ACCURACY EVALUATION OF RATIONAL POLYNOMIAL COEFFICIENTS SOLUTION FOR QUICKBIRD IMAGERY BASED ON AUXILIARY GROUND CONTROL POINTS

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1 ACCURACY EVALUAION OF RAIONAL POLYNOMIAL COEFFICIENS SOLUION FOR QUICKBIRD IMAGERY BASED ON AUXILIARY GROUND CONROL POINS Yu Zha a Chu Liu a,b Gag Qiao a a Departmet of Surveyig ad Geo-Iformatics, ogji Uiversity, Shaghai, Chia b Key Laboratory of Advaced Egieerig Surveyig of State Bureau of Surveyig ad Mappig, Chia Commissio VI, WG VII/6 KEYWORDS:Ratioal polyomial coefficiets, Batch iterative least-squares solutio with regularizatio, Icremetal discrete Kalma filterig, Geo-positioig accuracy ABSRAC: Relative to the rigorous physical model, ratioal polyomial coefficiet (RPC) has bee adopted as a alterative commo sesor model data for image Geometric correctio exploitatio. I this paper, based o collected QuicBird imagery i Shaghai regio, the iterative least-squares solutio with regularizatio(ilsr) is derived to determie the RPCs by usig 5 fair distributed groud cotrol poits(gcps)firstly. wo methods are the used to refie determied RPCs uder differet circumstace as: ) whe both the origial ad the additioal GCPs are available, the RPCs will be recomputed usig the batch iterative least-squares solutio with regularizatio (BILSR) method; ad 2) whe oly the ew GCPs are available, icremetal discrete Kalma filterig (IDKF) method has bee described. Meawhile, chec poits are used to evaluate their geo-positioig accuracy, ad their compariso is coducted. Fially, some coclusio is the achieved whe hadig the high resolutio imagery i metropolita area.. INRODUCION Satellite Imagery such as QuicBird, IKONOS has bee widely used with the developmet of high resolutio satellite techology. Colliearity based rigorous sesor model is the basis of geometric positioig for high resolutio satellite imagery (HRSI). Depedece o physical parameters ad satellite orbit parameters maes the rigorous sesor model much more complicated, thus hard to be applied with. he RPC, a mathematical model which is sesor idepedet ad ot rigorous, has bee used widely by satellite compaies for the survey process of HRSI ad as the alterative of the rigorous sesor model. he image coordiates are deoted as the third polyomial expressio i RPC. RPC, by providig a simple ad exact relatio for vedors ad customers to describe the relatioship of object ad image, has bee successfully employed i the terrai modelig, orthographic projectio ad feature extractio. A lot of research wor has bee doe about the geometric correctio ad D recostructio of IKONOS imagery usig RPC(ao ad Hu,2, 22(),22(2); Dowma,2;Fraser,22;Clive,22). wo methods are used for the calculatio of RPC, terrai depedat approach ad terrai idepedet approach (Yog Hu et al.,24). he terrai depedet approach, without settig up grids, is to obtai GCP through topographical measuremet or field survey to fit the imagery geometry usig sufficiet parameters. Its accuracy is determied by hypsography ad the GCP umber ad distributio (Fraser,26; Liu,26). he relativity betwee the RPC parameters may result i the sigularity of desig matrix for ormal equatio. he regularizatio method ca improve the coditio umber of the desig matrix, thus avoidig the umerical istability of least square solutio (ao ad Hu,2). he RPC direct correctio method was put forward to improve the positioig accuracy ad meet the demad of high accuracy users. Differet mathematical methods were applied for the RPC accuracy improvemet whe the physical sesor model was uow. Whe the origial ad auxiliary GCP were both available, BILSR was used to recalculate the RPC (Hu ad ao,22; Di et al.,2). Here the origial GCP deotes the GCP used for calculatig the origial RPC, while the auxiliary GCP meas the auxiliary collected GCP that ever used for RPC calculatio. he correctio process is to iclude all the GCP ito the RPC solutio with differet power to the ew ad origial GCP. Whe there are oly auxiliary GCP, the IDKF ca be employed to improve the RPC accuracy (Hu ad ao,22; Bag et al.,2), which meas the accuracy of RPC is improved through the iclusio of ew GCP with proper power. Based o the QuicBird imagery i Shaghai, Chia, this paper maily discusses the solutio of RPC ad accuracy after correctio applied i metropolita area without obvious hypsography. Experieces of applicatio for similar data could be leared from the models ad data this paper employed. 2. RPC MAHEMAICAL MODEL he RPC of QuicBird imagery deotes the image coordiates as the ratio of polyomials based o the variable of logitude, latitude ad height, which is as equatio (): 287

2 he Iteratioal Archives of the Photogrammetry, Remote Sesig ad Spatial Iformatio Scieces. Vol. XXXVII. Part B7. Beijig 28 r p ( P, L, H ) = = p2( P, L, H) c p ( P, L, H ) = = p4( P, L, H) () i= j= = i= j= = i= j= = i= j= = aplh i j t bpl H i j t cplh i j t dplh i j t he image coordiate ( r, c ) ad groud coordiate ( P, L, H ), whose value are withi [-,+], are both the stadardizatio coordiates through traslatio ad scale, for the purpose of reduce the roudig errors i calculatio because of quatitative differece. he uit of ( r, c ) is pixel; P, L are the coordiates of WGS84 with uit degree; H is the geodetic height with uit meter. ( P, L H ca be expressed as equatio (2):, p = a + a L + a P + a H + a L P (2) a6lh a7ph a8l a9p ah apl H a2l alp a4lh a5l P a6p a7ph a8l H 2 a9p H a2h a t is the polyomial coefficiets ( t =,2,...2; i, j, =,,2,) the similar expressio. ), other polyomials have he terrai depedet approach is to calculate the 8 parameters of RPC usig GCP from field survey. For coivace i the followig expressio, the ( P, L, H) are shorted for ( PLH,, ), ad ( r, c ) for (,) rc, so equatio () ad (2) ca be expressed as (): r = c = 2 ( L P H L P H H ) ( a a2 L a2) 2 ( L P H L P H H ) ( b L b9) 2 ( L P H L P H H ) ( c c2 L c 2) 2 ( L P H L P H H ) ( d L d9) () 2 2 L P H P H H rl rp rp H rh r v r = L L J B B B B B B B B B B B 2 2 L P H P H H cl cp cp H ch c v c = L L K D D D D D D D D D D D B= Z Y X L Y X b L b (4) ( ) ( 9) ( L L ) J = a a a b b b ( L ) ( L 9) D= Z Y X Y X d d ( L L ) K = c c c d d d (5) Suppose there are GCP, the error equatio is: V = WI WG L L B M O M M O M L L B W =, L L D M O M M O M L L D L rh L M O M M O M L rh L = L L ch M O M M O M L L ch I= a L a2 b L b2 c L c2 d L d2 [ ] [ ] G = r L r c L c (6) If the observatios are uit weight observatios, the the ormal equatio is: Liearizatio of equatio () (ao ad Hu, 2) we could get equatio (4): 2 2 WI WG= (7) he we get the coefficiet matrix I: 288

3 he Iteratioal Archives of the Photogrammetry, Remote Sesig ad Spatial Iformatio Scieces. Vol. XXXVII. Part B7. Beijig 28 = (8) 2 2 ( I W) WG. REGULARIZAION MEHOD. Regularizatio Model I =, W = W( I ) = E () () () I = I + ( W + h E) W v ( s) ( s ) ( s ) ( s ) ( s ) W = W( I ), v = G I ( s) ( S) ( s) ( S) () he deomiators Bi ad Di ( i =,... ) chage quicly i quatity whe the GPS iput distributed uevely i calculatio, which maes matrix ill-coditio i the equatio, ad thus the 2 matrix W sigularity. his case ofte happes whe the ra of RPC polyomial is higher (eg, more tha 2), which may result i ot coverget durig iteratio. A uit matrix E could be added by the regularizatio method to 2 improve the coditio umber of matrix W. 2 W is a symmetric oegative defiite matrix, ad the 2 2 eigevalue of matrix is i the rage of 2 2 h, h + W+ he 2 W h + W )/ h, so the coditio umber of which will o bigger tha (, o the cotrary, it will reduce with the icrease of. A regularizatio rule method is employed i this paper to improve the coditio umber of matrix to obtai stable umerical solutio. he ormal equatio is calculated through iteratio ad ca be eded whe the coditio be met(neumaier,998) ( W+ hei ) WG= (9) h 2 E is the uit matrix, h is the regularizatio parameter, s is the umber of iteratio. Whe the pixel measuremet error is ow, the covariace matrix P of parameter I could be calculated from the followig equatio: P= W + h E W R W W + h E ( ) G ( ) () here are may ways to obtai the regularizatio parameter h. Differet h will get differet result, of which the optimal value is attaied by trial method, here L curve method (Neumaier, 998) is employed. o get the optimal value of h, differet value was tested i formula (9). he third power RPC was employed i the test, 5 GCP ad their correspodig image poits were selected for calculatio of RPC, while 29 CP were used for accuracy assessmet. he result was show as figure. Based o the test, less tha iteratio times may get good covergece uder most coditios whe h was i the rage of.9 ad.. We also foud that as log as h was i the rage, the accuracy was ot sesitive to specific h, that is to say the results were withi. pixel (see table ). So h=.5 was used i the followig text. 6 4 RMSE of CKPs (Pixels) Regularizatio Parameter h Figure Calculatio for h----l Curve Figure 289

4 he Iteratioal Archives of the Photogrammetry, Remote Sesig ad Spatial Iformatio Scieces. Vol. XXXVII. Part B7. Beijig 28 h Value Matrix Sigularity (Y/N) Iteratio Number CP Poit Positio Error (pixel) h Value Matrix Sigularity (Y/N) able Positioig Accuracy with differet h values Iteratio Number CP Poit Positio Error (pixel) <.4 Y Not N.5 N Covergece.4 N N N N N.5.2 N N.29. N N N N N Calculatio of RPC he test imagery is QuicBird imagery of Shaghai area, of which the collectio time is February 5, 24, the coordiates of lower left corer (.4796, ),the coordiate of upper right corer ( ,2.659 ). he image collectio azimuth is 55. degree; the elevatio agle is 68.2 degree. 5 GCP distributed evely were selected from the overall 9 surveyed poits to calculate the iitial RPC, the distributio of which was show i figure poits were selected radomly from the rest surveyed poits as CP for accuracy aalysis, of which the distributio was as figure. Figure 2 GCP Distributio Figure CKP Distributio aig h =.5 groud poits ( PLH,, ) ito formula () to get 8 RPC parameters, we ca get the image coordiated by addig the parameters ad image poits of CP were show i table 2: Results ad Coclusio of tests: of the 26 CP ito formula (). he differeces betwee calculated image coordiates ad correspodig Absolute Error Error ype Accuracy (pixel) Error ype Accuracy (pixel) Max Row Error 4.92 Relative Error Row Error.59 Row Mea Square.864 Colum Error 2.55 Deviatio Max Colum Error 7.6 Poit Positio Error 2.2 Colum Mea Square Deviatio 2.65 Poit Positio Error.24 able 2 he positioig accuracy of D method 29

5 he Iteratioal Archives of the Photogrammetry, Remote Sesig ad Spatial Iformatio Scieces. Vol. XXXVII. Part B7. Beijig 28 D grid poits caot be established without physical model, so traditioal methods (eg, field survey, map measuremet or DEM) should be employed for the obtaiig of GCP ad CP. Uder this case, the result depeds o the hypsography, umber ad distributio of GCP. It is a popular positioig method whe the rigorous sesor model is uavailable or the accuracy is ot demadig. 4. RPC CORRECION MEHOD Here the coorectio method meas to apply differet mathematical models without chagig the RPC model to the 8 parameters to get the updated RPC parameters. If both the GCP calculatig the RPC ad the auxiliary GCP are available, BILSR is applied for a group of ew RPC, otherwise if oly the auxiliary GCP available, IDKF is employed. 4. BILSR Method Both the origial ad ew GCP are used i this method i batch process for the updated RPC. All the GCP are used i the equatio with differet power for each poit. time, there will be flexibility for the calculatio if a o-zero is provided (Hu ad ao,22). Q he process (equatio ) ad liearizatio (equatio 4) are realized by addig ew GCP usig icremet based o Kalma Filter to improved the iitial RPC accuracy. raditioally, Kalma Filter is used for the complicated time problems. Kalma Filter is used to space domai is based o its recursio character for the ew GCP are obtaied i sequece. ) Calculatio of iitial value ad covariace matrix I = I P P Q = K + (4) 4.2 IDKF Method Icremet is used i this method for the accuracy improvemet whe the 8 parameters ad the matrix P (i equatio ) are both available. With the ew GCP, the RPC accuracy ca be updated usig this method. he value of RPC iteratio is stable, of which the process ad expressio are as followig (Hu ad ao, 22) I = I + + w (2) G = I v = I + v ( =, 2...) () Equatio is trasformed from equatio 7, which deotes the liearity relatio betwee the observatios ad parameters. w is the oise vector, or white oise with ow covariace Q v matrix ; is the measure error of image poits, it is cosidered to be white oise with the ow covariace matrix R of ew GCP. Vector w ad v are idepedet. R is usually based o experieces ad tests i calculatio. est results show that eve though RPC chages very little every GCP are divided ito several groups, the repeated processes are eeded. From the above we ca see that the iitial value of RPC covariace is very importat for it decides the sesibility of ew GCP ad its covariace. 5. RPC CORRECION AND ACCURACY ANALYSIS he image is the same as metioed above. 5 GCP distributed evely were selected from the overall 9 surveyed poits to calculate the iitial RPC, the distributio of which was show i figure poits were selected radomly from the rest surveyed poits as GCP ad CP for accuracy aalysis, of which the distributio was as figure 4. Where - meas the value is the previous result of the ew oe. 2) Calculatio of icremet of Kalma Filter K P P R = ( + ) (5) ) Updatig I by addig ew GCP I = I + K v, v = G I (6) 4) Calculatio of updated I covariace P = ( E K ) P (7) here will be oly oe process from step (2) to (4) for the RPC updatig if all ew GCP are cosidered to a whole group. If the Figure 4 Distributio of 49 Auxiliary GCP 29

6 he Iteratioal Archives of the Photogrammetry, Remote Sesig ad Spatial Iformatio Scieces. Vol. XXXVII. Part B7. Beijig 28 9 of the 49 poits are selected auxiliary GCP, the remaiig 4 as CKPs. Equatio was used for calculatio of RPC. 5 GCP were used to get the iitial value of RPC, ad the BILSR ad IDKF were both used to improve the RPC accuracy. he poits were added ito the equatio oe by oe i IDKF for the use of 9 GCP (Because the auxiliary GCP usually have higher accuracy, they have higher power i calculatio). I this test, all the GCP were collected at the same time with the same way, so they all had the same power. So the test here is to determie improvemet efficiecy of BILSR ad IDKF for RPC accuracy after icrease of GCP. he covariace matrix is set to 6 Q = to test the accuracy of updated RPC. Image poits of 4 CP were obtaied usig updated RPC accordig to equatio (). he errors of calculated image poits ad measured oes were used for aalysis. he origial row mea square deviatio, colum mea square deviatio ad poit positio error were.56,.4,.76 pixels, respectively. he updated row mea square deviatio, colum mea square deviatio ad poit positio error with BILSR usig 9 GCP were.467,.5,.625 pixels, respectively. he updated errors with IDKF usig 9 GCP were.494,.2,.64 pixels, respectively. he RPC accuracy was improved withi.2 pixels after the additio of 9 GCP, for 9 GCP was ot otable compared with 5 GCP whe it comes to adjustmet (figure 5). he results of BILSR ad IDKF are as table, where the mea square deviatio, max absolute error ad poit positio error of CP i both row ad colum directios are give. BILSR:pixel IDKF:pixel Number of New Row Colum Poit Row Colum Poit GCP Positio Positio RMS MAX RMS MAX RMS MAX RMS MAX Error Error able Results of BILSR ad IDKF by addig GCP by sequece.8 Positio RMSE of BILSR Positio RMSE of IDKF Positio RMSE (pixels) Number of GCPs Figure 5 Compare of Poit Positio Error betwee BILSR ad IDKF 292

7 he Iteratioal Archives of the Photogrammetry, Remote Sesig ad Spatial Iformatio Scieces. Vol. XXXVII. Part B7. Beijig 28 From the test we ca see, both the BILSR ad IDKF ca improve the RPC accuracy with little amout, for RPC is maily iflueced by the distributio of GCP, ad it is very hard to improve the accuracy solely based o these two mathematical methods. able also shows that ot all addig GCP could result i accuracy improvemet, idicatig the ucertaity of these methods. Distributio of auxiliary GCP caot be tested for the fittig of terrai. Icrease of GCP umber caot esure the accuracy improvemet. 6. CONCLUSIONS Relativity amog coefficiets whe calculatig RPC results i the ill-coditio ad istability of matrix, which ca be solved by may approaches. Regularizatio method is used to esure the stability of the calculatio, durig which the selectio of regularizatio parameters is the ey to this issue. BILSR ad IDKF ca improve RPC by mathematical meas; they do ot modify the model, but resolve the ew RPC via calculatio. he covariace matrix of parameters must be provided i IDKF. here are little improvemet of accuracy withi pixel, so both of them are ot recommeded i calculatig of RPC. Yog Hu ad C. Vicet ao. updatig solutios of the ratioal fuctio model usig auxiliary cotrol iformatio. Photogrammetric Eigieerig& Remotig Sesig,Vol.68,No.7,pp.75-72,22. Di,K.,Ma,R.,Li,R.. Ratioal fuctio ad potetial for rigorous sesor model recovery, PE&RS,69(),pp.-4,2. Bag,K.I.,Jeog,S.,Kim,K.. Modificatio of sesor model parameters with a few GCPs, ASPRS Aual Coferece,-9 May,Achorage,AK,6p,2. Neumaier,A..Solvig ill-coditioed ad sigular liear system,siam Review,4(): ,998. ACKNOWLEDGEMEN he wor described i this paper was supported by Natioal 86 High-tech Project Complicated Features Extractio ad Aalysis from Cetral Districts i Metropolis ad Natioal Natural Sciece Foudatio of Chia (Project No.456) REFERENCE Vicet C,Yog H.A Comprehesive Study of the Ratioal Fuctio Model for Photogrammetric Processig[J]. Photogrammetric Egieerig & Remote Sesig,2,vol. 67 No.2 : Vicet C, Yog H.A Comprehesive Study of the Ratioal Fuctio Model Usig Auxiliary Cotrol Iformatio[J]. Photogrammetric Egieerig & Remote Sesig,22,68(7): Vicet C, Yog H.A D Recostructio Methods Based o the Ratioal Fuctio Model[J]. Photogrammetric Egieerig & Remote Sesig,22,68(7): Dowma I, Dolloff J.A Evaluatio of Ratioal Fuctios for Photogrammetric Restitutio[J]. Iteratioal Archives of Photogrammetry ad Remote Sesig,2,(B/ : Fraser C S, Baltsavias E, Grue A. Processig of Ioos Imagery for Submeter D Positioig ad Buidig Extractio[J]. ISPRS Joural of Photogrammetry &Remote Sesig,22,56(2): Clive S F, Happy B, Haley Y.hree-dimesioal Geopositioig Accuracy of IKONOS Imagery[J].Photogrammetric Record,22,7(99): Yog Hu, Vicet ao, Arie Croitoru.uderstadig the ratioal fuctio model: method ad applicatios ISPRS Cogress,2-2 July 24, Istabul,urey,vol.XXXV-B4. C.S. Fraser,etl.. Sesor orietatio via RPCs,ISPRS Joural of Photogrammetry & Remote sesig,vol.6,pp.82-94,26. Liu J, Y. Zhag, D. Wag, Precise Positioig of High Spatial Resolutio Satellite Images Based o RPC Models, Acta Geodaetica et Cartographica Siica, Vol.5,o.,26.pp:

8 he Iteratioal Archives of the Photogrammetry, Remote Sesig ad Spatial Iformatio Scieces. Vol. XXXVII. Part B7. Beijig