ESTIMATIO OF OPTO-MECHAICAL PARAMETERS FOR CALIBRATIO OF LASER SCAERS Qing FU a, Hongchao FA b, *, Aoysis WEHR c, Uwe STILLA b a Institte of Geodesy and Geophysic, Vienna nivesity of technoogy, Gsshasst.7-9/E8-, A- 4 Vienna, Astia, f@ai.twien.ac.at b Institte of Photogaety and Catogaphy, Technische Univesitaet Menchen, Acisst., 8 Mnich, Geany, {fan, stia}@bv.t.de c Institte of avigation, Univesity of Stttgat, Beitscheidst.,774 Stttgat, Geany, weh@nav.ni-stttgat.de Coission III, WG III/ KEY WORDS: caibation, teestia ase scanne, ay tacing, Accacy, coodinate syste, adjstent. ABSTRACT: Within this pape an appoach of caibation fo a teestia ase scanne with two ios diven by two gavanoetes is pesented. This incdes a atheatica ode fo the inteio of the ase scanne as we as the deteination fo the paaetes to be caibated. The atheatica ode is deived fo the fnctionaity of the ase scanne, wheeby the pincipe of Ray Tacing was sed. Accoding to the inne stcte of the scanne device and the fnctionaity of the scanning syste the ase ay wod be divided into thee pats. Evey pat cod be epesented by a etic ength and a nit vecto which cod be dedced by sing the agoith in tianga anges and the atix of efection espectivey. The atheatica ode is then the est of cobination of these thee vectos of ase pats. It was sspected that the engths of the ase ay and the anges of the io otations wee obtained inexacty ding easeent. Howeve they cod be coected by adding offsets and factos fo the scae cased by not exact othogona configation of the io axes. In ode to estiate these paaetes the adjstent ode was estabished sing the Gass-Makov-Mode. Rests fo the test with siated data showed that the atheatica ode and the adjstent ode wee adeqate to the pesented appoach.. ITRODUCTIO In ecent yeas ase scanning has fond wide-spead se fo acqiing geoetic infoation fo appications in cta heitage docentation, foesty, foensics, defoation easeents, etc. The sccess of this eative new technoogy can be attibted to his advantages: (i) D Lase scanning ecods f atoaticay dense points cods fo the object sfaces with ess effot than othe D acqisition techniqes; (ii) D Lase scanning is an active easing syste which aows the easeent independenty of both the texte of the taget object sface and the nata ight conditions. Fo these easons Lase scanning is becoing oe and oe popa nowadays. Bt ost of avaiabe Lase scannes on the aket ae concening the accacy to the iiete ange. One of the ipotant easons is the ack of exact caibation fo the opto-echanica defection of the Lase scanning io axes. In the pesented appoach an expeienta teestia ase scanne with two ios diven by two gavanoetes was bit p. Expeients have shown that the accacy of this ase scanne in ange is bette than. Athogh a vey high pecise opoto-echanica defecto fo the two ios is sed. Fo obtaining accate D coodinates of the objects an additiona scanne caibation is sti eqied. Fo easing one point on the object sface, thee obsevations ae ade, naey the ange, the anges γ of the io on the hoizonta axis, and the ange γ of the io on the vetica axis. oay, point cods ae pocessed in a Catesian coodinate syste which has to be convesed fo the spheica coodinate syste. In o wok we sed a atheatica ode fo the scanning syste, in which additiona paaetes wee intodced. This ode consides the inne stcte of the ase scanne. This ode is based on the pincipe of ay tacing. The ase ay cod be divided into thee pats: (i) in the fist pat the ase ight ns fo the ase cente to the idde of the fist io, it wod be epesented by, (ii) the ase ight is io efected on the fist io sface with a cetain otation ange γ, then it ns to the second io. This pat is the second pat of ase ay, and wod be epesented with, (iii) the ase ight is efected on the second io sface with a otation ange γ and ns to the taget object, this is the thid pat of the ase ay. Ths the popagate path ests in five paaetes, which consist of thee distances, and and two otation anges γ and γ. * Coesponding atho. fan@bv.t.de; phone +49 89 89-586; fax +49 89 8-957; www.cato-t.de. 55
The Intenationa Achives of the Photogaety, Reote Sensing and Spatia Infoation Sciences. Vo. XXXVII. Pat Bb. Beijing 8 In the pesented appoach we assed that the ength of the fist pat of the ase ight is known and constant and both of the anges between the optica axis and the ase path fo the ase cente ae zeo. It was sspected, that the distances and wee not exacty eased and the anges of the ios otation γ and γ wee incoect, too. These fo paaetes shod be coected by the caibation, i.e.: 4 offsets ( d, d ) to these fo paaetes (, ) shod be fond ot by the caibation. Additionay to these offsets we intodced the paaetes,, and fo the scae becase of γ γ effects given by diffeent Fied of Views (FOV). In ode to estiate the eight paaetes ( d, d,,,, ) the eqations in γ γ the atheatica ode wee inea zed. Then the adjstent ode was estabished sing the Gass-Makov-Mode. In the next step we siated sets of eased data accoding to the fnctionaity of the ase scanne. The data sets diffe in FOV (6 degee and 9 degee). The siated data wee expoited to estiate the eight paaetes and to vaidate the adjstent ode. The ests showed that the pocess of the adjstent is adeqate to the pesented appoach. The eainde of this pape is stcted as foows: in the second section it wi be descibed how the ase scanne woks. Accoding to this fnctionaity the atheatica ode is deveoped in the section thee. In the foth section the paaetes to be caibated wi be intodced and the fifth section pesents the test with siated data. Finay, the concsions ae pesented in the section six.. FUCTIOALITY OF LASER SCAER The teestia ase scanne sed in this wok was bit p at the Institte of avigation, Univesity of Stttgat, Geany. The fige beow shows the inne stcte of this device. Gavanoete Mio Fige. ase scanne with two gavanoetes Fige shows the device withot cage. And it can be abstacted as that shown in the fige. The ain constitents Lase cente of the ase scanne ae two otation ios and ase cente, in which the ase bea cod be eitted and eceived. The two ios ae attached to oto shafts of eectic otos which have oto capabe of pefoing a iited otation. The scanne device has aso a position senso of eectostatic capacitance type aanged aond the oto shaft of the oto, and otpt fo this senso is fed back to the instction signa given to the oto, theeby to position the io. And the positions of the ios ae given in anges. In the ange easeent the CWase pincipe is appied which obtains the ange by eans of easing the phase diffeence between the tansitted and the eceived signa backscatteed fo the object sface. Mio Gavanoete Lase cente Lase ay Fige. abstacted ode of ase scanne. The ed ine epesents the ase ay. The coodinate syste (in be) is the defat coodinate syste in the oca scanning syste. The abstacted ode shows the path of the ase ay ceay. At fist ase ight is eitted fo the ase cente. Then it ns to the fist io. Thee it is efected with a cetain otation ange of the io. Shoty afte that it is efected by the second io with anothe otation ange. Finay, it ns to the taget and then back to the ase cente in the sae way. The technica data of this ase scanne can be cited in a tabe. Tabe. Technica data of the ase scanne in the pesented wok. Lase otpt powe:.5 W Lase waveength: 67 n Instantaneos Fied of. (apete by sending: ) View (IFOV): Fied of View (FOV), Max. x vaiabe: Apete by eceiving 4 Saping: -di. Line scanning be of pixes pe x o 4 x 4 (defat iage: setting) Abigity inteva: >. (fo diffse efectivity 6% Resotion Range: and ange) Tone of easeent MHz and 4 MHz KHz (woking with one ton) Measeent ate 6 Hz (woking with two tone) Scanning ate 4 sec/scan As shown in the fist and second figes the teestia ase scanne eqies distaction echaniss in two diffeent diections fo sveying a cetain egion of an object of investigation, wheeas the two ios wee diven by two 56
The Intenationa Achives of the Photogaety, Reote Sensing and Spatia Infoation Sciences. Vo. XXXVII. Pat Bb. Beijing 8 gavanoetes. A ase scanne of this type is aso caed a XY- Scanne. (Fige.) five paaetes, which consist of thee distances, and and two otation anges γ andγ. In the pesented appoach we egaded these thee pats of ase path as vectos. In ode to descibe these vectos detaied and obviosy, it is necessay to intodce thee coodinate systes, in which the thee vectos ay espectivey. (See. Tabe.) Tabe. Coodinate systes in atheatica ode Fige. XY-Scanne The fige denotes the basic pincipe of the XY-Scanne. The two gavanoetes shod be instaed othogonay. The ase bea is initiay efected by the io diven by Y- Gavanoete. Aftewads, it is efected by the io diven by X-Gavanoete. Coodinate syste Oigin coodinate syste (syste). io coodinate syste (S- syste). io coodinate syste (S- syste) Oigin X-axis Y-axis Z-axis Lase cente point O Cente of the fist io O Cente of the second io O Paae to the diving second io Pependic a to his Y- axis on the io pan the diving second io Paae to the diving fist io the diving fist io Pependic a to his X-axis on the io pan Pependica to his XYpan Pependica to the fist io pan Pependica to the second io pan Fo easing one point on the object sface, thee obsevations ae ade, naey the ange, the anges γ of the io on the hoizonta axis, and the ange γ of the io on the vetica axis. oay, point cods ae pocessed in a Catesian coodinate syste which has to be convesed fo the spheica coodinate syste. In the pesented wok we estabished a atheatica ode fo the scanning syste, in which additiona paaetes wee intodced. This ode consides the inne stcte of the ase scanne which was descibed as the fnctionaity of ase scannes in this section. ote: a of the thee coodinate systes ae ight hand syste. The coodinate systes incding the vectos of the thee pats of the ase path cod be istated in the gaphic beow. (Fige. 4). MATHEMATICAL MODEL The atheatica ode pesented in this pape is estabished accoding to the pincipe of ay tacing. This incding the invoved coodinate systes and vectos of ase ay wi be descibed in this section.. Coodinate systes and vectos The atheatica ode is based on the pincipe of ay tacing. The ase ay cod be divided into thee pats: (i) the fist pat epesents the ase ight nning fo the ase cente to the idde of the fist io; (ii) the second pat epesents the ase ight between the two ios (it is efected on the fist io sface with a cetain otation angeγ, then it ns to the second io); (iii) the thid pat epesents the ase ight between the second io and the taget (it is efected on the second io sface with a otation ange γ and ns to the taget object). Ths the popagate path ests in Fige 4. coodinate systes and vectos of the ase path In the fige 4 A is the incidence of the ase ight on the fist io and A epesents the incidence of the ase ight on the second io. 57
The Intenationa Achives of the Photogaety, Reote Sensing and Spatia Infoation Sciences. Vo. XXXVII. Pat Bb. Beijing 8 In diffeent coodinate systes the vectos shod be epesented diffeenty, as yo can see in the tabe beow. Tabe. Expaining fo vectos vecto In - In io coodinate syste syste S (in S-syste) S S (in S-syste) (in S-syste) S (in S-syste) S (in S-syste) S (in S-syste) S Δ Δ Δ (in S-syste) S Δ (in S-syste). Rotation atices expaining Vecto to the cente of the fist io (foo to O ) Vecto to the cente of the second io (foo to O ) The fist pat of ase vecto (foo to A ) The second pat of ase vecto (fo A to A ) The thid pat of ase vecto (fo A to object) Vecto fo O to A Vecto fo O to A Fo the atheatica ode two types of otation atices wee needed. They wod be defined and descibed in the foowing... Matix fo efection Accoding to the aw of the efection the ase ay is efected at the io pan. Becase the incident ase vecto and the efected ase vecto have opposite diections in no diection of the io pan, the atix fo efection is then: S ().. Rotation atices fo io coodinate systes Using the otation atices the vectos can be tansfoed fo the -syste to the io syste (S- o S-syste). In photogaety (Kas 4) the otation and oientation of the ios cod be descibed by dint of affine tansfoations. R S R R R () S Whee R is the otation atix fo the fist io (S- syste) with the otation ange γ of the io on the S hoizonta axis, and R is the otation atix fo the second io (S-syste) with the otation ange γ of the io on the vetica axis. x y z. The known vectos Aiing to sipy the sitation to a cetain extent we assed in the pesented appoach that both of the anges between the optica axis and the ase path fo the ase cente ae zeo. On the othe hand we got soe known paaetes fo the device fo anfacte as foowing: The distance between the ase cente and the cente of the fist io is.65. The distance between two ios is.6. Then we have: and x.65 y () z x.65 y (4) z.6 and the nit vecto fo the fist pat of the ase vecto.4 The ase vectos x y (5) z The fist pat of the ase vecto was epesented by convotion of the ength of the fist pat of ase and its nit vecto. Unde the assption in section. vecto eqivaence of the vecto.4. The second pat of the ase vecto. (6) is then the Anaog to the eqation 6 the second pat of the ase vecto cod be cacated with the eqation beow: (7) whee is the ength of the second pat of ase and is its nit vecto. Fo the fige 4 we see that is the efected vecto of the incident vecto, hence we can get by sing the atix fo efection (eqation ). This pocess cod be ipeented in thee steps: (i) tansfo fo -syste to S-syste S S R (8) 58
The Intenationa Achives of the Photogaety, Reote Sensing and Spatia Infoation Sciences. Vo. XXXVII. Pat Bb. Beijing 8 (i i) se the atix of efection in S-syste S S (9) (iii) tansfo fo S-syste to -syste () T he cacation of the ength was eaized by disposing of tiange geoety: R (i) in the tiange ( S S S T S (R ) S R ΔO A A S ) + Δ we have the eqation S S S (ii) pt eqations 6 and 7 into we can get S ( ) () S S + Δ R () The eqation is eqivaent to the eqation beow: S T S S ( R ) ( R ( + Δ ) ( ) ) ().4. The thid pat of the ase vecto The thid ase vecto cod be dedced in the sae way as the second ase vecto. (4) Since the thid pat of ase ay is the est of the io efection of the second ase ay, the atix of efection wod be sed on the second io. Then the nit vecto of the thid pat of ase cod be deived: S T S (R ) S R (5) On the othe hand, the ength of the thid pat of ase was vey sipe to be cacated finay: w hee is the tota ength of the ase fo easeent. (6) ow the atheatica can be estabished by cobining the thee pats of ase vectos. 4. PARAMETERS FOR CALIBRATIO It was sspected, that the distances and wee not exacty eased and the anges of the ios otation γ and γ wee incoect, too. These fo paaetes shod be coected by the caibation, i.e.: fo offsets ( d, d ) to these fo paaetes (, ) shod be fond ot by the caibation. Additionay to these offsets we intodced the paaetes,, and fo the scae becase of γ γ effects given by diffeent Fied of Views (FOV). In ode to estiate the eight paaetes ( d, d,,,, ) the eqations γ γ (eq.7.) of the atheatica ode wee ineaized. Then the adjstent ode was estabished sing the Gass-Makov- Mode. 5. TEST OF THE MATHEMATICAL MODEL AD THE ADJUSTMET MODEL In this section the atheatica ode and the adjstent ode wod be tested by sing siated easeent data. Fo this poposa a set of easeent data was siated in Catesian accoding the fnctionaity of the ase scanne (section ). They wee then tansfeed in anges and distances by sing: X γ (actan Y γ (actan X + Y )/ + Z )/ + Z Z Y (8) In ode to test the sensitivity of the atheatica ode and the adjstent ode we added known offsets and scae factos on the anges and the distances obtained by sing eqation 8. In fige 5 the two types of points ae shown gaphicay, wheein the geen points wee the ests of adding known offsets and scae factos on the siated data (the ed ones). A of the siated points ay on a pan and the distance between neighbo points is thee c. Afte the adjstent the sae vaes as we had added wee eceived fo the paaete of the caibation, as shown in the fige 7, the geen points (ests afte the adjstent) ae aost identica with the ed points (siated data). This eans that the atheatica ode and the adjstent ode ae appopiate to the caibation and they ae sensitive on the othe hand. X Y Z + + + + (7) 59
The Intenationa Achives of the Photogaety, Reote Sensing and Spatia Infoation Sciences. Vo. XXXVII. Pat Bb. Beijing 8.5 Z-AXIS -.5 - Fige 5. siated data befoe adjstent. The geen points wee the ests of adding known offsets and scae factos on the siated data (the ed ones)..6.6 Y-AXIS.59 -.6 -.4 -. X-AXIS Fige 7. Meased data (in geen) and thei appoxiated pan (in ed)..4.6.5 Z-AXIS -.5 -.64.6.6.598.596 Y-AXIS -.6 -.4 -. X-AXIS. Fige 8. Rests of the adjstent.4.6 6. COCLUSIO Fige 6. Data afte adjstent in siation Afte the test fo sensitivity sing siated data, a sa white boad was eased. The boad was ocated veticay to the Y -Syste. And the distance between the boad and the XZ-pan of the -Syste was aond.6 etes. The easeent asted aond intes. And then the data and the known technica paaetes of the device wee stoed in a text fie. De to the iitation that we did not have ease instent with highe accacy, conto points fo the caibation wee not avaiabe. In this case we choose sevea points in the idde of the easeent fied to fit a pan. The points on this fitted pan wee viewed as the conto points of the eased points espectivey. Fige 7 pesents the eased points with geen coo and the fitted pan with ed coo. The est of the adjstent is shown in the fige 8. Copaing the geen points between the eased data and the data coected by adjstent it wod be fond that (i) the cvate of the oigina data was eiinated; (ii) the aveage distance between the coected data and the fitted pan wee significanty sae than that befoe adjstent; and (iii) the points afte the adjstent had a noa contibte. These cod ean that the adjstent is abe to coect the eased data in geat extent. This wok pesented a atheatica ode fo the ase scanne diven by two gavanoetes accoding the pincipe of ay tacing. In the atheatica ode the ase ay was teated as thee pats of ase vectos epesented by etic engths and nit vectos. And they cod be dedced espectivey by sing the agoith in tianga anges and the atix of efection. Then the atheatica ode is a cobination of the thee vectos. On the base of the atheatica ode the paaetes fo caibation cod be deived. The siation denoted that the deveoped ode was sensitive. And the test showed that the adjstent ode cod coect the eased data in geat extent, athogh thee wee not conto points fo the adjstent. 7. REFERECES Kas, K. 4. Photogaety. Wate de Gyte, Bein. ew Yok, pp. 5-6. Vozikis, G., Haing, A., Vozikis, E., Kas, K., 4, Lase Scanning: A ew Method fo Recoding and Docentation in Achaeoogy, Pape ID WSA.4 Weh, A., Thoas, M. 6. Cose ange Lase scanne, Stttgat, Geany. http://www.nav.ni-stttgat.de/navigation/ foschng/ nahbeeich_asescanne/ 6