ANALYSIS OF RATIONAL FUNCTION DEPENDENCY TO THE HEIGHT DISTRIBUTION OF GROUND CONTROL POINTS IN GEOMETRIC CORRECTION OF AERIAL AND SATELLITE IMAGES

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1 ANALSIS OF RATIONAL FUNCTION DEPENDENC TO THE HEIGHT DISTRIBUTION OF GROUND CONTROL POINTS IN GEOMETRIC CORRECTION OF AERIAL AND SATELLITE IMAGES M. Hosseii, Departmet of Geomatics Egieerig, Faculty of Egieerig, Tehra Uiversity, (Cetre of Excellece for Geomatics Egieerig & Natural Disasters Maagemet) ABSTRACT: Oe of the existece mathematical models is Direct Liear Trasformatio (DLT). These equatios are beig regarded because of their simplicity as they are direct. Whe the height distributio of groud cotrol poits (GCPs) is iappropriate, height accuracy of DLT is low. This problem was ot obvious about ratioal fuctios. To assess this case, the accuracy of ratioal fuctios has bee tested i three differet cases of GCPs distributio icludig over samplig, optimum samplig ad uder samplig. At last we have come to coclusios that the accuracy of ratioal fuctios i over samplig ad optimum samplig are more tha uder samplig. But the accuracy of over samplig has ot a sigificat differece with the accuracy of optimum samplig. I all cases, to compare the direct ad idirect solutio of ratioal fuctio, we solved ratioal fuctios with both metioed methods. At last it was clear that the accuracy of direct solutio is more tha the accuracy of idirect solutio. All doe tests are i terrai-depedet case of ratioal fuctios. 1 INTRODUCTION There are a lot of mathematical models for photogrammetric processigs. These models show the geometrical relatioship betwee 2D image space ad 3D groud space. Geerally these models are divided to rigorous ad geeric models (McGloe, 1996). Selectig oe of these models depeds o the required accuracy ad the available sesor ephemeris rigorous models are based o colliearity equatios. Oe of the difficulties of rigorous models is their depedecy to sesor. I other words these models have chaged for differet sesors. Because the umber of differet aerial ad satellite sesors like frame, pushbroom ad their applicatios are icreasig, it is ecessary that existed software be chaged for the aalysis of their differet data. Also for usig rigorous models it is ecessary that imagig parameters like orbital parameters, satellite ephemeris, earth curvature, atmospheric refractio ad les distortio be kow. It is essetial that liearize these models because of their o-liearity. But geeric models are i because of its idepedece from positio ad رايج تر orietatio of sesor. Geerally it is t essetial to kow sesor s geometry for usig geeric models ad it is possible to use them for differet types of sesors. I geeric models, relatioship betwee image space ad object space is makig by ratioal fuctios. 2 RATIONAL FUNCTIONS I ratioal fuctios, image pixel coordiates (r,c) are ratio of polyomials of groud coordiates (,,) (OGC, 1999): P1(,, ) m1 i= j= k = = = P2(,, ) r P7( r, c = P8( r, c,, ) ) i= m2 j= m3 k = a b ijk ijk i i j j k k c P3(,, ) m1 i= j= k = = = P4(,, ) i= m2 j= m3 k = Where r ad c are ormalized row ad colum pixel coordiates i image space ad, ad are ormalized coordiates i groud space. For miimizig calculatio errors, two iage coordiates ad three groud coordiates are ormalized such tha beig i (-1,1) (NIMA, 2). A ijk,, b ijk, c ijk ad d ijk are polyomial coordiates ad were amed ratioal fuctio coefficiets. For ormalizig coordiates we ca use below relatios (OGC, 1999): r r =, rs r c =, s c c =, cs = s c d ijk ijk i i =, s Where r ad c are image coordiate shifts ad rs ad cs are image coordiate scale umbers. Similarly,, ad are groud coordiate offsets ad s, s ad s are image coordiate scale umbers. Iverse ratioal fuctios are trasformatios from image space to groud space (Tao ad Hu, 21b): P5( r, c = P6( r, c,, ) ) j j k k 1149

2 The Iteratioal Archives of the Photogrammetry, Remote Sesig ad Spatial Iformatio Scieces. Vol. VII. Part B1. Beijig 28 I these equatios plaimetric groud space coordiates (,) are the ratio of polyomials of image pixel coordiates (r,c) ad vertical groud coordiate (). Ratioal fuctio coefficiets ca be solved by sesor physical model or without it. If physical sesor model be kow the we make a grid i image space. The we used this grid ad physical sesor models to produce aother grid i 3D object space. Grid dimesios deped o groud dimesios ad groud object s height differeces. I other words grid dimesios fill all 3D groud space. This grid has may layers. Each layer poits i each layer have the same elevatio. Number of layers should be more tha three to avoid rak deficiecy of desig matrix (Tao ad Hu, 2). After makig the grid we had used groud coordiates with their similar image coordiates to calculate ratioal fuctio coefficiets by least square method. I this method there is t ay eed to true groud iformatio ad it is amed groud idepedet (Tao ad Hu, 2). This method were used for geometric correctio of high resolutio satellite images (Paderes et al., 1989; Madai, 1999; ag et al., 2; Baltsavias et al., 21; Tao ad Hu, 2). We should kow physical sesor model to produce 3D groud grid. For solvig ratioal fuctios coefficiets, we should used groud cotrol poits (GCPs) that were collected by geeral methods like map ad DEM ad calculatig ratioal fuctio coefficiets. This method of solvig ratioal fuctios was amed terrai depedet (Tao ad Hu, 2). We used this method i remote sesig whe physical sesor model is ukow (Touti ad Cheg, 2; Tao ad Hu, 21a, b). There is limited research o solvig ratioal fuctios by terrai depedet method that had doe by Tao ad Hu. 3.2 Aerial data Figure1. Simulated groud poits I the ext step, we used true aerial data for testig the models. These images are stereo that show a part of Germay. We used Softcopy for measurig groud cotrol poits coordiates. These poits are early a منظم grid ad fill the etire image surface. The we used colliearity equatios with iterior ad exterior parameters to calculate image coordiates of groud poits o stereo images. Calibrated focal legth of the camera for take aerial images is ad the approximate scale of these images is 1:15. Figure 2 shows a 3D view of groud surface ad extracted poits. Maximum height differece of existed poits is 12 meters. 3 EPERIMENT AND RESULTS 3.1 Simulated data set For makig simulated data set, first we suppose of a grid i groud space. Number of poits should be sufficiet. For calculatig left ad right image coordiates of groud poits, we used colliearity equatios. Simulated is related to 1:1 image scale. There totally 96 poits that make a 12*8 grid. Figure 1 shows a 3D view of groud surface ad these groud poits. Heights of poits had bee choose such that the groud be approximately کوهستانی. Heights of poits are betwee 1-1m. Simulated data is related to 1:1 image scale. There are totally 96 poits that make a 12*8 grid. Figure 1 shows a 3D view of groud surface ad these groud poits. Heights of poits have bee chose such that the groud be early. They are betwee 1-1m. systematic error that کوهستانی have اعمال شدن is 1μm. Figure2. Extracted groud poits from aerial images 3.3 Space data Stereo images had take by IRS-1C satellite. These images show Mashhad city of Ira. Size of each image is 496*496 pixels ad the overlap area of two images is 9 percets. There are 53 groud cotrol poits ad their similar poits i the images. Height of these poits are betwee m. 3.4 Results of simulated data All the experimets have doe i three differet cases of cotrol poit s height distributio. We used terrai depedet ratioal fuctios i these experimets. 115

3 The Iteratioal Archives of the Photogrammetry, Remote Sesig ad Spatial Iformatio Scieces. Vol. VII. Part B1. Beijig 28 Whe we used optimum samplig we had GCPs i all places that there was high slope chage. 3Ddistributio of cotrol poits were show i figure 3. Number of cotrol poits are 56 ad umber of check poits are 4. Model accuracies are show i table 1. Regards to this table we ca coclude: The accuracy of direct ratioal fuctios is more tha the iverse oe. But third order iverse ratioal fuctios have high accuracy too. Whe are usig oversamplig there are GCPs i all places that there are high chage slopes ad i places betwee them. Number of cotrol poits are 88 ad umber of check poits are 8. Distributio of cotrol poits were show i figure 4. Results of table 2 show that whe we used oversamplig the accuracies of the models are a little more from whe we used optimum samplig but these differeces are t high. Figure 3. Groud cotrol poits distributio i optimum samplig Whe deomiator of ratioal fuctios are t equal ( ) the accuracies are more tha whe they are equal. Higher degree ratioal fuctios have more accuracy tha low degrees. Figure 4. Groud cotrol poits distributio i over samplig N.O.GCPs N.O.CKPs RMSECKP RMSECNP 1orderRFM orderRFM orderRFM e e-5 2orderRFM(P2=P4) orderRFM(P2=P4) Iverse 1orderRFM Iverse 2orderRFM Iverse 3orderRFM Table 1. Results obtaied from simulated data i optimum samplig N.O.GCPs N.O.CKPs RMSECKP RMSECNP 1orderRFM orderRFM orderRFM e e-5 2orderRFM(P2=P4) orderRFM(P2=P4) Iverse 1orderRFM Iverse 2orderRFM Iverse 3orderRFM Table 2. Results obtaied from simulated data i over samplig 1151

4 The Iteratioal Archives of the Photogrammetry, Remote Sesig ad Spatial Iformatio Scieces. Vol. VII. Part B1. Beijig 28 Whe we use uder samplig there is t groud cotrol poits i all places of high slope ad height distributios of cotrol poits are restricted. Number of groud cotrol poits are 44 ad umber of check poits are 52. Figure 5 shows distributio of cotrol poits. Results of the experimets were show i table 3. Regards to this table we ca coclude: Accuracies of check poits have decreased i all models because cotrol poits have t good height distributio ad all poits are placed i low height levels. This shows that i uder samplig case accuracies for check poits are t high. Approximately accuracies of cotrol poits have improved i all models because all cotrol poits are early i the same height level ad fittig of ratioal fuctios to these poits was better. By icreasig ratioal fuctios order, their accuracies have improved such that third order ratioal fuctios ( ) have the best accuracy. Figure 5. Groud cotrol poits distributio i uder samplig N.O.GCPs N.O.CKPs RMSECKP RMSECNP 1orderRFM orderRFM orderRFM orderRFM(P2=P4) orderRFM(P2=P4) Iverse 1orderRFM Iverse 2orderRFM Iverse 3orderRFM Table 3. Results obtaied from simulated data i uder samplig For the better aalysis of residuals, first order ratioal fuctios error vectors of, ad elemets i optimum samplig, over samplig ad uder samplig were show i figure 6 ad 7. For seeig error vectors more obvious, some of these vectors were show. I all cases of samplig, ratioal fuctios errors i directio are much more tha ad directios, but the errors i directio are the same as directio. Regardig these figures it ca be see that maximum error is high, but we saw i previous tables, the average error of first order atioal fuctios is approximately 6cm for optimum samplig ad over samplig ad the error is early 1cm for uder samplig. (a) (b) (c) Figure 6. Plaimetric error vectors of first order ratioal fuctios i a) optimum samplig b) over samplig c) uder samplig 1152

5 The Iteratioal Archives of the Photogrammetry, Remote Sesig ad Spatial Iformatio Scieces. Vol. VII. Part B1. Beijig 28 (a) (b) (c) Figure 7. Height error vectors of first order ratioal fuctios i a) optimum samplig b) over samplig c) uder samplig 3.5 Results of aerial images I optimum samplig, we have chose 71 GCPs ad 234 check poits. Distributio of cotrol poits were show i figure 8. Regardig table 4 we ca coclude: Accuracy of ratioal fuctios i is more tha Accuracy of direct ratioal fuctios are more tha iverse ratioal fuctios for first ad secod orders but the differeces are ot high for third order ratioal fuctios. Figure 8. Groud cotrol poits distributio i optimum samplig N.O.GCPs N.O.CKPs RMSECKP RMSECNP 1orderRFM orderRFM orderRFM orderRFM(P2=P4) orderRFM(P2=P4) Iverse 1orderRFM Iverse 2orderRFM Iverse 3orderRFM Table 4. Results obtaied from aerial data i optimum samplig 1153

6 The Iteratioal Archives of the Photogrammetry, Remote Sesig ad Spatial Iformatio Scieces. Vol. VII. Part B1. Beijig 28 Figure 9. Groud cotrol poits distributio i over samplig Number of poits that we have chose i over samplig are 185 GCPs ad 12 check poits. 3D distributio of groud cotrol poits were show i figure 9. Regards to table 5, accuracies of all models are more tha optimum samplig. Other results are the same as optimum samplig Figure 1. Groud cotrol poits distributio i uder samplig Number of poits i uder samplig are 6 GCPs ad 245 check poits. Distributio of GCPs were show i figure 1. Results of experimets were reported i table 6. Regards to this table, accuracies of models are much degraded o check poits respect to over samplig ad uder samplig but accuracies are icreased o GCPs. Other results are the same as other samplig cases.. N.O.GCPs N.O.CKPs RMSECKP RMSECNP 1orderRFM orderRFM orderRFM orderRFM(P2=P4) orderRFM(P2=P4) Iverse 1orderRFM Iverse 2orderRFM Iverse 3orderRFM Table 5. Results obtaied from aerial data i over samplig N.O.GCPs N.O.CKPs RMSECKP RMSECNP 1orderRFM orderRFM orderRFM orderRFM(P2=P4) orderRFM(P2=P4) Iverse 1orderRFM Iverse 2orderRFM Iverse 3orderRFM Table 6. Results obtaied from aerial data i uder samplig First order ratioal fuctios error vectors of, ad elemets i optimum samplig, over samplig ad uder samplig were show i figure 11 ad 12. Some of these vectors were elimiated to the figures be more obvious. For aerial data like simulated data, ratioal fuctios errors i directio are much more tha ad directios for all three samplig cases but the amouts of errors i ad directios are the same. As ca be see, maximum error is high but by regards prvious tables, check poits average errors of first order 1154

7 The Iteratioal Archives of the Photogrammetry, Remote Sesig ad Spatial Iformatio Scieces. Vol. VII. Part B1. Beijig 28 ratioal fuctios are early 5.5cm for optimum samplig, 5cm for over samplig ad 15cm for uder samplig. (a) (b) (c) Figure 11. Plaimetric error vectors of first order ratioal fuctios i a) optimum samplig b) over samplig c) uder samplig (a) (b) (c) Figure 12. Height error vectors of first order ratioal fuctios i a) optimum samplig b) over samplig c) uder samplig 3.6 Results of space data Because of there was t eough cotrol poits, we have doe the experimets for oly uder samplig. The results were show i table 7. Number of poits are 3 GCPs ad 14 check poits. Regards to table 7 we ca coclude: Accuracy of first ad secod order ratioal fuctios are more tha third orders. Accuracies i is much more tha Direct ratioal fuctios are دقيقتر tha iverse oes N.O.GCPs N.O.CKPs RMSECKP (m) RMSECNP (m) 1orderRFM orderRFM orderRFM orderRFM(P2=P4) orderRFM(P2=P4) Iverse 1orderRFM Iverse 2orderRFM Iverse 3orderRFM Table 6. Results obtaied from space data i uder samplig Plaimetric ad height error vectors were show i figures 13 ad 14. For seeig error vectors more obvious, some of these vector were elimiated. 1155

8 The Iteratioal Archives of the Photogrammetry, Remote Sesig ad Spatial Iformatio Scieces. Vol. VII. Part B1. Beijig Paderes, Jr., F. C., Mikhail, E. M., Fagerma, J. A., Batch ad o-lie evalutio of stereo SPOT imagery, Proceedigs of the ASPRS-ACSM Covetio, Baltimore, April, pp ag,., 2 : "Accuracy of ratioal fuctio approximatio i photogrammetry," Proceedigs of ASPRS Aual Covetio, Washigto D.C. May Figure 13. Plaimetric error vectors of first order ratioal fuctios 4- Baltsavias, E., Pateraki, M., ad hag, L., 21, Radiometric ad geometric evaluatio of IKONOS Geo images ad their use for 3D buildig modelig. Proceedigs of Joit ISPRS Workshop High Resolutio Mappig from Space 21, September (Haover: Iteratioal Society of Photogrammetry ad Remote sesig (CD-ROM)), pp Touti, T., Cheg, P., 2. Demystificatio of IKONOS Earth Observatio Magazie (EOM), 9(7): Figure 14. Height error vectors of first order ratioal fuctios 4 CONCLUSION For the aalysis of depedece of ratioal fuctios ad height distributio of cotrol poits, their accuracies were aalyzed i three differet cases of cotrol poits height distributio. At last we saw that ratioal fuctios are depedet to height distributio of cotrol poits ad accuracies i uder samplig are much degraded but because of the accuracies i over samplig ad optimum samplig are t too differet, usig over samplig for ratioal fuctios are ot ecessary REFERENCES 1- McGloe, C., 1996 :"Sesor modelig i image registratio," Digital Pgotogrammetry : A Addedum (C. W. Greve, editor), America Society for Photogrammetry ad Remote Sesig, Bethesda, Marylad, pp OpeGIS Cosortium, The OpeGIS Abstract Specificatio Topic 7: The Earth Imagery Case, 7- NIMA., 2 : "The compedium of cotrolled extesios (CE) for the atioal imagery trasmissio format (NITF), versio Tao, C. V., Hu,., 2: "A Comprehesive Study o the Ratioal Fuctio Model for Photogrammetric Processig," Photogrammetric Egieerig ad Remote Sesig. 9- Tao, C. V., Hu,., 21a : "The ratioal fuctio model a tool for processig high-resolutio imagery," Earth Observatio Magazie (EOM), 1(1) : Tao, C. V., Hu,., 21b : "3-D Recostructio Algorithms based o the Ratioal Fuctio Model," Proceedig of Joit ISPRS Workshop o "High Resolutio Mappig from Space 21," Haover, September Madai, M., 1999 : "Real-time sesor-idepedet positioig by ratioal fuctios," Proceedigs of ISPRS Workshop o "Direct versus idirect Methods of Sesor Orietatio," November, Barceloa, Spai, pp