends on he conex. The presened ideas are relaed o acual work in Maine (Bruns and Egenhofer, 1996, Egenhofer e al., 1997). They describe spaial scene similariy by disances aken from concepual neighborhood graphs of differen qualiaive relaions beween regions. Such a disance is discree and qualiaive; as a weighed mean over he differen relaions i is absrac and canno be inerpreed geomerically. In conras (and complemening) we use a coninuous measure: he disance funcion beween he wo boundaries (Winer, 1996). The disance funcion can be inerpreed geomerically, and addiionally we are able o caegorize he resuls for qualiaive disincions. Addiionally, we compare he inernal opological srucure of he associaed regions, which complemens also a comparison of relaions. 2.2 Spaial Daa Qualiy Descripions Qualiy of spaial daa (ses) is an acual opic in sandardisaion as well as in research. The CEN meadaa sandard (Comié Européen de la Normalisaion), conaining also spaial daa qualiy descripions, is pending for resoluion, and also ISO TC 211 is finalizing wih a sandard. Sandards provide a common se of parameers o describe a phenomenon, here spaial daa qualiy, which is necessary for daa caaloguing as well as for daa exchange. The acual need for sandards is beyond doub, for reasons of marke developmen. Bu he process of specificaion ook several years, and he difficuly o se sandards in he field of spaial daa qualiy indicaes ha heory is no well developed. Research sared wih collecions of qualiy aspecs (Gupill and Morrison, 1995, Aalders, 1996), which influenced direcly sandardisaion. A es wih he pending CEN sandard concluded wih criique on pracical applicabiliy, especially from he users poin of view (Timpf e al., 1996). A heoreical framework for spaial daa qualiy is lacking up o now. In absence of such a framework invesigaions of spaial daa qualiy will lead o conex-specific mehods. In principle, one has o reference (a) o ground ruh, (b) o a reference daa se, (c) o a large number of oher daa ses, or (d) o qualiy descripions exraced from knowledge abou daa capure and experience (Baarda, 1967). In his paper we presen a sysemaic analysis of differences beween regions from wo daa ses, applying he approach (b). Based on his analysis we propose mehods o deec and idenify he differences. Wihou prior knowledge abou he compared daa ses, one is able o describe he differences beween he daa ses. If one daa se can be used as reference, he saemens can be urned ino qualiy descripions. 3 INVESTIGATION OF THE PROBLEM In his secion we specify our ask, discuss represenaional issues regarding comparisons, and caegorize he differences ha exis beween maching regions from differen daa ses. 3.1 The Task We assume ha wo ses R j, j = 1 2, of regions r are given: R j = fr i g j. Each region r ij is described geomerically. The regions wihin one se may be relaed, e. g. by specifying a neighbourhood relaion. These relaionships are represenable in a region adjacency graph (RAG) G j R j N j, consising of he regions of R j as nodes, and of he neighbourhood relaions N j as edges. For simpliciy, we assume he geomery of he regions are described by polygons. This resricion does no resul in concepual limiaions wihin our conex. The ask is o characerize he srucural and geomerical differences beween such wo ses, and o develop ools for esing he equivalence of he wo ses, or for deecing and idenifying he differences, respecively. 3.2 Represenaion of Areal Objecs Spaial represenaions in GIS are classified as vecor and raser, which are dual in he sense of space bounding and space filling (Peuque, 198). Boh conceps describe locaion in a reference frame, e.g. in IR 2 or Z 2. In a vecor model, regions are represened by heir boundary curves, which usually are regular closed polygons of nodes and edges. In conras, in a raser model regions are represened by he se of raser elemens which belong a leas o a dominan par of each elemen o he objec. The shape of regions is independen from heir represenaion, bu he shape as well as he represenaion deermine he kind and number of disinguishable opological relaionships (Egenhofer e al., 199, Winer, 1995). Shape may be arbirarily complex: simply conneced, muliple conneced, i.e. wih holes, or no conneced,

3 D. Frisch, M. Englich & M. Seser, eds, 'IAPRS', Vol. 32/, ISPRS Commission IV Symposium on GIS - Beween Visions and Applicaions, Sugar, Germany. Lemonia Ragia and Sephan Winer 3 i.e. aggregaions of disjunc pars. Addiionally, in vecor represenaion he inner geomery may be decomposed ino pars. Tha may be caused by pure geomeric reasons, like a decomposiion ino cell complexes, e.g. as in finie elemen mehods, or by semanic reasons, when he objec is aggregaed from disinguishable pars. The laer ofen happens for man-made objecs, like buildings (Gülch, 1997). The pariion beween wo daa ses may differ significanly. I urns ou ha comparison of opology and of geomery requires a scale (Sevens, 196) and a common spaial represenaion. The scale defines conex and a measure, which is inroduced boh in he nex secion. Deerminaion of opological differences is performed easily using a vecor represenaion. For he analysis, in Secion.1 a mapping is proposed from geomeric polygons o he dual, a region adjacency graph, which keeps only opological informaion. In conras, measures for differences in geomery will be based on vecor properies as well as on a discree space (Secion.3). For ha reason, geomery is invesigaed using a hybrid raser ha preserves opological properies of IR 2. The hybrid raser represenaion consiss of a decomposiion of he plane IR 2 ino uniform cell complexes, where he grid of Z 2 is chosen as he se of -cells (or nodes), which are conneced by axis-parallel 1-cells (or edges). Boh ogeher form an 1-skeleon, which encloses he 2-cells (or faces) (Kovalevsky, 1989, Winer, 1998a). I is presumed ha he daa ses are opologically consisen. Such a propery of a daa se can be checked auomaically in spaial daabases (Joos, 1996). Neverheless, one has o define in deail wha is opological consisence. Consider, e.g., a sysem for building exracion, based on volumeric primiives. An inersecion of he primiives is possible and allowed (consrucive solid geomery works wih unificaion of elemens). In he measuring process he operaor adaps he form and pose parameers of he primiives. S Complex buildings b are composed by some primiives p: b = p i i. Even when buildings are ouching inersecions occur, guaraneeing o avoid small gaps beween he buildings. Especially qualiaive mehods of comparison have o be robus o rea effecs from imprecise geomery. Tha is reached by olerances, and by weighed opological relaionships (Secion.3). 3.3 Classificaion of Differences of Ses of Regions Differences beween ses of regions can be classified ino opological differences, which are qualiaive, and geomerical differences, which are quaniaive. In Fig. 2 hey are specified ino inner srucure, boundary srucure, and locaion. Topological differences Inerior Srucure: Pariioning of he RAG Boundary Srucure: Aribues of he RAG Geomeric differences Locaion Figure 2: Classificaion of differences beween wo mached regions (RAG: region adjacency graph). Differences in he Inner Srucure Differences in he region adjacency graphs (Secion.1) of wo ses of regions require a mapping beween he wo graphs. This correspondence may be esablished by invesigaing he spaial relaionships beween all regions r i1 of he firs se wih all regions r i2 of he second se. We assume his mapping be esablished, e.g. by reaing regions r i1 and r i2 as corresponding if hey overlap srongly, coincide or conain each oher (cf. T 2 in Eq. 2). In he mos simple case we may idenify four ypes of differences (cf. (Fuchs e al., 199)): A region is missing. A region is spurious. Two or more regions are merged ino one. A region is spli ino wo or more regions. In general, however, his mapping may be n : m, which resuls in a complex correspondence relaion beween he wo ses of regions. Differences in he Boundary Srucure Assuming he boundaries of regions o be represened as polygons, or more general as possibly labeled sequence of boundary elemens, we can idenify differences beween wo maching srucures wihou analysing geomery. Only in case of srucural differences, we need o check he equivalence of he geomery of he boundaries. E.g., if a region has poins and he corresponding region has 5 poins, he geomery may be idenical, in case poins are idenical and he oher lies on one of he sraigh edges. In case he wo srucures are idenical, a direc comparison of he aribues of he boundary elemens may be sufficien, e.g. by jus comparing he coordinaes of corresponding polygon nodes. This ype of analysis can be applied also o he boundary of wo corresponding ses of regions fr k g 1 and fr l g 2, assuming ha he maching process has idenified hese wo ses as corresponding. For example, if r a1 and r b1 from R 1 are corresponding o r a2 from R 2, one can deermine he difference in he srucure of he boundaries of r a1 [ r b1 and r a2. Differences in Locaion The deerminaion of differences in he locaion of wo corresponding regions or wo corresponding ses of regions requires a measure of similariy. This may be, e.g., he mean or he maximal disance (Hausdorff disance (Serra, 1982)) beween corresponding poins of he boundary. In order o be able o esablish such a measure, he expeced uncerainy of he region boundaries needs o be smaller han he regions hemselves. As he geomeric differences beween wo boundaries may be arbirarily complex here is no canonical way o measure his difference. However, in case of quie similar and smooh boundaries, his disance may be easily esablished (Secion.3). Also here ses of regions may be compared. For example, a complex building may be pariioned ino r a1 and r b1 in R 1, and in r a2 and r b2 in R 2. Maching should have idenified r a1 corresponding o r a2, and r b1 o r b2, however r a1 6 = r a2 and r b1 6 = r b2. Then he common boundary of r a1 [ r b1 is o be compared o r a2 [ r b2. The following differences are of special ineres: wo mached regions differ in single boundary poins; wo mached region differ in pose (shif, or roaion); wo associaed regions differ in form parameers. USED TECHNIQUES In his secion he echniques are presened which are used laer o characerize he discussed opological and geomeric differences in wo ses of regions. Especially we inroduce he region adjacency graph, a refinemen of opological relaionships, and a local disance funcion wih is hisogram.

4 D. Frisch, M. Englich & M. Seser, eds, 'IAPRS', Vol. 32/, ISPRS Commission IV Symposium on GIS - Beween Visions and Applicaions, Sugar, Germany. Lemonia Ragia and Sephan Winer.1 The Region Adjacency Graph The region adjacency graph (RAG, Fig. 3) is a concep used in image processing. I assumes a complee pariion of he plane. I is defined as he dual o he cell graph of conneced regions. In our conex he RAG describes an arbiray se of regions. The nodes of he RAG represen he componens of he regions; heir aribues are he number of nodes of he bounding polygon of he represened componen. Two nodes of he RAG are conneced by an edge if he componens are in any opological relaion differen from disjunc. The aribues of he edges specify his opological relaion; hree ypes of relaions are considered ouch, weak overlap, and srong overlap (cf. Fig. 3). This se of relaionships is no purely opological; here are meric influences, and groups of opological relaionships are combined (he meaning of he noions differs from he nex secion): wo regions ouch if hey have common pars in heir boundaries, bu do no overlap; hey ouch weakly if a remaining gap is lower han a olerance; wo regions overlap weakly if heir inersecion se is smaller han 5% of he smaller region (Eq. 1); and (in he conex of he RAG:) srong overlap sums up all opological relaions of he cluser T 2 (Eq. 2). Consider Fig. 3 for an example. This figure describes he RAG of wo ses of regions. The graph represens he righ-hand par of he scene in Fig. 1, which corresponds o he image in Fig.. There are one complex and hree single buildings in he firs se of regions, and one complex and four single buildings in he second se of regions. In he firs RAG, he number of boundary poins of he componens of he buildings is always four, and he relaions beween he componens are weak overlap and ouch. In he second RAG, he number of he boundary poins varies, and he opological relaion is only ouch. The complex building consiss of eigh segmens in he firs se and of five componens in he second se. There exis hree single buildings in he firs se as in he second one, bu one of he firs daa se canno be mached. w w w w Figure 3: The region adjacency graph; : ouch, w: weak overlap, s: srong overlap. 5.2 Topological Relaions beween Regions In his secion a shor reference o opological relaionships is given, and a refinemen is presened which groups he relaionships ino wo clusers. The clusers are he basis for a disance funcion laer. In principle, i depends on represenaion of he space which (families of) opological relaionships can be disinguished. In vecor models a poin se heoreic approach allows o define he following ses of a region X wih he usual opology of he IR 2 : he inerior of X, he boundary of and he exerior of X, :X. Refering o hese ses, opological relaionships beween wo regions can be defined by heir nine inersecion ses (Egenhofer and Franzosa, 1991). In raser models only X and :X are disinguishable; bu mapping a raser o he cied hybrid raser preserves full opology of IR 2 (Winer, 1995). A second facor influencing he number of relaionships is he complexiy of he regions (Egenhofer e al., 199). Relaionships can be ordered in a concepual neighborhood graph. For deails we refer o he cied lieraure. Wih regard o he vecor and he hybrid raser represenaion, he disinguished relaionships are: disjunc, ouch, overlap, coverage, conainmen, and equal. In (Winer, 1996) a refinemen was inroduced which is usefull also here: he relaion overlap can be splied ino a srong overlap and a weak overlap, by a hreshold of 5 % in he weigh of overlap: ka \ Bk = mina B This refinemen was used o pariion he concepual neighborhood graph ino wo relaion clusers T 1 and T 2, which are conneced only by he crossing from weak o srong overlap: T 1 = fdisjunc ouch weak overlapg T 2 = fsrong overlap coverage conainmen equalg (2) Wih he assumpion of small errors and uncerainies agains he size of he regions, he assignmen of a relaion beween wo regions ino a cluser is safe, having in mind ha weak and srong overlap refer o he same opological relaionship overlap..3 A Deailed Disance Funcion Beween Regions In his secion a disance funcion is described in deail, which characerizes meric differences beween wo similarly locaed regions 1 or unions of regions. The cenral idea is o observe disances of he wo boundaries locally. A reference axis is required locaing such disances uniquely. In he following we inroduce a zonal skeleon for ha reference axis, give a definiion of a local disance, derive he disance funcion and is convoluion, he disance hisogram. Consider Figure 5. On he lef, wo regions overlap considerably (relaion 2 T 2 ). In such cases le us concenrae on hree inersecion ses: (1) R = A \:B S = B \:A T (3) Through hese (always conneced) ses a medial axis is calculaed (Serra, 1982), which we call a zonal skeleon. For an algorihm i is usefull o inroduce here he complemenary ses P and Q: P = A \ B Q = :A \:B () Figure : The corresponding image ( cdetemobil 1998). The zonal skeleon is he collecion of cener poins of all circles ha ouch P as well as Q wihou inersecing P or Q. Then he local disance a any poin of he skeleon can be defined by he (signed) diameer of is defining circle. The local disance is defined o be posiive in R and negaive in S (andint ); i is no a meric. I is realized in he hybrid raser represenaion by a disance ransform on P [ Q, on he nodes of he hybrid raser. The 1 A non-meric difference measure is presened in (Winer, 1998b).

5 D. Frisch, M. Englich & M. Seser, eds, 'IAPRS', Vol. 32/, ISPRS Commission IV Symposium on GIS - Beween Visions and Applicaions, Sugar, Germany. Lemonia Ragia and Sephan Winer 5 disance funcion beween A and B is he sequence of local disances along he skeleon (Fig. 5, righ). If only he shape of he graph of he funcion ineress, is origin (x =)isarbirary. Oherwise one could define a sar poin a he skeleon poin neares o he origin of he reference sysem, or, in a raser, a he firs occurence in search order. A B Figure 5: Two srong overlapping regions r p1 = A and r q2 = B (of square ouline each) and he zonal skeleon (lef), and he graph of local disances along he skeleon (righ). The described mehod works correc only for regions wih a relaionship in T2. In oher cases, i.e. for relaionships in T1, he zonal skeleon has o be defined hrough oher inersecion ses, bu looks very similar; for deails see (Winer, 1996). For complex regions, i.e. regions wih holes, a hierarchic approach could be applied. Several characerisics of differences beween regions refer o he hisogram of he disance funcion. The hisogram is he densiy funcion of he disances. Le us assume a discree disance funcion, as derived in he hybrid raser represenaion; for a coninuous disance funcion i is necessary o resample he funcion in discree seps. Then he hisogram characerizes he frequency of local disances d along he skeleon: d h(d) =kfx j d(x) =dgk (5) The hisogram is basis for oher characerisics, like he disribuion funcion, he mean, and momens of higher order. s ha exceeds parly he olerance +b; ha indicaes ha he associaed regions differ in pars significanly, namely in R (Eq. 3): A ouranges B in his area. In conras, f2 represens a disance funcion ha is somewhere in beween significan decisions. A las, f3 represens a disance funcion ha differs from only by noise; he wo regions can be classified as equal in geomery. The same inerpreaion can be aken from he hisogram of local disances (Fig. 7), because up o now only he exremal values of he disance funcions are considered. Furhermore, addiional informaion can be aken from he form of he disance curve and he hisogram curve. In Fig. 6, f1 exceeds one ime he olerance b: A ouranges B in one par (his informaion is los in he hisogram). If he hisogram shows wo peaks symmeric o he y-axis and he disance funcion is smooh (like f2), a shif exiss beween he wo regions (Fig. 8). If he hisogram shows wo peaks, bu one is cenered, hen suppor for he hypohesis exiss ha one building has addiional pars (Fig. 9, lef). If he hisogram shows one cenered peak bu one-sided a cerain amoun of oher disances, hen probably one boundary node differs, and we classify o an oulier (Fig. 9, righ). Oher classificaions can be made similarly. b a -a -b Disance along he skeleon f 2 f 3 f 1 skeleon curve Figure 6: Three differen classified disance funcions. f1: a pair of regions which mos probably refer o differen real world saes (or are erroneous); f2: a pair of regions wih differences ha are no auomaically assessable; f3: a pair of regions which is nearly equal. 5 CHARACTERIZING DIFFERENCES BETWEEN TWO REGIONS In his secion differences beween wo regions are invesigaed in wo direcions: opological differences are described qualiaively, and geomerical differences are described quaniaively. f 3 Frequency of disances 5.1 Checking Topology f 2 f 1 Basis for a comparison of he srucure of mached, possibly complex regions are heir region adjacency graphs (Secion.1). Then a check of wo region opologies means a comparison of wo region adjacency graphs. Resuls are differences in he number of graph nodes and edges he inner srucure, or he level of pariioning, and differences in node and edge aribues he boundary srucure, or level of deail (Fig. 2). Problems in his procedure occur if some disjunc regions in one RAG mach o he same region in he reference RAG. However, in his case he disjunc regions are o be conneced, and he edges are o be labeled by ouch. 5.2 Checking Geomery Checking geomery is done by local disances in he hybrid raser represenaion (Secion.3). See Figure 6: here are wo olerance levels inroduced, a and b. Local disances inside of [,a +a] are considered as no significan, due o rounding, imprecision or noise, and local disances ouside of [,b +b] are considered as indicaing significanly differen (pars of) regions. Inermediae values are reaed as undecided. f1 represens a disance funcion -b -a a b Disances Figure 7: The hisograms of he hree differen classified disance funcions in Fig. 6. The deailed geomeric differences beween associaed regions can be generalized for he whole daa se. 6 EMPIRICAL TEST In his secion he developed qualiy characerisics are applied o a comparison of wo real daa ses, where one is a reference for he oher. Compleeness of he second daa se, differences in opological srucure and in geomeric locaion are invesigaed

6 D. Frisch, M. Englich & M. Seser, eds, 'IAPRS', Vol. 32/, ISPRS Commission IV Symposium on GIS - Beween Visions and Applicaions, Sugar, Germany. 6 Lemonia Ragia and Sephan Winer Frequency of disances d max Disances Figure 8: The hisogram of a region shifed o he reference region. differences beween a sereo-ploing daa acquisiion and a semiauomaic, model-based sysem. Typically, sereo-ploing yields less pars bu more boundary poins per par han he consrucion by primiives. Sereo-ploing suppors opological daa models; for ha reason he only relaionship beween pars found in he RAG edge aribues is ouch. Consrucion by primiives requires unificaion, and herefore also oher opological relaionships beween pars are observed in he region ses from he model-based sysem. Someimes he pariioning level differs also by absracion; Figure 11 gives an example of a simple building from sereoploing (below) ha is composed by five primiive volumes in he second daa se (above). The building is shown in Fig. 1. Frequency of disances Frequency of disances Disances Disances d max d max Figure 9: The hisogram of a region covering he reference, i.e. wih addiional pars (lef), and of a region wih one boundary poin differen o (ouside of) he reference (righ). Figure 1: The building of Fig. 11 ( cdetemobil 1998). and presened. I urns ou ha he qualiy descripions are usefull o assess he second daa se, and, by he way, is acquisiion mehod. 6.1 Se-Up of he Tes Basis for he es are wo daa ses conaining abou 6 buildings of he same area. The firs daa se, R 1, was produced from an image pair on an analyical ploer. We consider his daa se as he reference because of he experience of he professional operaor in image inerpreaion. The second daa se, R 2 is based on he same image pair, bu was regisered on a semi-auomaic, model-based sysem for building exracion which has been developed a he Insiue of Phoogrammery in Bonn (Gülch, 1997). Primiive volumeric models suppor image analysis, bu could fi less o realiy. For ha reason we will describe he qualiy of his daa se. I is supposed ha here is no sysemaic error beween he daa ses (roaion, ranslaion, scale), and boh daa ses are of a similar granulariy. The daa ses conain simple, large and complex buildings from urban and indusrial areas on he images. In deail, hree areas are chosen and measured compleely. The variaion should minimize sysemaic influences of perspecive, of building ypes or building complexiy ono he qualiy descripions. The image pair is in a scale of 1:15.. For he semi-auomaic building exracion he images were scanned wih 7m pixel size; ha corresponds o a ground resoluion of 1 dm/pixel. Wih hese numbers, le us assume a sandard deviaion of =1m, i.e. 1 pixels. The olerances a and b (Sec. 5.2) can be chosen wih regard o he sandard deviaion. The olerance a, represening a hreshold describing wha is considered sufficienly near o be reaed as equal, is se o, and he olerance b, represening a hreshold describing wha is considered significanly differen, is se o Resuls of he Tes Topology. The srucural differences are aken from he region adjacency graphs. We have found ou ha here are sysemaic Figure 11: Two RAGs of a building: sereo-ploing (below) ypically leads o less pars which are more deailed han model-based consrucion (above). Geomery. For he geomery we have compared all he simple buildings and some complex buildings neglecing he pariioning. The differences are classified using he exremal values in he disance hisograms and he olerances a and b. The resuls for he daa se are: Correc facor of % [3%-58%]: his facor is he percenage of mached regions wih locaional deviaions inside of [,a a]. The regions are considered as sufficienly idenical, and he deviaions are only due o numerical rounding effecs, resoluion higher han of pracical ineres, admied inaccuracy, and so on. Semi-error facor of 37% [23%-52%]: his facor is he percenage of pairs of regions wih locaional deviaions inside of [,b b]. For an auomaic sysem i is no possible o decide wheher he compared regions represen he same real world objec or no. Error facor of 19% [8%-32%]: his facor is he percenage of mached regions wih locaional deviaions of a magniude ouside of [,b b]. I is mos likely ha hey do no refer o he same real world objec, may be due o errors in daa capure, may be due o changes in real world beween wo daa capures (no in his es), or for any oher reason. The resuls in brackes indicae he 99% confidence region of he esimaed percenages.

7 D. Frisch, M. Englich & M. Seser, eds, 'IAPRS', Vol. 32/, ISPRS Commission IV Symposium on GIS - Beween Visions and Applicaions, Sugar, Germany. Lemonia Ragia and Sephan Winer 7 5 2_85dis.gnu 1 2_85his.gnu 5_2193dis.gnu 25 5_2193his.gnu disance funcion 1 5 h(d) 2 15 disance funcion 2 1 h(d) zone-skeleon Disance d zone-skeleon Disance d Figure 13: A significan difference in absracion. Figure 12: Oulines (firs), skeleon (second), local disances (hird), hisogram (las) of a correc building (jdj < 1 pixels). Buildings which fulfill he condiion 8d : jdj < ahave no (significan) geomeric differences (Fig. 12). Oher buildings can be classified by he ype of he error, analyzing he form of he disance funcion and of he hisogram. If here is a secondary peak in he hisogram hen we have a difference in he locaion of a leas a boundary poin. Where he secondary peak is conneced wih he main peak by a consan densiy of significan lengh, he buildings differ in pars (Fig. 13). The sign of he local disance indicaes which building is conained by he oher one. Two or more secondary peaks in he hisogram indicae several differences in pars of he buildings. I depends on he disance value of he peak wheher he difference is significan (Fig. 1). An oscillaing disance funcion indicaes a comparison of a high-resoluion building wih a generalized one, which also may be caused by a model ha does no fi o complex realiy (Fig. 15). 7 SUMMARY AND CONCLUSIONS We presened a sysemaic classificaion of differences beween independenly acquired regions, and we proposed mehods for idenificaion and quanificaion of hose differences. Following he paradigm in spaial daa base queries we invesigae opology before analysing geomery in order o speed up, and also o clarify he analysis. Topological differences firsly are differences in he adjacency graph of he given regions. The geomeric srucure of he graph, e. g., he number of polygon poins of he region boundary, is analysed nex. This analysis, as well as he geomerical analysis of he boundaries, may refer o single regions as well as o corresponding ses of regions. The subsequen geomerical analysis uses he zone skeleon and is hisogram. We esed he procedure on 6 buildings acquired wih wo differen mehods and found i o be suiable for specifying he correcness or he differences of he wo daa ses. Saemens abou he compleeness of he acquired daa ses can be made addiionally, afer he opological and geomerical differences are idenified. The geomeric analysis using he zone skeleon has been auomaed, whereas he maching of he wo region adjacency graphs sills wais for implemenaion. The final goal is o characerize he differences of wo daa ses as compleely as possible, in order o idenify ypical failures in he acquisiion procedures, and o reliably evaluae he differences wih respec o given specificaions. This may lead o clearer specificaions of daa acquisiion procedures and increase he fideliy of auomaic evaluaion procedures. Acknowledgemen We graefully acknowledge he righ of use of daa ses from DeTe- Mobil, Bonn. REFERENCES Aalders, H. J. G. L., Qualiy merics for GIS. In: Advances in GIS Research, Taylor & Francis, Delf, The Neherlands, pp. 5B.1 5B.1. Baarda, W., Saisical Conceps in Geodesy. Vol. 2, Neherlands Geodeic Commission, Delf. Bruns, H. T. and Egenhofer, M. J., Similariy of spaial scenes. In: M.-J. Kraak and M. Molenaar (eds), Advances in GIS Research, Taylor & Francis, Delf, pp Burrough, P. A. and Frank, A. U. (eds), Geographic Objecs wih Indeerminae Boundaries. ESF-GISDATA, Vol. 2, Taylor & Francis. Egenhofer, M. J. and Franzosa, R. D., Poin-se opological spaial relaions. Inernaional Journal of Geographical Informaion Sysems 5(2), pp

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