Measurement Indices of Positional Uncertainty for Plane Line Segments Based on the ε

Transcription

1 Proceedngs of the 8th Internatonal Smposum on Spatal ccurac ssessment n Natural Resources and Envronmental Scences Shangha, P R Chna, June 5-7, 008, pp 9-5 Measurement Indces of Postonal Uncertant for Plane Lne Segments Based on the ε Model Guoqn Zhang and Changqng Zhu Informaton Engneerng Unverst, Zhengzhou 45005, Chna Mnstr of Educaton Ke Laborator of VGE, Nanjng 0046, Chna bstract Frstl, eght cases of the random lne segments e dscussed Secondl, to the eght estng cases of lne segment, the analtc epressons of the error band for the ε uncertant model of lne segment e deduced, the pameter equatons of the error band bound e gotten, and t s proved that the bound of the error band s contnuous Thrdl, the average error band wdth and the algebrac epressons of the uncertant regon ea bounded b error band e calculated Fnall, the vsual graphcs of the error band e drawn wth analtc epresson of the error band b eamples Thus, three ndees e gven to measure the precson of the lne uncertant: the average error band wdth, the uncertant regon ea and the vsual graphcs Kewords: plane lne, uncertant, ε model, error band Introducton Spatal data s one of the fundamental pts of GIS The qualt of spatal data drectl determnes the ftnessfor-use of GIS and affects the results of GIS applcatons Therefore, accurac analss of spatal data s [,] regded as one of the fundamental theoretcal resech b ssues nternatonall Pont, lne segment and plane segment e the fundamental elements of vector spatal data So, resech on uncertant of pont, lne segment and plane segment s the man resech n the uncertant of spatal data Especall the resech on lne segment s mportant n GIS, because lne segment s not onl the bass of the uncertant resech on plane segments but also the basc element n overla analss and buffer analss There e man studes for the error band model of lne segment Perkal buld ε band model that s the strp band based on the buffer of lne segment wth constant ε Usng the ε band model, Blakemore descrbed the spatal uncertant of lne segment Sh et al developed a method to buld a generalzed error model(g-band) that descrbes the postonal uncertant of lne segment G-band s composed of the envelop comng from a group of error ellpses and the pt error ellpses of the etreme ponts The analtc epresson of G-band s comple Based on above models, Lu et al descrbed a general ε model for the postonal uncertant of lne segments Zhu et al proved the geometrcal chacter of the error band of the ε model based on algebrac method For theε model, the wdth of error band s represented b the root mean sque error n normal drecton of the lne, and the error band s composed of half of the error crcle at the etreme pont In ths paper, the ε model s studed further Frstl, eght cases of the random lne segment estng e dscussed Secondl, the analtc epresson of error band bound lne for the uncertant ε model of lne segment for ts eght cases s deduced Thrdl, the average error band wdth and the algebrac epressons of the uncertant regon ea bounded b error band e calculated wth the analtc epresson Fnall, the vsual graphcs of the error band e drawn wth analtc epresson of the error band b eamples Thus, three ndees e gven to measure the precson of the lne uncertant: the average error band wdth, the uncertant regon ea and the vsual graphcs The uncertant metrcs ndees can make sure the relablt of the spatal analss and the applcaton n GIS So, the users can know the sze the ISBN: , ISBN3:

2 dstrbuton and the spatal structure of the lne segment postonal uncertant drectl, and can understand and use the GIS product better The Error Band naltc Epresson of The ε Model for Lne Segment Uncertant The error band of theε model for lne segment Suppose Z Z be a lne segment and the coordnates of the endponts e Z (, ) and Z (, ) respectvel (as Fgure shows) The vance and co-vance matr of the lne segment e represented as the followng respectvel [] : D =, D =, D = Suppose θ s the angle between the lne Z Z and -as It s known easl that θ = ctan( ( ) ( )) and θ Suppose Z (, ) be an pont at the lne segment Z Z Then there e the followng relatons among the coordnates of Z, Z and Z [] : = ( r) r () = ( r) r where r = s s, s s the dstance of Z to the end pont Z, and s s the length of the lne segment Z Z It s obvous that 0 r Known from reference [], for theε model, the wdth of error band s represented b the mean sque error at the pont Z n normal drecton of the lne, and the error band s composed of pt error crcles at the etreme ponts (as Fgure shows) In order to get the mean sque error n normal drecton of the lne, the orgnal coordnate sstem O s rotated θ antclockwse about the coordnate orgn pont O Then the new coordnate sstem O s obtaned, and n the new coordnate sstem, O as s n a accordance wth lne segment Z Z B the co-vance propagaton rule, the vance and co-vance matr of the pont Z n the new coordnate sstem O can be obtaned 0 Fg : Error band of a lne segment From reference [], t s known that the error band wdth epresson of s as the followng: where a = s the functon of the pameter r, and the = br c (), b = ( ), c = Z θ B Known from epresson [], when the error of the end ponts s determned, for theε model, the wdth of error band at an pont Z on the lne s related to the pameter r onl Therefore, the error band can be epressed n the functon about the pameter r ( 0 r ) The error band analtc epresson of the ε model for lne segment uncertant Defnton : When one s walkng from the frst etreme pont to the after etreme pont along the lne segment, the error band bound lne located at the left s named Left Bound Lne, the error band C Z D

3 bound lne located at the rght s named Rght Bound Lne, half of the error crcle at the frst etreme pont s named Left Error Semcrcle, and half of the error crcle at the after etreme pont s named Rght Error Semcrcle From the defnton of the ε model of lne segment uncertant and defnton, t s known easl that the error band bound lne s composed of 4 pts: left bound lne = f ( ), rght bound lne = f ( ), left error semcrcle = f 3 ( ) and rght error semcrcle = f 4 ( ) Resech shows that there e 8 cases of the random lne segment: () <, < ;() >, < ;(3) <, > ;(4) >, > ;(5) =, < ; (6) =, > ;(7) <, = ;(8) >, = The analtc epresson of the error band bound lne wll be dscussed n the followng sectons to the 8 cases of the random lne segment for the ε model The analtc epresson of the error band bound lne when Frstl, the analtc epresson of the error band bound lne for the ε model of lne segment s dscussed when < and <, shown as Fgure () f P Z Q Z f () O θ Z Fg : The bound lne of the error band when < and < ( ⅰ) The analtc epresson of the left bound lne = f ( ) Suppose P(, ) s an pont at the left bound lne = f ( ) Draw a lne through the pont P perpendcul to the lne segment Z Z, and the perpendcul foot s Z (, ) Suppose Z PQ = θ It s known from the defnton of the error band that PZ = So we have PQ = = cos θ, QZ = = snθ From formula (), (), and the formula mentoned above, the coordnate epresson of an pont P (, ) at the left bound lne s obtaned: = cosθ = ( r) r f( ) = snθ = ( r) r br c cosθ br c snθ (0 r ) θ and a, b, c e related to the coordnates and the vance-covance of the etreme ponts, that s, the e constant So the coordnate of an pont P (, ) at the left bound lne s the functon about the pameter r Therefore, the pameter equaton of the left bound lne about r can be get, where the range of r s ( 0,) (ⅱ) The analtc epresson of the rght bound lne = f ( ) Smll, the analtc epresson of the rght bound lne = f ( ) can be obtaned: = cosθ = ( r) r f( ) = snθ = ( r) r br c cosθ br c snθ (ⅲ) The analtc epresson of the left error semcrcle = f ( ) 3 (0 r ) (3) (4)

4 It s known that the center of the error crcle at the frst etreme pont Z s Z and the radus s The left error semcrcle can be got b cuttng the error crcle through the lne D whch perpendcul to the lne segment Z Z So, the pameter equaton of the left error semcrcle s : = sn t 3 f3( ) ( θ t θ ) (5) = cost (ⅳ) The analtc epresson of the rght error semcrcle = f ( ) Smll, the pameter equaton of the rght semerror crcle s: = sn t f4( ) = cost 4 ( θ t θ In the same wa, we have the analtc epresson of the error band bound lne when >, <, =, <, <, = or >, =, and the analtc epresson of the error band bound lne s the same as the analtc epresson when <, < To conclude, the analtc epresson of the error band bound lne when s formula (3)-(6) The analtc epresson of the error band bound lne when > Same as the dervaton process n the secton, we can get the analtc epresson of the error band bound lne when <, >, >, > or =, > (That s > ) The analtc epresson of the left bound lne = f ( ) : = cosθ = ( r) r f( ) = snθ = ( r) r The analtc epresson of the rght bound lne = f ( ) : = cosθ = ( r) r f( ) = snθ = ( r) r The analtc epresson of the left error semcrcle = f 3 ( ) : = sn t f3( ) = cost ) br c cosθ br c snθ br c cosθ br c snθ ( θ t θ The analtc epresson of the rght error semcrcle = f 4 ( ) : = sn t f4( t) = cost ) 3 ( θ t θ ) (0 r ) (0 r ) Known from the analtc epresson of the error band bound lne (3)-(0), the equaton that an pont at the error band bound lne satsfes s determned b the coordnaton and ts vance-covance of the lne segment, s endponts Z, ) and Z, ), that s, the error band s determned b the relevant ( ( nformaton of the lne segment, s endponts Vsualzaton model can be drawn easl wth the analtc epresson of the error band bound lne 3 The contnut of the error band bound lne for the ε model of lne segment Known from secton and the defnton of theε model, the functon of the error band bound lne s composed of 4 pecewse functons: the left bound lne = f ( ), the rght bound lne = f ( ), the left error semcrcle = f 3 ( ) and the rght error semcrcle = f 4 ( ) From formula (3)-(0), the 4 pecewse functons s contnuous n the pecewse nterval, so, to resech on the contnut of the error band (6) (7) (8) (9) (0)

5 bound lne, the stud on the contnut at the pecewse ponts s enough That s, the contnut at the pecewse ponts B C D s need to studed (known from fgure ) When, calculated from formula (5), the pameter t = θ at the pont n the left error semcrcle and the coordnaton of the pont s: sn( ) = θ = cosθ () = cos( θ ) = snθ Known from the contnut of the crcle equaton, the functon of the error band bound lne s leftcontnuous at the pont Now we stud the rght-contnuous at the pont From formula (3), judgng the rght contnut of functon s to calculatng lmt : lm = lm[( r) = = = r br c cosθ ] c cosθ cosθ r 0 r 0 () lm = lm[( r) r br c snθ ] = c snθ = snθ = r 0 r 0 So, the functon s rght-contnuous at the pont Therefore, the functon s contnuous at the pont when It s the same when > Provng b the same method, the functon s contnuous at the ponts B C D Therefore, the error band bound lne for the ε model of lne segment s a contnuous closed curve 3 Measurement ndees of postonal uncertant for lne segment based on ε model Now the analtc epresson of the error band bound lne for the ε uncertant model of lne segment for ts eght cases have been deduced Wth the analtc epresson of the error band bound lne, we can draw the vsual graph and calculate the average error band wdth and eas of the error band Thus, three ndees e gven to measure the precson of the lne uncertant: the vsual graph, the average error band wdth and the uncertant regon ea The uncertant metrcs ndees can make sure the relablt of the spatal analss and the applcaton n GIS So, the users can know the sze the dstrbuton and the spatal structure of the lne segment postonal uncertant drectl, and can understand and use the GIS product better 3 The vsual graphcs of the error band for the ε model of lne segment Vsualzaton s a knd of technolog wa, through whch some data can be converted nto graphcs and new knowledge can be gotten Wth the analtc epresson of the error band bound lne defned n secton b formula (5)-(), we can draw the vsual graphcs of the uncertant for lne segment through programmng The vsual graph can epress the error band well 3 The algebrac epresson of average error band wdth s The average error band wdth ε s calculated b the followng formula: ε s = dr = 0 0 s a b br cdr = 4a a b c b 4a ε 4ac b c 3 8 a a b a( a b c) ln b ac (3) 33 The algebrac epresson of the ea of the uncertant regon bounded b error band Obvousl, when the endponts nformaton of lne segment s certant, the uncertant regon bounded b error band s bgger and the uncertant of the lne segment s more So t s sgnfcant to calculate the ea of the uncertant regon bounded b error band and make t a matr to measure the postonal uncertant of lne segment Known from the defnton of the ε model, t s composed of 3 pts: two error semcrcles at the endponts and the mddle regon between the left bound lne and the rght bound lne, so the ea of the error band s the ea sum of, 3 and and 3 denote the ea of the left error semcrcle and the rght error semcrcle respectvel, and denote the ea of the mddle regon between the left bound lne and the rght bound lne We have the followng formulas: 3

6 = l ε s a b = l a =, 3 =, (4) b a b c a where l s the length of the lne segment 4ac b c 3 4 a a b a( a b c) ln b ac, (5) From formulas (4) and (5), the algebrac epresson of the ea of the uncertant regon bounded b error band s obtaned as followng: = (6) The uncertant regon ea of lne segment s related to the etreme pont coordnates and ther vance-covance onl 4 Eamples and analss Now lne segments s dscussed as eamples, and ther coordnates e Z (500,00), Z (550,30) ; Z (50,80), Z (30,30) Ther vance-covance e gven n tab Lne segment Tab Orgnal data of the vance and co-vance of the lne segments Wth the analtc epresson (3)-(0), we can draw the vsual graph of lne segment and b programmng Graph 3-3 show results Fg 3: The error band when < and < Fg 3: The error band when < and > The ea of error band for lne segment and s : and It s shown n the vsual graph that the error band bound lne s a contnuous and closed curve From the vsual graph, the average error band wdth and the ea of error band, t s shown that the uncertant regon ea of lne segment s bgger when the error s bgger, the uncertant regon ea of lne segment s smaller when the error s smaller, and error band wth the shape lger at etreme ponts and smaller at mddle The result has been proofed n reference [4] 5 Concluson Error band uncertant ε model for lne segment can represent the error of lne segment Measurement ndees of postonal uncertant for lne segment based on ε model e studed n ths paper Eght cases of the random lne segment estng e dscussed frstl, then the analtc epresson of the error band bound lne for the uncertant ε model of lne segments for ts eght cases s deduced The functon of the error band bound lne s composed of 4 pecewse functons 4

7 It s proofed that the error band bound lne for the uncertant ε model of lne segments s a contnuous and closed curve 3 Wth the analtc epresson of the error band bound lne, the vsual graph of uncertant for lne segment can be drawn and the average error band wdth and the algebrac epressons of the uncertant regon ea bounded b error band can be calculated Thus, three ndees e gven to measure the precson of the lne uncertant: the vsual graph, the average error band wdth and the uncertant regon ea The uncertant metrcs ndees can make sure the relablt of the spatal analss and the applcaton n GIS So, the users can know the sze the dstrbuton and the spatal structure of the lne segment postonal uncertant drectl, and can understand and use the GIS product better 6 cknowledgement The work s supported b the fund from NSFC (Project No: ) 7 References [] Lu DJ, Sh WZ, Tong XH, Sun HC, ccurac nalss and Qualt Control of Spatal Data n GIS Shangha: Shangha Press of Scence and Technolog, 999 [] Sh WZ, Prncple of Modelng Uncertantes n Spatal Data and nalss Bejng: Press of Scence, 005 [3] Lu DJ, Hua H, More Dscusson to Modelng Uncertant n Lne Prmtves n GIS cta Geodatca et Ctographca Snca, 998, 7: [4] Zhu CQ, Zhang GQ, Sh WZ lgebrac Resech on Geometrcal Chacterstcof error Band for D Lne Segment Uncertant ε Model cta Geodatca et Ctographca Snca, 007, 36: [5] Zhang GQ, Zhu CQ, Zhou B, etc Uncertant ε Model of Spatal Crcul Curve eeatures n GIS cta Geodatca et Ctographca Snca, 005, 34: [6] LIU WB, DI HL, The naltc epresson of geometrc fgure on plan lnes error band cta Geodatca et Ctographca Snca, 998, 7: 3~37 [7] Lu C, Zhan Y, Probablt nalss of error Band of Plane Lne Segments n GIS Journal of Tongj Unverst, 007, 35: [8] Sh WZ, Lu WB, Stochastc Process-Based Model for Postonal error of Lne Segments n GIS Internatonal Journal of Geographcal Informaton Scence, 000, 4(000),