IMPROVEMENT OF THE STABILITY SOLVING RATIONAL POLYNOMIAL COEFFICIENTS

Transcription

1 IMPROVEMEN OF HE ABIIY OVING RAIONA POYNOMIA COEFFICIEN Xanyong n a,*, Xuxao Yuan a a chool of Remote ensng and Informaton Engneerng, Wuhan Unversty, 19 uoyu Road, Wuhan 40079, Chna [email protected] Commsson I, WG I/5 KEY WORD: Hgh Resoluton atellte Imagery; ensor Model; Orentaton; Regularzaton; Accuracy ABRAC: he ratonal functon model (RFM) utlzed for hgh resoluton satellte magery (HRI) provdes a transformaton from mage to obect space coordnates n a geographc reference system. Compared wth the rgorous model based on the collnearty condton equaton or the affne model, the RFM wth 80 coeffcents would be over parameterzed. hat would result n an ll-condtoned normal equaton. khonov regularzaton s often used to resolve ths problem, and many applcatons have verfed ts servceablty. hs paper wll detal the method for regularzaton parameter selecton. However, khonov regularzaton makes the two sdes of equaton unequal, resultng n a based soluton. An unbased method - he Iteraton by Correctng Characterstc Value (ICCV) was ntroduced, and a strategy to resolve the ll-condtoned problem for solvng ratonal polynomal coeffcents (RPCs) was dscussed n ths paper. he tests wth PO-5 and QuckBrd magery were accomplshed. he emprcal results have shown that our methodology can effectvely mprove the condton of the normal equatons. 1. INRODUCION nce the launch of the IKONO II satellte, the ratonal functon model (RFM) has ganed consderable nterests n photogrammetrc communty. paceimagng Company provdes the RFM to users nstead of the physcal sensor model, subsequently, DgtalGlobe Corporaton provdes the RFM together wth the strct geometrc model n order to satsfy dfferent users. he RFM has been unversally accepted, and valdated, as an alternatve sensor orentaton model for hgh resoluton satellte magery (HRI). he RFM s an approxmaton of the rgorous sensor model, va a number of control ponts. hen t could be utlzed n the photogrammetrc process nstead of the complex rgorous sensor model. It would be a part of the standard mage transfer format, and t s becomng a standard way for economcal and fast mappng from remotely sensed magery. he key of the RFM s to gan accurate ratonal polynomal coeffcents (RPCs). Compared wth the rgorous model based on the collnearty model or the affne model, the RFM wth 80 coeffcents would be over parameterzed (Fraser et al., 005). hat may cause the desgn matrx to become almost rank defcent because of the complex correlaton among RPCs. It may result n numercal nstablty n the least squares adustment, or even producng wrong solutons. he regularzaton technque was often suggested to tackle the possble ll-condtoned problem durng the adustment (ao and Hu, 001a). It has been proved to effectvely mprove the condton of the normal equatons. But the determnaton of the regulaton parameter has stll been consdered to be a challenge. Regularzaton parameter selecton s crucal to the regularzaton technque. here are several methods for the optmal parameter determnaton, ncludng rdge trace method, -curve crteron, generalzed cross valdaton (GCV) method, ordnary cross valdaton (OCV) method, and so on. he effects may be totally dfferent when we use dfferent methods. he -curve, GCV and OCV were compared by Cho et al. (007). In practce, rdge trace method s wdely used for ts smpleness, where solutons are computed for a large number of dfferent regulaton parameters, selectng the best one by sutable heurstcs (ao and Hu, 001a). However, rdge trace method can not obtan the optmal parameter, and t s nconvenent for automatc computaton. he authors try dfferent regulatons for the RPCs computaton. he -curve crteron has been proved to be effcacous. Despte regularzaton technque gves a good result, t makes the two sdes of equaton unequal by mposng constrants to the dagonal elements of the normal equaton matrx, resultng n a based soluton. o, we wll ntroduce an unbased method, the Iteraton by Correctng Characterstc Value (ICCV). hs method s smple, and t was put forward more than ten years ago, but stll not wdely used. he ntal values wll be the man factor that affects the result. And ths paper wll suggest two ways to set ntal values, ust for RPCs computaton. Accurate RPCs are crucal to the RFM model, whch drectly determnes whether t could replace the physcal sensor model to accomplsh the photogrammetrc process. And the ll-condtoned normal equaton would be the man problem. hs paper amed at fndng a proper method to resolve the possble ll-condtoned problem, gettng an accurate soluton. Intally, we wll revew the basc model and the methods ncludng the RFM soluton, the terran-ndependent and terran-dependent computatonal scenaros. he regulaton technque and the method of regularzaton parameter selecton are then addressed, focusng on the -curve method. he unbased ICCV s followed. he results of expermental tests wth PO-5 and QuckBrd magery are then dscussed. Fnally, we wll suggest a strategy for RFM computaton accordng to the experments and comprehensve analyss of the characterstcs of the varous methods. 711

2 he Internatonal Archves of the Photogrammetry, Remote ensng and patal Informaton cences. Vol. XXXVII. Part B1. Beng 008. HE RAIONA FUNCION MODE. RFM oluton.1 he RFM Model he RFM relates obect pont coordnates to mage pxel coordnates n the form of ratonal functons that are ratos of polynomals. For the ground-to-mage transformaton, the defned ratos have the forward form (OGC, 1999): Den Den ( U, V, W ) l n = (1) Den ( U, V, W ) ( U, V, W ) s n = Den ( U, V, W ) 1 ( U, V, W ) = a1 V U W VU 6VW 10W 1VU + 16U 19U W + ( U, V, W ) = b1 + bv + ( U, V, W ) = c1 + c V ( U, V, W ) = d + d V d UW 8V 9U 11UVW 1V a14vw 15V U 17UW 18V W a 0W K + b19u W + b0w + K + c19u W + c 0W + K + d 19U W + d 0W Here, a and are the 80 RPCs; and are commonly, b, c b d1 set to 1. ( l n, s n ) are the normalzed lne and sample ndex of the pxels n mage space, whle ( U, V, W ) are normalzed obect pont coordnates. hat s: l neoffset ln = necale s ampleoffset sn = amplecale φ attudeoffset U = attudecale λ ongtudeoffset V = ongtudecale h HeghtOffset W = Heghtcale () () wo methods have been developed to solve for the RFM, drect and teratve least-squares solutons (ao and Hu, 001a). Here, the drect least-squares soluton of RFM s gven as follows: Fs = ( U, V, W ) s Den ( U, V, W ) = 0 (4) F = ( U, V, W ) l Den ( U, V, W ) = 0 l V = BX, P (5) 1 X = ( B PB) B P (6) where Fs Fs Fs Fs a b c d B =, Fl Fl Fl Fl a b c d ( = 1,,0; =,,0) [ a ] b c d [ d s b l] = [ s ] X =, =. 1 1 l P s the weght matrx, and t s usually set as dentty matrx.. Approaches of Determnng RPCs here are terran-ndependent scenaro usng known physcal sensor model and terran-dependent scenaros usng ground control ponts. he terran-ndependent scenaro s to use the onboard ephemers and atttude data. Wth the physcal model avalable, a vrtual control grd coverng the full extent of the mage and the entre elevaton s generated. he RPCs are estmated usng a least-square soluton wth the mage grd ponts and the correspondng obect grd ponts (ao and Hu, 001a, 001b). For the terran-dependent scenaro, a number of ground control ponts are collected for the RPCs computaton. At least 9 ground control ponts are needed per mage to solve 78 RPC coeffcents, excludng the constant parameters b and. And 1 d1 the soluton s hghly dependent on the actual terran relef, the dstrbuton and the number of GCPs (ao and Hu, 001a, 001b). Here, ( l, s) are the mage lne and sample coordnates; ( φ, λ, h) represent lattude, longtude, heght; the offsets and scales normalze the coordnates to [-1,1], mnmzes the ntroducton of errors durng computaton. he RFM has nne confguratons wth some varatons, such as subset of polynomal coeffcents, equal or unequal denomnators. Also, t has forward and backward forms (ao and Hu, 001a). Generally speakng, the RFM refers to a specfc case that s n forward form, has thrd-order polynomals wth unequal denomnators, and s usually solved by the terran-ndependent scenaro..1. REGUARIZAION ECHNIQUE he RPCs may dsplay very hgh correlaton between coeffcents. hat would be a potental problem for obtanng a stable soluton. he desgn matrx s usually ll condtoned n the experments (ao and Hu, 000). Even for the well condtoned observaton equatons, regularzaton can mprove the accuracy of the RPCs, and help produce well-structured RPCs, especally for the thrd-order RFM (Hu and ao, 004). Rdge Regresson Rdge regresson (Rdge estmate), a part of regularzaton technque, s a based estmaton for nonorthogonal problems (Hoerl and Kennard, 1970). It carres out by addng a small postve quantty to the dagonal of B B. Rdge regresson 71

3 he Internatonal Archves of the Photogrammetry, Remote ensng and patal Informaton cences. Vol. XXXVII. Part B1. Beng 008 obtans based estmates wth smaller mean square error. Rdge regresson s defned as follow: Xˆ 1 ( k ) = ( B PB + ki ) B P (7) k s rdge parameter or regularzaton parameter, usually a small postve quantty; I s dentty matrx; X ˆ ( k ) s rdge regresson estmaton... Rdge trace method for parameter determnaton olutons are computed for a set of dfferent k values. And the best k s selected by sutable heurstcs, for the least error at check ponts (ao and Hu, 1970). hs method s very smple, and t s wdely used. -curve crteron for parameter determnaton he -curve s a log-log plot of the norm of a regularzed soluton versus the norm of the correspondng resdual (fttng error) as the regularzaton parameter s vared (Hansen, 199; Rodrguez and hes, 005). -curve s presented as: ( η ( k), ξ ( k)) = (lg BXˆ,lg Xˆ ) (8) he curve s -shaped: approxmately vertcal for small k, and approxmately horzontal for large k, wth the corner provdng the optmal regularzaton parameter. o the obect s to fnd out the pont wth bggest curvature: ξ η ξ η ξ η k = arg max (9) ( ξ + η ) ξ = the frst and second dervatve of ξ on k; η = the frst and second dervatve of η on k. In practcal computaton, curve fttng s often used to obtan the -curve. he -curve crteron s able to recognze correlated errors, whle the GCV method may fal to do so. hat s essentally because the -curve crteron combnes nformaton about the resdual norm wth nformaton about the soluton norm, whereas the GCV method only uses the nformaton about the resdual norm. he research done by Cho et al (007) shows us that the -curve method performed better than OCV or GCV, partcularly for hgh nose levels. he -curve method s found to be less susceptble to producng large reconstructon errors but t tends to over-regularze the soluton n the presence of low nose, leadng to under-estmates of the forces. 4. HE IERAION BY CORRECING CHARACERIIC VAUE Regularzaton technque mposes constrants to the dagonal elements of the normal equaton, resultng n a based soluton. o, we wll ntroduce an unbased method - the Iteraton by Correctng Characterstc Value (Wang et al. 001) for RPCs computaton. k k Here s the norm functon, B PBXˆ = B P Add Xˆ to both sdes, ( B PB + I)Xˆ = B here are mode: Xˆ P + Xˆ on both sdes, so t should be resolved n teratve ˆ ( k ) 1 ( ) ( ˆ ( k 1 X = B PB + I B P + X ) ) (1) 1 If we set q = (B PB + I), hen the (1) could be wrtten as: ˆX (0) ˆ ( k ) k k ˆ (0) X = ( q + q + + q ) B P + q X (14) = ntal values of the solutons. Eqs. (1) and (14) are the expressons of the teraton by correctng characterstc value. he convergent and unbased propertes are dscussed by Wang Xnzhou et al. (001). he ICCV carred by teraton, ntal values should be offered for the teraton, and they have an mportant mpact on the result or even determne the success of the method. he drect least-squares solutons are usually used as the ntal values. Unfortunately, when the ll-condton happens, t s possble that the soluton s so bad that the teraton s unconvergence. o, here we suggest another way specal for the RPCs soluton. Consderng that the thrd-order RPCs are closed to zero, the ntal values may set to zero. We wll test t n the experments. 5. E REU AND EVAUAION 5.1 Desgn and ests he tests have been desgned for these purposes: o evaluate the numercal stablty of the drect least squares soluton. he condton number s cursorly employed to measure the condton of the desgn matrx. he number s much bgger when the functon s ll-condtoned. o compare the performances of the regularzatons for the RPCs computaton. We choose the wdely-used rdge trace method and the -curve crteron. Manly to evaluate the potental of the unbased ICCV method for the RPCs computaton, and to test the mpact of the ntal values. Intal values are set by zero and least square soluton respectvely. o fnd out an effectve strategy to tackle the possble ll-condtoned problem. Here we confne the experments to the thrd-order RFM wth 80 coeffcents, based on the terran-ndependent scenaro. Wth the rgorous sensor model establshed and the elevaton range obtaned from a cursory DEM, the -D grd of obect ponts was generated, wth 5 constant elevaton planes each wth 10 by 10 grd ponts. Whle the check grd conssts of 10 constant elevaton planes each wth 0 by 0 grd ponts. o there are 500 control ponts and 4000 check ponts. he fttng accuracy s measured n mage both at control ponts and check ponts. Frstly, the mage poston of the grd ponts s 71

4 he Internatonal Archves of the Photogrammetry, Remote ensng and patal Informaton cences. Vol. XXXVII. Part B1. Beng 008 calculated by the obtaned RFM. hen the dfferences between the pxel coordnates of the orgnal grd ponts and those from the RFM are calculated for evaluaton. he accuracy determnaton s qute the same as mentoned by Grodeck and Dal (001). where PO-5 magery s employed as an example. In the experment, RPCs are computed for a number of k wth 10 dfferent orders of magntude varyng from 10 to 10 1, to determne the order of magntude. hen employ more k around the order of magntude, and choose the one that has the smallest error at check ponts. hs method can not select the best parameter, and t s not convenent for automatc computaton. For the -curve method, the curve s shaped lke. And the corner pont on the -curve that has maxmum curvature corresponds to the optmal parameter. hs method can offer an exact parameter automatcally, wthout the need to plot the -curves. In the experments, the parameter determned by 7 -curve crteron s for PO-5 data, and for QuckBrd data. Fg. 1 D obect grd generated for solvng RPCs 5. est data sets ao and Hu (001a) tested wth the aeral photograph data and PO data wth szes of 6000 by In order to evaluate the fttng accuracy of the dfferent methods for HRI, we choose PO-5 and QuckBrd magery. Respectve ground pxel szes for testfeld magery were 5 m for PO-5, and 70 cm for the QuckBrd. Further detals regardng the test-range are gven n able 1. Data set Ground pxel ze (m) Image sze (pxel) Elevaton range (m) PO ~7 Fg. Determnng rdge parameter usng rdge trace method QuckBrd ~1194 able 1. Informaton of the data sets 5. Results and evaluaton All the methods are tested on both the PO-5 and QuckBrd magery. he RME and the maxmal errors n the magery at the control ponts and the check ponts are lsted n able for PO-5 data, able for QuckBrd data. he condton numbers of the norm functon, before and after regulaton, are also lsted n the tables. here s not an absolute crteron for exactly udgng that the norm functon s ll-condtoned or not, and how ll-condtoned t s. Generally speakng, the condton number s helpful, the bgger t s, the worse the condton s. Based on able and able, the condton numbers are bg for both mages, for PO-5 data, and for QuckBrd data. he drect least square solutons are not very good, out of sub-pxel, especally for the QuckBrd data, the RME at check ponts arrves at pxels, and the maxmal error s as bgger as pxels. herefore, the drect least square solutons here could not be the fnal RPCs whch would substtute the physcal sensor model. Determnng regularzaton parameter usng rdge trace method s shown n Fg., and -curve method s shown n Fg., Fg. Determnng rdge parameter usng -curve method By comparson, the regulaton by -curve crteron, made very sgnfcant mprovements n terms of accuracy. After the use of regulaton, the condton numbers are smaller than the orgnal 14 9 one, for the PO-5 data, reducng from 10 to 10, and for 1 8 the QuckBrd data, reducng from 10 to 10. Except the hgh accuracy, -curve method shows very strong stablty based on more data sets. he ICCV carres out by teraton. In the experments, the ntal values are set by zero and least square soluton respectvely. he teratve threshold value set as For the zero ntal values condton, the results are pretty good based on the tables that the accuracy s so close to and even better than the results 714

5 he Internatonal Archves of the Photogrammetry, Remote ensng and patal Informaton cences. Vol. XXXVII. Part B1. Beng 008 of the -curve method. It converged after a few tmes teraton, even the teratve threshold value set to 10 9, t converges quckly. For the least squares soluton ntal value condton, we can note that t s good for PO-5 data but nvald for the QuckBrd data. It converged only after one tme teraton even that the teratve threshold value s strct. Coercve teraton s also nvald for the mprovement of the accuracy. More tests should be done wth the ICCV. It s worth of pontng out that the computaton of the ICCV s very smple and fast, that s because there s no need to determne the regulaton parameter, and no need to nverse the matrx every tme durng the teraton. And the structure of the soluton by ICCV s as good as that by regulaton. Anyway ICCV s a potental way to overcome the ll-condtoned problem for the RFM soluton. Approaches Condton number Iteraton tmes Errors at CNPs (pxels) Errors at CKPs (pxels) RME Max RME Max Rdge Estmate east squares () -curve crteron ICCV solutons as ntal values Zero as ntal values able. RME and Max errors n mage wth the PO-5 data Approaches Condton number Iteraton tmes Errors at CNPs (pxels) Errors at CKPs (pxels) RME Max RME Max Rdge Estmate east squares () -curve crteron ICCV solutons as ntal values Zero as ntal values able. RM and Max errors n mage wth the QuckBrd data he DgtalGlobal Corporaton provde the RPC fle to the users. As a comparson, we choose 9 ground control ponts to checkout the RPCs, calculatng the dfferencec between the mage coordnates of the GCPs and that from the RPCs. And the errors at the GCPs are lsted n able 4. From the table we can see the RPCs by the -curve and the ICCV are so close accordng to the accuracy, and they are slghtly better than the RPC provded by the corporaton. RPC By -curve Errors at GCPs (pxels) m l m s m ls RME Max By ICCV RME Max Provded by RME DgtalGlobal Max able 4. RM and Max errors n mage at GCPs 6. CONCUION nce the paceimagng Company provded the RPCs to the end users and the servce provders, the RFM has been wth us for eght years, and a lot of researches show us that t s a useful tool for explotng hgh resoluton satellte mages. ubsequently, the DgtalGlobal Corporaton provdes the RPCs together wth the physcal sensor model, and more magery vendors may adopt the RFM, provdng a way for economcal and fast mappng from HRI. he am of ths paper s to suggest a proper way to resolve the ll-condtoned problem for the RFM soluton. Regulatons mprove the stablty of the nverse matrx evdently and produce a well structure RPCs. And the -curve method performs well n the experments, beng accurate and stable. he ICCV s unbased, smple, fast, and accurate, and the dea that set the ntal values as zero acts well for the RFM soluton. Both the methods show good effects, mprovng the accuracy of the solutons, and ameloratng the RPCs structure. Consderng that the -curve method has the rsk of over-regularzng the soluton when the ll-condton s slght, though not happened n the experments, the ICCV should be the frst choce. Fnally, we suggest a strategy, that -curve method work for hgh level ll-condton and ICCV for low. Even for the well condtoned 715

6 he Internatonal Archves of the Photogrammetry, Remote ensng and patal Informaton cences. Vol. XXXVII. Part B1. Beng 008 desgn matrx, the ICCV s really helpful to mprove the accuracy. ACKNOWEDGEMEN hanks for the supportng from the 97 Program of the People s Republc of Chna under Grant 006CB7010 and the Natonal Natural cence of Chna under Grant REFERENCE Cho, H.G., hte, A.N., hompson, D.J Comparson of methods for parameter selecton n khonov regularzaton wth applcaton to nverse force determnaton. Journal of ound and Vbraton, 04, (-5), pp Hoerl, A.E., Kennard, R.W., Based Estmaton for Non-orthogonal Problems. echnometrcs, 1(1), pp Hansen, P.C., 199. Analyss of Dscrete Ill-posed Problems by Means of the -curve. IAM Revew, 4(4), pp Hu, Y., ao, C.V., Crotoru, A., 004. Understandng the ratonal functon model: methods and applcatons, IAPR, 1- July, Istanbul, vol. XX, 6 p. OpenGI Consortum (OGC), he opengi abstract specfcaton-opc 7: Earth magery. Rodrguez, G., hes, D., 005. An Algorthm for Estmatng the Optmal Regularzaton Parameter by the -curve. Rendcont d Matematca, 5(1), pp Fraser, C.., Dal, G., Grodeck, 005. ensor orentaton va RPCs. IPR Journal of PR, 60, pp ao, C.V., Hu, Y., 000. Investgaton of the ratonal functon model. Proceedngs of APR Annual Conference, Washngton, DC. May pages. protected/procs/asprs/asprs000/pdffles/papers/09.pdf (accessed 1 Mar. 008). ao, C.V., Hu, Y., 001a. A comprehensve study on the ratonal functon model for photogrammetrc processng, Photogrammetry Engneerng and Remote ensng, 67(1), pp ao, C.V., Hu, Y., 001b. he ratonal functon model: a tool for processng hgh resoluton magery. Earth Observaton Magazne (EOM). 10(1). pp Grodeck, J., Dal, G., 001. Ikonos geometrc accuracy, Jont IPR Workshop on HRM from pace, 19-1 ept., pp Wang Xnzhou, u Dngyou, Zhang Qanyong, Huang Halan he Iteraton by Correctng Characterstc Value and ts applcaton n surveyng data processng. Journal of Helongang nsttute technology. 15(), pp