SEMANTICALLY CORRECT INTEGRATION OF A DIGITAL TERRAIN MODEL AND A 2D TOPOGRAPHIC VECTOR DATA SET

Transcription

1 SEMANICALLY CORREC INEGRAION OF A DIGIAL ERRAIN MODEL AND A 2D OPOGRAPHIC VECOR DAA SE A. Koch Ititute of Photogammety ad GeoIfomatio (IPI), Uiveity of Haove Niebuge Stae 1, Haove koch@ipi.ui-haove.de KEY WORDS: GIS, Adutmet, Itegatio, Viuaizatio, Modeig, DM ABSRAC: he mot commoy ued topogaphic vecto data, the coe data of a geogaphic ifomatio ytem (GIS), ae cuety twodimeioa. he topogaphic obect ae modeed a poit, ie o aea with additioa attibute cotaiig diffeet ifomatio about fuctio ad dimeio of the obect, poiby icudig height vaue. Height vaue may ao be added to the obect by uig a digita teai mode (DM), i.e. by itegatig vecto data ad DM. I geea, howeve, dicepacie exit betwee the two diffeet data et. hee dicepacie ae caued by diffeet obect modeig ad diffeet uveyig ad poductio method. I the vecto data et, fo exampe, a oad i ofte modeed a a ie o a poyie. he attibute cotai ifomatio about the oad width, the oad type, etc. If the oad i ocated o a dam, the coepodig pat of the DM ca be modeed a a eogated, hoizota pae of ome width with two ope at the ide. Diffeece betwee the attibute oad width of the vecto data et, ad the width of the hoizota pae i the DM may caue dicepacie whe itegatig thee data et. Additioay, the data ae ofte poduced idepedety. he DM may be geeated uig ae caig o aeia photogammety. he topogaphic vecto data may be baed o digitized topogaphic map o othophoto eutig i diffeece i 2D ocatio. A aeady metioed, the topogaphic obect of the vecto data ae two-dimeioa. But thee exit vaiou obect which cotai impicit height ifomatio. A ake, fo exampe, epeet a hoizota pae, ad a height vaue of the ake aea mut be the ame, the eighbouig bak, howeve, have to be highe tha the wate. o give aothe exampe, the ope of a oad aog ad aco the oad diectio mut ot exceed a cetai maximum vaue. hi pape decibe a agoithm which ue thi ematic ifomatio fo a ematicay coect itegatio of DM ad 2D topogaphic vecto data. It i baed o a iequaity cotaied eat quae adutmet fomuated a the iea compemetay pobem (LCP). he agoithm eut i impoved height vaue of the DM. Fit ivetigatio wee caied out by uig two diffeet data et epeetig two cae of obect. he fit imuated data et epeet a tited pae, the ecod data et i a ea data et of the Gema AKIS. Fit eut uig thee data et ae atifyig. 1. INRODUCION he mot commoy ued topogaphic vecto data, the coe data of a geogaphic ifomatio ytem (GIS), ae cuety twodimeioa. he topogaphic obect ae modeed a poit, ie o aea with additioa attibute which may cotai diffeet ifomatio o fuctio ad dimeio of the obect. I cotat a digita teai mode (DM) i mot cae i a 2.5D epeetatio of the eath uface. Each poitio i paimety cotai oe height vaue. he poit ae ditibuted ieguay ad/o i a gid, oy poit epeetig tuctue eemet cotai ematic meaig. hey epeet, fo exampe, beak ie o ige igificat height vaue. Itegatio of thee two data et may be udetood i tem of a itegated data modeig fo coideatio of height ifomatio i a GIS. he 2D topogaphic obect ae competed by height ifomatio of the DM, ad afte the itegatio both data et epeet a itegated 2.5D topogaphic data et. hee ae diffeet eao fo itegatio of the two data et. Fit, data acquiitio i moty the mai cot facto fo geeatio of geogaphic ifomatio ytem (Bi & Fitch, 1994). Itegatio of DM o othe pecific data et with epect to the geogaphica data bae may educe cot becaue thee i o eed fo ew data acquiitio i 2.5 o 3D. Secody, itegatio wi eut i coitet data et. Icoitecie ae caued by diffeet obect modeig ad diffeet uveyig ad poductio method. Vecto data et ofte cotai oad modeed a ie o poyie. he attibute cotai ifomatio o oad width, oad type etc. If the oad i ocated o a dam, the coepodig pat of the DM ca be modeed a a eogated, hoizota pae of ome width with two ope at it ide. Diffeece betwee the attibute oad width of the vecto data et, ad the width of the hoizota pae i the DM may caue dicepacie whe itegatig thee data et. Additioay, data ae ofte poduced idepedety. he DM may be geeated by uig ae caig o aeia photogammety. opogaphic vecto data may be baed o digitized topogaphic map o othophoto. A thid eao fo the itegatio i veificatio of the DM. I may cae topogaphic vecto data ae amot up to date becaue obect a teet ad aiway poe mao pioity i GIS. A DM, howeve, may be ode tha oe decade. It i tue that height chage appea e fequety tha chage i the hoizota poitio of obect. Nevethee, itegatio of both data et coideig the ematic meaig of obect wi how dicepacie, ad wi aow to daw cocuio o the quaity of the DM. hi pape peet a agoithm fo a ematicay coect itegatio of a DM ad a 2D topogaphic vecto data et. he foowig ectio decibe the meaig of ematic coecte uig ome exampe. Sectio 3 epeet the agoithm ad ectio 4 how fit eut uig imuated ad ea data et. he pape cocude with a hot ummay ad ome futue wok.

2 2. SEMANIC CORRECNESS hee ae evea appoache fo itegatio of a DM ad a topogaphic vecto data et. Some method ae baed o the ue of height vaue a attibute, i.e. the -coodiate i iked to the obect. I cae of aea baed obect poyomia face may be added. Othe appoache ue tiage o gid fo the combiatio of topogaphic obect ad DM ad fo competio of the tuctue of a tiaguated o gidded DM, wheea the atte method ocay deifie the gidded tuctue of the DM by ao uig tiaguatio. A oveview about exitig method i give by (Lek, 2001). A thee method do ot coide ay ematic meaig of the topogaphic obect. Fo exampe, if a teet coe a idge, the itegatio of a teet ito a exitig tiaguated DM may caue a icoect teai mophoogy, if o additioa poit i added o the idge, which wi eut i a teai iectio. O, if a ake o a ubet of a ake i ituated o a ope, the wate of a imuated tadig wate body fow. Figue 1 how a DM ad a topogaphic data et with thee ake. he height at the bode of the ake ae ot the ame a iide the ake, thee i o cotat height eve, the ematic ae icoect. Repeetatio Hoizota pae ited pae with maximum ope Height eatio Obect Spot fied Race tack Ruway Dock Caa Lake, poo Steet, Road Path Raiway, tamway Rive Bidge Udecoig, coove abe 1. opogaphic obect ad thei epeetatio i the teai fom the AKIS Bai-DLM featue cataogue he at ca how i tabe 1 decibe obect which have a height eatio to othe obect. Bidge, udecoig ad coove may cotai a cetai eatio to thei eighbouig teai. hee eatio maiy ae compoet of the 2D vecto data et. 3. AN ALGORIHM FOR HE SEMANICALLY CORREC INEGRAION Figue 1: A viuaizatio of a DM (AKIS DGM5) ad a topogaphic vecto data et with thee ake (AKIS Bai-DLM) he topogaphic obect ae two-dimeoa. Nevethee, obect exit which cotai impicit eative height ifomatio o the teai. abe 1 how ome of thee obect ad thei epeetatio i the teai. he tabe cotai thee obect cae. he fit ca cotai obect which epeet hoizota pae. Fo exampe, pot fied (footba o occe fied) ae maiy hoizota. Futhemoe, uway o ake epeet amot hoizota pae, i additio, ake have bak at thei ide, ad the teai i acedig. he ecod ca decibe obect epeetig tited pae. Steet o oad epeet tited pae with ome ope i divig diectio ad aco. Pepedicua to the oad diectio the iciatio of the teet i eceay becaue of wate daiage. Additioay, i cuve the ope wi educe the foce which affect the vehice. I topogaphic vecto data et teet ad aiway ae moty modeed by ie. heefoe, thee obect have to be buffeed uig additioa ifomatio o the width of the obect whe itegatig them with a DM. A DM i compoed of eguay o ieguay ditibuted poit with it coodiate X,Y, ad a itepoatio fuctio to deive vaue at abitay poitio X,Y. Additioay tuctue eemet ae icuded which cotai ifomatio o the mophoogy of the teai (beak ie, igificat poit, etc.). he topogaphic vecto data cuety ae two-dimeioa. he topogaphic obect ae modeed by poit, ie o aea. I thi pape the paimetic coodiate of the topogaphic obect wi be itoduced a eo-fee. Ay kid of ytematic o adom eo wi be egected, i.e. eo of the itepoated height vaue caued by iaccuate paimetic coodiate of the vecto data ae ot coideed up to ow. he aim of itegatig both data et i a itegated data modeig fo coideig height ifomatio i GIS. he data et mut be itegated i uch a way that the topogaphic obect have to fufi cetai cotait which aie fom the ematic of the obect. he fit tep of the agoithm i a cotaied Deauay tiaguatio of a poit of the DM (ectio 3.1.). he, diffeet obect cotaiig impicit height ifomatio ae itoduced, ad thei height ae itepoated uig the height vaue of the tiaguated DM. Equaity ad iequaity cotait ae the coideed i a optimizatio poce, eutig i impoved height vaue ad i itegated data et which fufi the pedefied cotait. A pecoditio of the agoithm i that the teai mophoogy i the eighbouhood of the obect ha to be coideed but the impovemet of the height have to be ma. he obect ae fiay itoduced a cotait i a ew Deauay tiaguatio, poducig two coitet data et Cotaied Deauay tiaguatio he hybid DM wi be tiaguated uig a cotaied Deauay tiaguatio. he tuctue eemet ae itoduced

3 a cotait, ad the eut i a tiaguated iegua etwok (IN). hee ae may Deauay tiaguatio agoithm. I ou appoach the divide ad coque agoithm peeted by Guiba ad Stofi, 1985, wa ued becaue it pefom vey we (Shewchuk, 1997). he tuctue eemet wee itoduced i uch a way that the poit of the tuctue eemet iteectig the edge of the tiage wee added a ew poit of the tiaguatio Equaity ad iequaity cotait poit 214 iteect the bouday poygo, ee the fat dak bue ie). he coepodig iequatio i: k > whee k = 1,, ob ob Numbe of poit of the topogaphic obect Height vaue of the hoizota pae (3) Fo the topogaphic obect which cotai impicit height ifomatio (ee tabe 1), equaity ad iequaity cotait ae fomuated. Each ca cotaied i tabe 1 ha it ow cotait, which wi be deived i the foowig. Hoizota pae Height vaue of obect epeetig a hoizota pae mut be idetica, i.e. i cae of aea baed obect the poit with paimetic coodiate iide the obect poygo mut have the ame -coodiate. I idea cae the foowig equatio mut be fufied: i = (1) whee i = 1,, i i Numbe of poit with paimetic coodiate yig iide the poygo epeetig the obect Height vaue of the hoizota pae he height vaue of the obect bouday poygo poit mut i geea be itepoated fom the eighbouig DM height. Agai, thee height mut have the ame height eve : Figue 2: DM poit ad bouday poygo of a topogaphic obect ad the eighbouig teai he gee DM poit ae a poit outide the obect poygo which ae coected to oage DM poit of the bak. I figue 2, the DM poit 17 i coected to thee oage poit (214, 215, 216). he coectio wi be added by uig the height diffeece betwee the poit: ( X Y, u, v, w ), = (2) k = (4) k X,Y ae the paimetic coodiate of a obect bouday poit, u, v, w ae the height vaue of the tiage of the IN which cotai the paimetic coodiate of the obect poit. he height vaue ae computed uig the equatio of a tited pae. he eighbouig teai of the obect outide the poygo ao ha to be coideed i the agoithm. he fit eao i that ome of the obect have bak o ope at thei ide which have a cetai vaue with epect to the hoizota pae. Fo exampe, the height vaue of a bak of a ake hoe have to iceae a othewie the wate woud fow out. Secody, the obect i eated to the eighbouig teai: he futhe a DM poit ie away fom a obect, the mae houd be the ifuece of the obect o the height chage of thee poit. Figue 2 how ome poit of a ake ad it eighbouig teai. he diffeet coou of the poit idicate the equaity o iequaity cotait which mut be fufied afte appyig the agoithm. he gay ad the dak bue poit mut have the ame height vaue a they epeet the wate body. he - coodiate of the oage poit have to be highe tha the height eve of the ake. he oage poit ae a poit outide the obect poygo which beog to tiage of the IN whoe edge iteect the bouday poygo of the ake (fo exampe k it the height vaue of a DM poit of the bak, i the coepodig height vaue of the gee DM poit. hee height diffeece ead to a teai mophoogy which i eay the ame a befoe. ited pae he obect epeetig a tited pae ae maiy eogated obect (ee tabe 1). Aog the mai diectio thee obect ae ot aowed to exceed a pedefied ope vaue : m D m Additioay the ope diffeece d betwee eighbouig obect ectio, compaabe to the cuvatue of the obect i ogitudia diectio, i eticted: m D m D o o d (5) (6)

4 D m ad D o ae the hoizota ditace betwee the obect bouday poit P m, P ad P, P o, epectivey. I cae of obect which ae modeed by ie, thee poit beog to the obect poyie. he coepodig height vaue i pepedicua diectio mut have the ame height eve, becaue the ope of teet o aiway wi be egected (equatio 7). Obect modeed by ie have to be buffeed uig attibute epeetig the width of the obect. he, p epeet the height vaue of the buffeed eft o ight ide ad m i the height of the cete axi. p = m Fiay, the height vaue of poit with thei paimetic coodiate iide the obect have to be ituated o the tited pae. he ditace betwee the poit P q ad the pae mut be zeo: q d = 0 d i the ditace of the pae to the oigi of the coodiate ytem, i the oma vecto of the tited pae ad q cotai the coodiate of the poit P q. Agai the coectio to the DM poit outide the obect poygo wi be itoduced uig the height diffeece betwee thee poit ad poit epeetig the obect (ee equatio 4). Height eatio Bidge, udecoig ad coove have a cetai height eatio to othe obect (fo exampe teet, aiway, ive, etc.). he height vaue which beog to thee obect mut be highe o owe tha the eated obect: < > 3.3. Iequaity cotaied eat quae adutmet he defied cotait (equatio 1 to 9) have to be itoduced i a optimizatio poce. he agoithm ued i baed o a iequaity cotaied eat quae adutmet which i fomuated a the iea compemetay pobem (LCP). I the foowig the baic picipe of the agoithm wi be decibed, fo detai ee Lawo & Hao, 1995, Fitch, 1985 ad Schaffi, he aim of a eat quae adutmet i to miimize the quae um of the eidua v (L 2 appoximatio): (7) (8) (9) A x Jacobea matix of the patia deivatio of the obevatio equatio with epect to the ukow paamete x vecto of ukow paamete vecto of educed obevatio he equaity cotait 1, 2, 4, 7 ad 8 ca be ued to deive obevatio equatio of thi adutmet pocedue: = ˆ i i ( X, Y,, ) u v w (1.1) 0 + v ˆ = ˆ, (2.1) k + vˆ pm k = ˆ ˆ = ˆ ˆ p k m (4.1) (7.1) q d ditace = (8.1) Hee, a Gau-Makoff adutmet mode wa ued becaue the defied equaity cotait ae ot ticty adheed to. he height vaue Ẑ ae the ukow, the obevatio ae peudo-obevatio. By weightig them uig the weight matix P, the fufimet of the cotait ca be cotoed. he height of the gee DM poit outide the obect (ee figue 2) ae ued to fom a additioa obevatio equatio: = ˆ (11) If a height vaue ha eay to be uchaged, thi obevatio get a high weight. he adutmet pocedue i competed by itoducig the iequatio 3, 5, 6 ad 9 i the optimizatio poce: 0 > ˆ ˆ k ˆ ˆ (3.1) m (5.1) Dm d ˆ ˆ ˆ ˆ m o (6.1) Dm Do ˆ < ˆ, ˆ > ˆ (9.1) hee iequatio fom the foowig iequatio ytem the ukow paamete ao have to atify: whee v P v Pv Mi ( Ax ) P( Ax ) Mi vecto of eidua weight matix (10.1) whee B b B x b (10.2) matix of patia deivatio of the iequatio with epect to the ukow paamete x ight ide of the iequatio ytem

5 he LCP i fomuated a foow: ẑ = M uˆ + k (12) whee 1 ( ) 1 ( ) ( ) M : = B A PA B k: = b B A PA A P (13) Figue 3: IN of the imuated data et he outio (12) i caed a compemetay outio, becaue the vecto ẑ ad u ˆ ae compemetay to each othe, i.e. the foowig compemetay coditio i fufied: zˆ 0, uˆ 0 uˆ zˆ = 0 ie epeet the eutig pofie afte the itegatio of both data et. he height age of the ew pofie i mae tha the height age of the oigia data, the mea height of the cete axi of the oad peeted i figue 4 i eay the ame a befoe. he defied cotait ae fufied, a ope vaue ae mae tha 0.05 ad a cuvatue vaue ae mae tha he ukow paamete vecto x of the iitia eat quae adutmet with iequaity cotait (ee 10) ae computed uig the eut vecto û of equatio 12: 1 ( ) ( ˆ ) xˆ = A PA B u A P (14) I ou appoach the Liea Compemetay Pobem i oved uig the Lemke agoithm (Lemke, 1968). Figue 4: Eat-wet pofie befoe ad afte the agoithm wa appied 4. RESULS he fit eut wee detemied by mea of two diffeet tet data et. he fit data et i a imuated oe ad coit of a ma DM tip ad a obect epeetig a tited pae. he ecod data et i the aea fom figue 1. he topogaphic vecto data ae obect of the Gema AKIS Bai-DLM. he data et coit of thee ake bodeed by poygo with paimetic coodiate. he DM, the AKIS DGM5, i a hyid data et, cotaiig eguay ditibuted poit with a gid ize of 12,5 m ad tuctue eemet ited pae he imuated egua DM coit of 6 ow ad 31 coum with a gid ize of 10 m. he height wee cacuated uig a ie fuctio i X-diectio (eat-wet) with a additioa adom coectio. he obect i a taight oad with adom X- ad cotat Y-coodiate, it coit of 54 poit. he ie ha bee buffeed by uig a oad width of 4 m o the eft ad 4 m o the ight ide of the cete axi. Figue 3 how the tiaguated DM. I a computatio the vaue ad d of the iequaity cotait wee et to 0.06 ad 0.05, epectivey. he fit ivetigatio wee doe uig the ame weight fo a obevatio, i.e. the matix P fom equatio (10.1) i a uit matix. Figue 4 how the pofie of the itepoated height vaue of the oad befoe the agoithm wa appied (gay ie). he bue Pepedicua to the divig diectio of the oad the height diffeece betwee poit of the cete axi ad the coepodig height of the bode ae ot zeo, becaue the iequaity cotait (iequatio 5.1 ad 6.1) efe to the cete axi. Uig a highe weight fo thee kid of obevatio eut i hoizota co ectio. A highe weight of the gee DM poit ad of the height diffeece betwee the gee ad the oage poit outide the obect eut i iceaig icied co ectio. he oad adut to the teai, the impovemet of the poit height outide the obect become mae Hoizota pae he ecod exampe dea with AKIS data. he tet data et coit of about 3400 DM poit, thee ake ae epeeted by tota of 300 paimetic coodiate. Figue 5 how the DM afte the cotaied Deauay tiaguatio. he bue poit ae the itepoated height vaue of the topogaphic vecto data et. Obviouy, the bue poit do ot a have the ame height eve. Mot of them ae ituated o the bak of the ake. hu, the itepoated height ae highe tha the poit iide the ake. Additioay, the height vaue iide the obect poygo ao do ot have the ame height eve. Agai, the fit ivetigatio wee caied out by uig the ame weight fo a obevatio. he height vaue of the bak have to be at eat 1 cm highe tha the height eve of the ake (ee iequatio 3.1).

6 he agoithm eut i ake height which ae eay idetica to the mea vaue of a height vaue epeetig a ake. he iequaity cotait ae fufied. A highe weight fo the gee DM poit outide the ake (equatio 11) ad fo the height diffeece betwee thee poit ad the poit of the bak (equatio 4.1) eut i ake height which diffe fom the mea vaue. he teai outide the obect i eay the ame a befoe. Oy the height vaue which ae to ow wee impoved igificaty. Figue 6 epeet the IN of the itegated data et. he bue aea ae the ake: A height vaue iide the ake ad o the boudig poygo of the ake ae the ame. Figue 5: iagua iegua etwok (IN) of the DM, topogaphic vecto data et with itepoated height Figue 6: IN of the itegated data et, DM ad topogaphic vecto data et afte appyig the agoithm

7 5. SUMMARY AND FUURE WORK hi pape peet the fit eut which have bee deived uig a agoithm fo ematicay coect itegatio of DM ad a topogaphic vecto data et. Baed o a cotaied Deauay tiaguatio the height of the two-dimeioa vecto data et ae cacuated uig the eighbouig height of a tiagua iegua etwok. Some of the obect of the topogaphic vecto data cotai impicit height ifomatio. Fo exampe, ake decibe a hoizota pae, oad o aiway do ot exceed maximum ope ad cuvatue vaue. hi ifomatio ha bee ued to deive equaity ad iequaity cotait. he agoithm i baed o a iequaity cotaied eat quae adutmet fomuated a the iea compemetay pobem (LCP). he agoithm eut i impoved height vaue of the DM. Fit ivetigatio wee caied out by uig two diffeet data et: A imuated oe epeetig a tited pae ad a ea data et of the Gema AKIS. he cotait have bee fufied but big diffeece betwee the cotait ad the DM may caue a o-eaitic impovemet of the oigia height ifomatio. hu, bude o big diffeece have to be aayzed i the agoithm i the futue. Futhemoe, the paimetic coodiate of the topogaphic vecto data et wee itoduced a eo-fee. hi may caue a educed height eve of the topogaphic obect if the paimetic coodiate of the obect epeetig a oad o a dam ae ituated beeath the coepodig pat of the DM. O, if the paimetic coodiate of the boudig poygo of a ake ae ituated o the bak, the ake height aie. hat mea, the accuacy of the paimetic coodiate of the topogaphic data have to be coideed. I additio, paimetic coodiate of tuctue eemet have to be coideed i the adutmet poce. Othewie, tuctue eemet iide aea baed obect wi be deeted ad the mophoogy ca be eoeou. Nevethee, the agoithm how fit atifyig eut but futhe ivetigatio have to be cay out. Fitch, D., Some Additioa Ifomatio o the Capacity of the Liea Compemetay Agoithm, i: E. Gafaed & F. Sao, Ed., Optimizatio ad Deig of Geodetic Netwok, Spige, Bei, pp Guiba, L., Stofi, J., Pimitive fo the Maipuatio of Geea Subdiviio ad the Computatio of Voooi Diagam. ACM aactio o Gaphic 4(2), pp Lawo, C. L., Hao, R. J., Sovig Leat Squae Pobem. Society fo Idutia ad Appied Mathematic, Phiadephia. Lemke, C. E., O compemetay pivot theoy, i: G. B. Datzig, A. F. Veiott, Ed., Mathematic i the Deciio Sciece, Pat 1, pp Lek, U., D-GIS ud Geobaidate Itegatio vo Höheifomatio ud Digitae Situatiomodee, Wi. Ab. Fach. Vem. Uiveität Haove N. 244 ud DGK bei de Baye. Akad. D. Wi., Reihe C, N Di., Uiv. Haove. Schaffi, B., Augeichug mit Bedigug- Ugeichuge. AVN, 6, pp Shewchuk, J. R., Deauay Refiemet Meh Geeatio. PhD hei, Schoo of Compute Sciece, Caegie Meo Uiveity, Pittbugh. ACKNOWLEDGEMEN hi eeach wa uppoted by the uveyig authoity of Lowe Saxoy Ladevemeug ud Geobaiifomatio Niedeache (LGN). We ae expe ou gatitude to LGN fo povidig the data. REFERENCES Bi, R., Fitch, D., Gudage de Geo- Ifomatioyteme, Bd. 1, Hadwae, Softwae ud Date, Wichma Veag, Heidebeg.