CAMERA CALIBRATION ERROR ANALYSIS IN STEREO MEASUREMENTS

Transcription

1 CAMEA CAIBATION EO ANASIS IN STEEO MEASUEMENTS Sergiu Nedevschi 1, Tiberiu Mrit 1, du Dnescu 1, Florin Onig 1, Dn Frentiu 1, Ciprin Pocol 1 1 Technicl University o Cluj-Npoc, Computer Science Deprtment E-mil:{Sergiu.Nedevschi;Mrit.Tiberiu;du.Dnescu}@cs.utcluj.ro Abstrct: Cmer clibrtion is n importnt nd sensitive step in 3D environment reconstruction by stereovision. Smll errors in the estimtion o the cmer prmeters could rise to high errors in the 3D mesurements, s the working distnce increses. Thereore method or nlyzing the inluence o ech cmer s prmeter error in the ccurcy o the 3D mesurement is compulsory in order to minimize the overll 3D reconstruction error. In this pper we propose method to nlyze the inluence o ech cmer s prmeter o the stereovision system by pplying 3D stereo-reconstruction lgorithm on controlled 3D scene, hving depth rnge comprble with the working rnge o the 3D mesurement ppliction. 1. INTODUCTION Cmer clibrtion in the context o three-dimensionl mchine vision is the process o determining the internl geometric nd opticl chrcteristics (intrinsic prmeters) o the cmer nd/or the 3D position nd orienttion o the cmer coordinte system reltive to certin world coordinte system (extrinsic prmeters). The correct estimtion o these prmeters is crucil or pplictions bsed on rtiicil vision in order to perorm correct 3D reconstruction rom 2D imges o the environment. The perormnces o these pplictions re strongly ected by their precision especilly when stereovision lgorithms re used nd the mesurement working rnge is high (or exmple outdoor robot vision pplictions). Severl methods or generl-purpose geometric cmer clibrtion re presented in the literture. Some representtive ones re: Direct iner Trnsormtion (DT) [1], Tsi's lgorithm [2], hng s method [3], Heikkil s method [4] or the Bouguet (ClTech) method [5,7]. Some o their ssessment criteri re: utonomy, ccurcy, eiciency nd verstility [1]. Most o them re giving cceptble results or ner rnge mesurements. But smll errors in the estimtion o the cmer prmeters could rise to high errors in the 3D mesurements, s the working distnce increses. Some errors re systemtic ones due to the clibrtion lgorithm s limittions or mechnicl imperections o the stereovision system, others re non-systemtic ones cused by the exploittion conditions o the vision system during the 3D reconstruction/mesurement process, especilly when the system is mounted on mobile robot/vehicle. By nlyzing the inluence o ech cmer s prmeter error in the ccurcy o the 3D mesurement we will be ble to know witch prmeter is criticl or the mesurement ppliction nd to minimize the overll 3D reconstruction error in the ppliction context. In this pper we propose method to nlyze the inluence o ech cmer s prmeter o the stereovision system by pplying 3D stereo-reconstruction lgorithm on controlled 3D scene, hving depth rnge comprble with the working rnge o the 3D mesurement ppliction. For this purpose three tools were developed: clibrtion tool, 3D stereoreconstruction tool, nd n error nlyzer tool bsed on the previous one.

2 2. CAMEA CAIBATION For cmer clibrtion the pinhole cmer model is used [6]. The model is ugmented considering the lens distortions [4]: () (b) Fig. 1: (). The pinhole cmer model; (b). Set o imges or intrinsic clibrtion The ollowing cmer prmeters re considered: Intrinsic prmeters describing the internl geometric nd opticl chrcteristics o the cmer (those tht speciy the cmer itsel): ocl length, tht is, the distnce between the cmer lens nd the imge plne: ; the principl point - loction o the imge center in pixel coordintes: (u 0,v 0 ); the rdil nd tngentil distortion coeicients o the lens. Extrinsic prmeters - the 3-D position nd orienttion o the cmer rme reltive to certin world coordinte rme (re needed to trnsorm world coordintes to cmer centered coordinte rme or vice vers): the trnsltion vector T = [ Tx, Ty, Tz ] T ; the rottion vector r = [ rx, ry, rz ] T or its equivlent rottion mtrix. The trnsormtion between the coordintes o the point P in the cmer reerence rme(cf) C = [,, ] T nd its coordintes expressed in the world reerence rme (WF) W = [ W, W, W ] T is clled reconstruction or inverse mpping nd cn be perormed trough the ollowing mtrix eqution: W CW ( C TCW ) (1) where: T CW is the cmer to world trnsltion vector; CW is the cmer to world rottion mtrix: r11 r12 r CW r21 r22 r r31 r32 r (2) For cmer clibrtion tool bsed on the Bouguet (ClTech) method [5,7] ws used. The method estimtes the cmer intrinsic nd extrinsic prmeters in two dierent steps. As clibrtion pttern, plne plte with blck nd white squres o known sizes is used. Ech cmer is clibrted individully. For intrinsic prmeters set o t lest 20 imges must be tken t severl orienttions nd distnces, with the pttern covering the mximum re rom the imge, s possible (Fig. 1.). A good set o input imges gives smller uncertinties (less then 3 pixels) in the estimtion o the intrinsic prmeters nd determines the lgorithm to

3 converge in ewer steps. Then the extrinsic prmeters re estimted. For this tsk the cmers re mounted on stereo rig, nd the pttern is positioned to known position reltive to the cmers. For ech cmer single imge is tken nd the extrinsic prmeters o ech cmer, reltive to world reerence rme relted to the clibrtion pttern, re computed. The min drwbck o this method is tht is highly dependent on the clibrtion imge set (size nd position o clibrtion pttern, lighting conditions, cmer nd lens qulity, lens ocusing nd perture, number o imges). I the proper intrinsic clibrtion methodology is not kept, errors could be encountered especilly in the estimtion o the principl point coordintes. The principl point estimtion errors cn degrde drsticlly the qulity o the extrinsic prmeters nd the stereo reconstruction process. A method to improve the overll perormnce o the clibrtion lgorithm by inerring stereo-inormtion rom controlled 3D scene nd to correct the initil wrong estimted principl point nd consequently the extrinsic ones ws developed: the brut orce clibrtion method, s ws described in [8]. 3. THE 3D ECONSTUCTION AGOITHM The problem o mpping pir o correspondent imge points (p, p ) into unique 3D point P W ( W, W, W ) is clled 3D reconstruction. The stereo cmer conigurtion used is generl one, becuse it ws proved tht is very diicult to obtin cnonicl one due to the mechnicl imperections o the cmers nd the mounting rig. Solving the problem is stright orwrd: P W is positioned t the intersection o the two projection rys tht genertes p (x, y ) nd p (x, y ). First the 3D equtions o the two projection rys (which re pssing trough P W nd the two opticl centers o the two cmers) should be ound. This is widely covered subject [9,10]. Computing the intersection cn be done in geometric shion or n lgebric shion. Algebriclly, the intersection o the rys is obtined by solving eqution system (3). The distortion coeicients o the lenses were ignored. where: r x r11 r y r12 A r x r r y 11 r 12 r x r21 r y r22 r x r r y r W A W B W r x r31 r y r32 r x r r y 31 r B b 1 W b2 W (3) b3 W 43 b C C C C 43 C C r ij si r ij re the elements o the cmer-to-world rottion mtrices or the let ( ) nd the right cmer ( C ) respectively ( CW mtrix rom chpter 2)., verge ocl lengths o the let nd right cmer respectively: ( x + y )/2; (x, y ), (x, y ) coordintes o imge points p nd p reltive to the imge plne centers (principl points): x xl u0 y y v0 x x u0 y y v0

4 T = nd T C = C C C re the positions o the two cmer s CF in the WF (T CW vector). Becuse the number o equtions in system (3) is greter then the number o vribles nd due to the ct tht the intrinsic nd extrinsic prmeters re determined with some errors, this system must be solved using method tht should minimize the indetermintion (or exmple SE est Squres Estimtion). Geometriclly, this mens tht the projection rys do not intersect ech other nd the 3D point must be determined s the middle o the common orthogonl segment or the two rys [6]. 4. CAIBATION PAAMETES EO ANASIS In the ollowing the error nlysis o the inluence o the cmer prmeters in the ccurcy o the 3D reconstruction process is done. For this purpose controlled scene ws used with control points/objects with known depths ( distnce) rnging between 15 to 100 m The cmers re clibrted with the clibrtion method exposed in chpter 2. Then representtive control point ws chosen t m hving reltive estimted depth error o (0.3 m). The error nlyzer tool modiies in speciic rnge ech cmer prmeter round its initil estimted vlue rom the clibrtion procedure nd computes the depth ( coordinte o the control point). Only one prmeter o one cmer is modiied t time. The depth computtion is done using the 3D reconstruction lgorithm. In wht ollows the results o pplying the error nlyzer tool re shown. Only relevnt prmeters or the ccurcy o the depth estimtion were considered. Thereore we neglected orm the nlysis the ocl length which hd been proved the most stble nd less prone to errors intrinsic prmeter nd the trnsltion nd rottion long the xis (which hve insigniicnt inluence in the depth ccurcy). The min source o error originting rom the intrinsic prmeters is the horizontl (/u) coordinte o the principl point: 1 pixel vrition or one o the cmers determines 10% depth error; the inluence o the verticl (/v) coordinte is little bit smller: 3 pixels vrition gives 10% depth error (ig. 2): Fig. 2. Principl point coordintes errors: (). For /u xes - let cmer; (b). For /v xes right cmer.

5 egrding to the extrinsic prmeters, the horizontl component (Tc ) o the trnsltion vector (origin o the CF - Cmer eerence Frme in the WF - World eerence Frme) hs moderte, liner inluence: 30 mm vrition or one o the cmers determines 10% depth error; while the inluence o its verticl component (Tc ) is negligible: 100mm vrition gives 10% depth error (ig. 3); the inluence o the Tc ) is insigniicnt nd ws not shown: 10000m vrition rises to 10% depth error. Fig. 3. Trnsltion vector (CFs in the WF) errors: (). For Tx - let cmer; (b). For Ty right cmer. The rottion ngles o the origin o the CF (Cmer eerence Frme) reltive to the WF (World eerence Frme) hd gret inluence on the depth ccurcy: bout rdins (0.1 deg.) vrition o the pitch ngle o one cmer cn rise to dept error o 10%, while bout rdins (0.03 deg.) vrition o one cmer s yw ngle cn give depth error o 10% (ig. 4). The inluence o the roll vrition is less signiicnt: rdins (1 deg.) vrition determines 10% depth error nd ws not shown. Fig. 4. ottion vector (CFs reltive to the WF) errors: (). For rc - let cmer; (b). For rc right cmer.

6 5. CONUSIONS The precision o the cmer prmeters estimted in the cmer clibrtion procedure is essentil or stereovision bsed 3D reconstruction pplictions, especilly when high working rnges re used. Some o them re most criticl when we re interested in speciic mesurement. The proposed method llows determining which one is crucil or the cse considered in our exmple: the depth estimtion. The prmeters with the gretest weights in depth estimtion by stereovision re: the horizontl coordinte o the principl point, the bseline-length (Tx coordinte), nd the most criticl ones: the cmer pitch ngle (rcx) nd yw ngle (rcy). I smll errors rise in their estimtion the 3D reconstruction ccurcy will be drsticlly degrded. As urther improvements or the proposed error nlysis method the ollowing extensions re oreseen: Integrting more then one prmeter t time in the nlysis procedure; Considering ll 3D coordintes in evlution (, nd ). Anlysis o the inluence o the stereo-rig prmeters in the ccurcy o the reconstruction: the two cmers re considered s whole nd we re deling with the rig s trnsltion vector nd rottion ngles. This pproch will llow us to estimte which rig prmeters re criticl in the cse o dynmic mesurements (the stereovision system is mounted on moving robot/vehicle). EFEENCES [1] Abdel-Aziz,. I. & Krr, H. M. (1971), Direct liner trnsormtion into object spce coordintes in close-rnge photogrmmetry. Proc. Symposium on Close-nge Photogrmmetry, Urbn, Illinois, pp [2] Tsi,..(1987), A verstile cmer clibrtion technique or high-ccurcy 3D mchine vision metrology using o-the-shel TV cmers nd lenses. IEEE Journl o obotics nd Automtion, A-3(4)/1987: pp [3] hng (1999), Flexible Cmer Clibrtion By Viewing Plne From Unknown Orienttions. Interntionl Conerence on Computer Vision (ICCV'99), Coru, Greece, September 1999, pp [4] Heikkil, J.; Silven, O. (1997), "A our-step cmer clibrtion procedure with implicit imge correction," Proc. IEEE Computer Society Con., 1997 pp [5] Cltech Vision Group oicl site: [6] Trucco E., Verri A. (1998), Introductory techniques or 3D Computer Vision, Prentice Hll, [7] Intel Corportion (2001), Open Source Computer Vision ibrry. eerence Mnul, [8] S. Nedevschi, T. Mrit, M. Vid,. Dnescu, D. Frentiu, F. Onig, C. Pocol, Cmer Clibrtion Method or Stereo Mesurements, IEEE-TTTC Interntionl Conerence on Automtion, Qulity nd Testing, obotics, Cluj-Npoc, omni, My 2002, pp [9]. C. Gonzlez,. E. Woods, Digitl imge processing. pg Addison-Wesley Publishing Compny [10] J. mesh,. Ksturi, B. G. Schunk, Mchine Vision, McGrw-Hill, Inc., 1996.