Combined Bundle Adjustment with Distances and GPS Observations**

Transcription

1 GEOMATICS AND ENVIRONMENTAL ENGINEERING Volume 9 Number Jaub Kolec* Combned Bundle Adjustment wth Dstances and GPS Observatons** 1. Introducton Today, automatc mage orentaton and automatc mage-based dense reconstructon algorthms are often mplemented n professonal photogrammetrc software. At the same tme progress n computer vson brngs many advantages to everyone who s dealng wth mage-based object modelng. A number of free, often on-lne software for automatc sparse or dense reconstructon have appeared durng the last years [8]. The majorty of them allow only for fully automatc reconstructon [1, 7] whch s good for amateurs but bad for specalsts, as controllng some mportant aspects of processng, le camera calbraton s not avalable. As a result generated models are out of scale and proper angular orentaton, and cannot be used as metrc products. Addressed problems have been overcome n professonal photogrammetrc software where the bundle adjustment wth control ponts s a standard. However, control ponts collecton s a tme consumng process even for small objects. Besdes t nvolves usng addtonal survey equpment. As an alternatve approach the drect geo-referencng methods seem to be a very attractve soluton. Nowadays a number of cameras are equpped wth ntegrated GPS, or have the nterface open for an external recever. Smartphones and tablets as a standard are equpped wth GPS, MEMS nertal sensors and cameras whch stll mprove ther resoluton and mage qualty. Low-cost devces provdng mage drect geo-referencng could be potentally utlzed to mprove the accuracy of obtaned photogrammetrc products, brngng new functonalty to free, mage-based modellng software. The only problem wth low-cost GPS recevers s the low accuracy of postonng. The postonng errors can be n the consderable extend compensated by the photogrammetrc ervatons [2, 5] however not completely. GPS measurements of projecton centers, when conducted n the networ wth good controllablty, allow * AGH Unversty of Scence and Technology, Faculty of Mnng Surveyng and Envronmental Engneerng, Kraow, Poland ** Ths paper was supported wthn the Faculty s research project no /13 51

2 52 J. Kolec for the estmaton of angular elements of external orentaton usng only low cost nertal sensors [6] or even wthout usng nertal sensors at all [2]. Loong for cheap and smple approaches to mprovng accuracy n small-area combned mage networs one may thn about drect dstance measurements. Ellum [3] tred to nclude sngle dstance ervatons nto a small terrestral mage networ achevng some mprovement n fnal accuraces. Kolec [5], smlarly to Ellum, ddn t notce a consderable decrease of errors. However, both authors used survey-grade GNSS recevers n ther systems. Kolec [5] n hs smulatons proves that dstance ervatons have the potental to mprove accuracy of mage orentaton only f lower accuracy GPS s nvolved. In the case of usng a survey-grade recever, dstance measurements are smply redundant. In ths paper more experments and smulatons are addressed to gve deep nsght nto the nature of dstance ervatons and ther mpact on adjustment results. 2. Functonal Model The functonal model of appled bundle adjustment conssts of three nds of equatons. The majorty of ervatons are the mage coordnates, brngng nto the model equatons of collnearty (1) n ts usual form. The erved external orentaton parameters are ncluded usng equatons (2) and (3). The dstance ervatons are ncluded by addng nto the model equaton (4): r X X r Y Y r Z Z x v x c x x 0 r X X r Y Y r Z Z r X X r Y Y r Z Z y v y c y y 0 r X X r Y Y r Z Z X v X 0 X0 0 Y v Y 0 Y0 0 Z v Z 0 Z0 0 (1) (2) v v v (3) d v X X Y Y Z Z j d j j j (4) j

3 Combned Bundle Adjustment wth Dstances and GPS Observatons 53 where: X, Y and Z stand for projecton center coordnates, X 0 0 0, Y Z are the object pont coordnates, α, ν and κ are the angles of mage orentaton, the r coeffcents come from the rotaton matrx, c, x 0 and y 0 are the nternal orentaton parameters. As gven n the equaton (4), the dstances are always measured between object ponts that must also be mared n the mages as te ponts. 3. Testng 3.1. Data Preparaton The am of ths experment s the evaluaton of adjustment accuracy wth and wthout dstances n deal condtons. Ths means that the true external orentaton parameters and ther accuracy are nown. Besdes the dstance ervatons that are free of errors can be provded. However t should be notced that the nfluence of dstances may depend on the overall networ confguraton and the dstance spatal dstrbuton. The examned networ s shown n Fgure 1. It has a closed-loop confguraton to provde good controllablty of ervatons. Fg. 1. Test mage networ wth camera statons and dstances. Te and chec ponts are mared n blac and red respectvely

4 54 J. Kolec Durng the preparatory wors ths networ was adjusted wth 10 control ponts, to calculate the truth external orentaton parameters. Then the 11 dstances were calculated usng the adjusted coordnates of te ponts. As t can be seen n Fgure 1 these dstances are unformly dstrbuted. However t should be clear that some of them may be hard to determne by a feld survey. The adjusted projecton centers (PC) coordnates can be treated as true values. In ths experment these values are corrupted by the zero-mean random Gaussan nose wth the σ parameter changng from 5 to 300 mm n the ntervals of 5 mm. To mprove the relablty of the smulatons 40 sequences of such a random number were generated for each projecton center coordnate. Ths means that the networ can be adjusted wth the corrupted coordnates 40 tmes for each σ. The number of 40 seems to be a reasonable compromse between calculaton tme and the relablty of the results. Altogether, for all values of σ, there are 2,400 adjustments n each adjustment scenaro. For each adjustment the standard devatons for chec ponts are calculated and averaged wthn bns of 40. One should expect the averaged seres of errors wll form the ascendng lne n the σ over RMSE plot Smulaton Scenaros Havng the data prepared, the smulatons can be started. The followng scenaros of adjustment are gong to be carred out: A0: 2,400 adjustments wthout dstance ervatons and wth constant a pror error of projecton center coordnates, B0: 2,400 adjustments wthout dstance ervatons, wth a pror errors of projecton centers changng accordng to values of sgma n the random number seres, A1: 2,400 adjustments wth only one dstance and constant a pror error of projecton center coordnates, B1: 2,400 adjustments wth only one dstance and a pror errors consstent wth the values of sgma. A11: 2,400 adjustments wth 11 dstances and constant a pror error of projecton center coordnates, B11: 2,400 adjustments wth 11 dstances and a pror errors consstent wth the values of sgma. In scenaros A the a pror errors of PC coordnates are 4 cm, 4 cm and 5 cm respectvely for X, Y and Z. The aforementoned values seem to be the reasonable estmaton of the expected accuracy of the real-tme nematc GPS survey n the bult-up areas. The hghest possble value of σ n the B scenaros (.e. 30 mm) corresponds to the accuracy that may be acheved usng pos-processed, code-only GPS surveys provded the good satellte vsblty.

5 Combned Bundle Adjustment wth Dstances and GPS Observatons Results 4.1. Accuracy Plots showng the relatve accuraces of bundle adjustment for all adjustment scenaros are presented n Fgure 2 (scenaros A) and Fgure 3 (scenaros B). It can be notced (Fg. 2) that underestmaton of a pror errors of geo-referencng results n systematc accuracy drop. Includng dstances helps to get slghtly better results. However, the effect of usng 11 dstances nstead of 1 seems to be surprsngly nsgnfcant. To mae the overall concluson one can examne the 3D poston error (Fg. 2d) and notce that ncludng dstances provdes about 20 25% accuracy mprovement. However t s clear that ths rule may not hold n networs wth dfferent mage number and dfferent geometry. a) b) c) d) Fg. 2. Relatve accuracy of bundle adjustment wth constant errors of PC coordnates n scenaro A; blac: no dstances, red: one dstance, blue: eleven dstances In practce such consderable under (or over) estmaton of a pror errors s not lely to happen, but the experment shows some general phenomena, and llustrates the mportance of assessment of ervatons accuracy n the context of adjustment. The extent n whch a pror accuracy s over estmated s llustrated n Fgure 4, where the change of standard devaton of an ervaton of unt weght s shown.

6 56 J. Kolec a) b) c) d) Fg. 3. Relatve accuracy of bundle adjustment wth truth a pror errors of PC coordnates n scenaro B; blac: no dstances, red: one dstance, blue: eleven dstances When the weghs n adjustment process are consstent wth the true accuraces of PC coordnates, the dstance ervatons stop the chec pont errors from ncreasng (Fg. 3). Some remanng errors are stll present (about 3 mm for each coordnate) possbly due to ntal naccuracy of chec ponts, nevertheless the accuracy seems not to decrease, even though the PC errors reach 30 cm. However f the dstances were excluded from the adjustment, the chec pont errors would ncrease notceably (blac lnes n Fgure 3c, d). What s more nterestng, the number of erved dstances has no nfluence on accuracy. The blue and red lnes n the plots shown n Fgure 3 almost concde. It seems that n the assumed condtons of smulaton of scenaro B (true EO and true dstances consstent wth the ervatons, true a pror accuraces are nown), erved dstances do not help to correct the networ geometry locally. Instead the dstance ervatons correct the scale of the networ, whch n the case of no dstances, would be dsturbed by the erroneous erved PC coordnates. The rotaton of the networ may not be dsturbed suffcently to cause sgnfcant chec pont errors. In scenaro A the value of the standard devaton of unt weght s not nfluenced by ncreasng the errors of the erved PC coordnates (Fg. 4). However ts value s lower than 1.0 due to the overestmated errors of the mage coordnates of the te ponts.

7 Combned Bundle Adjustment wth Dstances and GPS Observatons 57 Fg. 4. Dependency of the standard devaton of an ervaton of unt weght from the true error of erved PC coordnates; red fxed a pror errors of PC coordnates, blue weghts of erved PC coordnates consstent wth true errors 4.2. Scale Further calculatons were carred out n order to show the role the erved dstance plays n the bundle adjustment of the test networ, especally ts nfluence on the scale. For scenaros B0 and B1 the 3D 7 degrees-of-freedom transformaton was calculated accordng to the Horn closed-form soluton [4]. Scale checng was conducted for 15 networs n scenaro B0 and for 15 networs n scenaro B1. For each scenaro checed networs were grouped n 3 bns, each contanng 5 networs wth the same errors of PC coordnates. The values of the PC coordnate error and correspondng average scale s gven n Table 1. Accordng to the termnology of Horn, as the rght (target) data set the true chec pont coordnates were utlzed. For scenaro B1 the devaton of the scale from the unty s ept close to the level of 10 4 rrespectve of PC error. If the dstance ervaton s removed (scenaro B0) ths devaton ncreases reachng the level of 10 3 for the 15 cm error of PC coordnates and 10 2 for the 30 cm error. Provded results show that usng the dstance ervaton can help to fx the scale of the networ, whch may be dsturbed to a large extend by erroneous drect geo-referencng. Table 1. Results of scale parameter estmaton for networs wth three dfferent error levels of PC coordnates adjusted wthn scenaros B0 and B1 B0 B1 PC coordnates error [mm] Average devaton from unty (absolute values) [untless]

8 58 J. Kolec 4.3. Redundancy Fgure 5 allows further examnaton of the role of the erved dstances n the adjustment. If the constant weghts are appled to PC coordnates, the redundancy of erved dstances (one dstance n case of Fgure 5a) are also constant (Fg. 5 red lne). a) b) Fg. 5. Dependency of mean erved dstance redundancy from the true error of erved PC coordnates: a) scenaros wth one dstance (A1, B1); b) scenaro wth 11 dstances (A11, B11); red fxed a pror errors of PC coordnates, blue weghts of erved PC coordnates consstent wth true errors In Fgure 5a t can be erved that the percentage of ervaton that s utlzed n adjustment (complement of redundancy to the value of 1.0), n the scenaro B1 ncreases from 10% to 99%. It means that for large PC coordnate errors, almost 100% of ths sngle dstance ervaton s utlzed n the adjustment and the proper scalng of the networ s almost fully dependent on the accuracy of ths measurement. In the Fgure 5b the mean redundancy has values closer to unty, as the total dstance redundancy s dstrbuted to sngle ervatons. The controllablty of each dstance s therefore better. What s qute nterestng, n Fgure 5a and n Fgure 5b the blue lne crosses the red lne around the abscssa 40 mm the value of PC X and Y coordnate error assumed for scenaros A. 5. Conclusons The experments presented n ths paper show some phenomena that occur n the adjustment of dstance-supported photogrammetrc networs wth drect geo-referenced mages. Presented smulatons allow for examnng the nteracton that taes place between erved projecton center coordnates and dstances for the mage networ wth favorable geometrcal confguraton (closed loop) and good controlablty. In that case the dstance ervatons are error-free, they can play a great role n stablzng the scale of networ, that n case of erroneous geo-referencng can be easly dsturbed. However one very mportant aspect should be ept n

9 Combned Bundle Adjustment wth Dstances and GPS Observatons 59 mnd, namely the proper weghtng of the erved PC coordnates (and potentally angles). Underestmaton of a pror errors may lmt the correctve role of the dstances and ncrease adjustment errors. Proper weghtng of certan groups of ervatons may be acheved usng the methods proposed by Zobro [9, 10] In practce photogrammetrc networs may not have as good controlablty as networ presented n ths paper (Fg. 1). Besdes the true accuracy of PC coordnates remans unnown and the accuracy of measured dstances s lmted. Often dstances cannot be measured n versatle drectons le those n scenaro B11. Smlar tests, also for networs conssted of strps, are presented by Kolec [5]. However, n those smulatons the true PC coordnate errors were unnown and the dstances were measured between terran ponts wth lmted accuracy. Under those condtons dstances also helped to mprove accuracy, however not n such a large extent as n the presented tests. Photogrammetrc processng, especally the close range measurements, can be supported by addtonal nformaton about objects. Ths may nclude not only dstances but also te lnes and mposed constrants le vertcal, horzontal, parallel and orthogonal drectons. Also, measurements wth geo-referencng sensors should help mprove the accuracy of created models openng fully-automated photogrammetry to many applcatons. References [1] ARC3D. [on-lne:] [access: ]. [2] Bartelsen J., Mayer H.: Orentaton of Image Sequences Acqured from UAVS and wth GPS Cameras. Internatonal Calbraton and Orentaton Worshop EuroCOW 2010, Castelldefels, Span, [3] Ellum C.M., El-Shemy N.: Portable Moble Mappng. Internatonal Conference New Technology for a New Century, FIG Worng Wee 2001, Seoul, South Korea, 2001 [on CD-ROM]. [4] Horn B.K.P.: Closed-form soluton of absolute orentaton usng unt quaternons. Journal of the Optcal Socety of Amerca, vol. 4 (4), 1987, pp [5] Kolec J.: Wyznaczane elementów orentacj zewnętrznej zdjęć nazemnych z wyorzystanem erwacj fotogrametrycznych nercyjnych oraz sateltarnego systemu pozycjonowana. Wydawnctwa AGH, Kraów [6] Kolec J., Kuras P.: Relablty of Terrestral Image Geo-referencng usng GPS and MEMS nertal measurement unt. Internatonal Calbraton and Orentaton Worshop EuroCOW 2012, Castelldefels, Span, [7] My3DScanner. [on-lne:] [access: ]. [8] Snavely N., Setz S.M., Szels R.: Modelng the World from Internet Photo Collectons. Internatonal Journal of Computer Vson, vol. 80 (2), 2008, pp

10 60 J. Kolec [9] Zobro J.: Przedwyrównawcze wyrywane błędów grubych w pomarze środów rzutów dla aerotrangulacj. Archwum Fotogrametr, Kartograf Teledetecj, vol. 16, 2006, pp [10] Zobro J.: Przedwyrównawcze wyrywane błędów grubych w pomarze ątów orentacj dla aerotrangulacj. Archwum Fotogrametr, Kartograf Teledetecj, vol. 24, 2012, pp