Accurate Calibration of Stereo Cameras for Machine Vision

Transcription

1 JCS&T Vol. 4 No. 3 Otober 004 Aurate Calibration o Stereo Camera or Mahine Viion Liangu Li, Zuren Feng, Yuanjing Feng Intitute o Stem Engineering, Xi an Jiaotong Unierit Xi an, ShanXi Proine, China ABSTRACT Camera alibration i an important tak or mahine iion, hoe goal i to obtain the internal an eternal parameter o eah amera. With thee parameter, the 3D poition o a ene point, hih i ientiie an mathe in to tereo image, an be etermine b the triangulation theor. Thi paper preent a ne aurate etimation o CCD amera parameter or mahine iion. We preent a at tehnique to etimate the amera enter ith peial arrangement o alibration target an the amera moel i aime at eiient omputation o amera parameter oniering len itortion. Built on trit geometr ontraint, our alibration metho ha ompenate the error or itortion ae b irular eature on alibration target, hih get oer the relatiit inluene o eer unknon parameter o traitional alibration a an make the error itribute among the ontraint relation o parameter, in orer to guarantee the aura an oniten o alibration reult. Eperimental reult are proie to ho that the alibration aura i high. Keor: Calibration, Camera, Parameter, Viion, Meaurement.. INTRODUCTION Reontruting 3D ene inormation rom D image ha been an important tak or man mahine iion appliation uh a iion guie robot or automate aembl [], iual ero ontrol, robot naigation, iion bae inutrial etetion an reognition, et. The goal o mahine iion i making mahine imulate the iion untion o mankin, to reontrut 3D inormation ith tereo image. Viion bae ening, alo alle optial gauging, i a tehnique or making iplaement meaurement bae on the relatie poition o ome tpe o pattern or eature in the iel o ie o a iion enor uh a a CCD amera []. Camera i the main tool or getting original 3D inormation o mahine iion. Viion bae tem ith amera mut etermine amera parameter, o amera alibration i a ruial tep to obtaining 3D inormation ith amera-bae iion tem. Camera alibration i to etermine the internal amera geometri an optial harateriti, along ith the 3D poition an orientation o the amera rame relatie to a ertain orl oorinate tem, hih are alle etrini parameter, in orer to etablih the onnetion beteen the poition o image piel an ene point. For an iion appliation here more aurate 3D orl oorinate houl be etermine rom their D image oorinate, the amera alibration ith higher aura mut be obtaine irt [3]. Due to ariou appliation, there are ierent requirement or amera alibration. In uh appliation a robot guiane, the alibration proeure houl be at an automati, but in inutrial metrolog appliation, the preiion i tpiall a more important ator. Muh ork ha been one in the amera alibration iel [4], [5], [6], [7], an e an ategorize thoe alibration metho generall into three lae: The traitional metho uing the 3D alibration target: Camera alibration i perorme b obering a alibration target oniting o to or three plane orthogonal to eah other, hoe 3D geometr imenion i knon ith goo preiion [4], but thi approah require epenie equipment an elaborate intallation. D plan bae alibration: Camera alibration require obering a planar pattern hon at a e ierent orientation, hile the knolege o the plane motion i not neear. Sine almot anone an make uh a alibration eie b himel, the intallation i er eaier than the traitional metho, but the amera parameter an't be etimate reliabl [5], [6]. Sel-alibration metho: Camera alibration oe not ue an alibration objet, jut b moing a amera in a tati ene, o onl image point orreponene are require [7]. Although no alibration objet i neear, a large number o parameter require to be etimate, reulting in a muh harer mathematial problem an unreliable reult. Our urrent reearh i oue on obtaining the i DOF poe o our moable robot. In the mot general ae, an objet poition an orientation an be eribe b i-dof like the robot name Steart platorm [8], o e ue tereo iion bae tem a hon in Fig. to meaure it aurate poe, aoringl ahiee iual ero ontrol. We eal ith the alibration o amera mounte on the all o our eperimente room, to perorm real time loalization o our moable robot. Ater e obtaine the moement inormation an error inormation o robot ith iion bae tem, e an ahiee preiel loe-loop eebak ero ontrol. L ' L 3 L O ' z' L 4 L 6 O z Fig. Viion bae Steart platorm In orer to arr out high preiion an high-pee iual ero ontrol o robot, ontruting an eat an proper moel i er important to the aura an eiien o CCD amera parameter. In thi paper e preent a ne amera alibration metho bae on aurate moel an rapi algorithm. Compare ith alibration a in the ' L 5 47

2 JCS&T Vol. 4 No. 3 Otober 004 literature, our metho an get einite, reliable an high aura parameter, hile the algorithm i er imple.. CAMERA MODEL Camera moel i the mathemati eription o the phi proe rom ene imaging to image plane. The perpetie projetion o ee an amera an be approimatel looke a a pinhole moel. Here e ill ue pinhole moel oniering len itortion to eribe the image orming in the amera. Let u irt ontrut an ieal pinhole amera moel a illutrate in Fig. to eribe perpetie tranorm, here (,, z i the oorinate o the target point P in the 3D orl oorinate tem, an (,, z i the oorinate o the target point P in the 3D amera oorinate tem, hoe orreponing image oorinate i ( u,. The image o a 3D point P [,,,] T z on the target o amera p [ u,,] T an be eribe a a perpetie projetion o P on p through the opti enter o, hoe itane to the image plane i the eetie oal length, epree in homogenou notation in the olloing matri equation: p CR T P ( here i a ale ator, R i a 3 3 orthonormal rotational matri, an T [ t ] T t tz i a 3 tranlation etor, hih repreent the relatie rotation an tranlation beteen the orl reerene rame an the amera oorinate rame repetiel. C i the amera intrini matri an an be enote a: N 0 u0 0 u0 C 0 N ( here N an N are the number o piel per unit itane in the ro an olumn repetiel, u 0 an 0 are the oorinate o the prinipal point in piel. P (, o i P (, X Y z o o z u ou (, 0 0 P (, u P( z,, Fig.. Camera moel The pinhole moel i onl an approimation o the real amera projetion, hoeer, it i not ali or the requirement o high aura. In the inutrial mahine iion appliation, raial itortion i the main ator that ha eet on it aura [4], o onl raial itortion nee to be oniere, an it an be epree approimatel a: ( kr (3 ( kr here (, i the atual image oorinate hih ier rom (, ue to len itortion, r an k i the len itortion oeiient. 3. CALIBRATION STRATEGY Thi etion proie the trateg on ho to eetiel ole the amera alibration problem or mahine iion. To improe the alibration aura, the irular eature hae been ue a alibration pattern or it propert o the noie immunit [9], o e ue alibration boar ith irular eature point. To moel the a that amera projet the 3D orl into D image e nee to irt kno here the amera image enter i, hoe aura ha eet on the aura o amera moel. R.G.Wilon an S.A.Shaer [0] hae ummarize 5 tehnique or meauring image enter, but almot all o them require epenie equipment an preie etup, hile the reult i unreliable. We ue a at an aurate tehnique to etimate the amera enter an other alibration parameter ith error ompenation or irular eature etetion. Preetermine the amera enter: To preetermine the amera enter, e an ue a peial alibration a. We arrange the alibration target an take to image at to ierent epth an rom the opti enter, both o hih are perpeniular to the optial ai. A hon in Fig.3, thi arrangement o alibration target an eliminate the unknon intrini parameter eept the amera enter b the geometr ontraint. The origin o the amera oorinate (,, z i aume to be loate at the opti enter o. I to alibration point hoe itane to the opti ai are h an h hae the image point height b an a repetiel, e an get the olloing equation through triangle geometr theorem: a h b h (4 a b o z h h Fig.3 Preetermine amera enter Then e an onlue that: a (5 b 48

3 JCS&T Vol. 4 No. 3 Otober 004 Let ( u0, 0 be the image enter an a u u0, b u u0, then e hae: a u u0 (6 b u u0 An the ame e hae: a 0 (7 b 0 So e an obtain ( u 0, 0 b oling the olloing equation ith leat-quare algorithm: u0 u u u u 0 (8 Calibration other amera parameter: Etrat ome eature point rom a 3D target, etet eah alibration point ( ui, i, i,, N, hoe reponing orl oorinate i ( i, i, z i, an take the alibration point the number i bigger than ie an it orreponing image point, e an get a out-table equation, then ole the ariable through the leat-quare olution: t / t, i ri / t, i,, 4, 5 (9 So e an obtain t ith: / [ 4( 5 4 ] t (0 ( 5 4 here 4 5. For eah alibration point, e an obtain: ( kr z ( ( kr z here r, an k i the len itortion oeiient. Without lo o the generalit, let z 0, uppoe that: E r r t E r4 r5 t ( Ez r7 r8 tz an let m k, G r7 r8, that i: E, E, z Ez, G Ez tz ; then e hae: E E r m tz G (3 E E r m tz G For N eature point hoe ( u, an it orreponing (,, z are knon, e an get the optimize, k, t z ith the leat-quare metho. Error ompenation o irular eature: Cirular eature a irle an ellipe are not onl the bai element in nature but alo er ommon hape in man man-mae objet, hih hae been ommonl ue in robot iion iel. The aurate enter o projete irle proie eat orreponene beteen ellipe in D image plane an irle in 3D plane. The perpetie projetion o a patial irle i ommonl an ellipe, hoeer, the enter o projete irle oe not orrepon to the ellipe enter, a illutrate in Fig.4. Let both the orl oorinate tem (,, z an the amera oorinate tem (,, z be entere in the opti enter o, an let z ai be orthogonal to the objet plane, alo let z ai be perpeniular to the CCD plane, hoe orreponing image ai u an are parallel to an repetiel. I the iretion angle o etor oo i,, in, an the itane beteen o an o equal to, then the projetion o the irle on the o plane i a irle 3 hoe raiu r i knon, hih an be gien b: o o ( z ( z ( r o o (4 z o z 3 o Fig.4. Perpetie projetion o irular eature So the ra oming rom the irle loate on the plane orm a kee one, hoe bounar ure an be epree a: ( za ( za ( za3 (5 o o r here a, a, a3 o o The relationhip beteen the orl oorinate tem an the amera oorinate tem an be gien b: b b b3 b4 b5 b6 (6 z b7 b8 b 9z here the etor [ b, b 4, b 7 ] T, [ b, b5, b 8] T, [ b3, b6, b9] T orm an orthonormal bai. The orthogonal itane beteen the opti enter o an the image plane i the oal length, o e an epre Eq. (5 in the amera oorinate tem: o 49

4 JCS&T Vol. 4 No. 3 Otober 004 ( g l p ( gh lm pq here g b ab 7 h b ab 8 k ( b3 ab 9 l b4 ab7 m b5 ab8 n ( b6 ab9 p a3b7 q a3b8 ab. ( h m q ( gk ln p ( hk mn q k n (7 A e kno that a ommon quarati ure an be epree a the orm o: A B C D E F 0 (8 an the projetion o a ommon irle on the image plane i an ellipe, then the image o the irle i an ellipe loate on the image plane. Thu the enter point oorinate o the ellipe an be alulate uing the olloing ormula: 3 4 u ( gqhp ( pmql ( gmlh 5 (9 3 ( gqhp ( pmql ( gmlh here: h m q ( ( gk ln p 3 ( ghlm pq 4 ( hk mnq 5 ( g l p. In the amera oorinate tem the equation o the line oo an be gie b: b bz 3 4 b5 bz 6 o o (0 7 b 8 bz 9 o The real oorinate o the ellipe enter i the interetion o the line oo an the plane z i: u 5 4 ( here: bo b7 o b o b o 8 ( b o b o b o b o b o b o b o b o The obere image oorinate ( u, houl be orrete ith the error ompenation: ( u un u u ( N( Ater the error ompenation, the amera parameter are ompute again, o e an get more aurate amera parameter, to guarantee high aura or mahine iion. 4. EXPERIMENT RESULTS In thi tem, e ue the amera name TM400, hih i mae in Pulni Compan o Unite State ith high reolution 39*040, an igital image boar name P-amlink, hih i mae in Coreo Compan o Canaa, to orm a high aura meaurement tem. We ue 64 alibration point o irular eature, to ahiee thee amera parameter a hon in table.. Table. Camera parameter uing real image Camera parameter Rotation matri R Tranlation T Intrini parameter Firt amera Seon amera r r r r r r r r r t t tz u k To hek the aura o our tem, e ue 3 tet point to meaure the ierene beteen their real oorinate an reontrute oorinate ith: N ( X X i N (3 here X i real 3D oorinate hile X i the meaurement alue, an N i the number o tet point. The epth rom the baeline to the alibration boar i 4.68m, an the aerage error o our meaurement tem i =0.6mm, o that the alibration aura i er high. We alo hae onute other eperiment. General peaking, thi metho i quite aurate. 8. CONCLUSIONS In thi paper, an aurate etimation o CCD amera parameter a preente or mahine iion appliation here high aura i neee. We ue a at tehnique to 50

5 JCS&T Vol. 4 No. 3 Otober 004 etimate the amera enter ith proper arrangement o alibration target, an obtain other parameter through logial organization o oling orer. Built on trit geometr ontraint, our alibration metho ha ompenate the error or itortion ae b irular eature on alibration target, hih get oer the relatiit inluene o eer unknon parameter o traitional alibration a an make the error itribute among the ontraint relation o parameter. Thi repreentation make the eompoing o all amera parameter poible, an lea to the parameter etimation one b one. Compare ith laial alibration tehnique that ue epenie equipment an ompliate mathematial. 9. REFERENCES [] F.G. King, G.V. Pukoriu an F. Yuan, "Viion guie robot or automate aembl", Pro. 988 IEEE Int. Con. Roboti an Automation, Vol. 3, April 988, pp [] Harr Dougla Garner, "Deelopment o a real-time iion bae abolute orientation enor", Ph.D iertation, Georgia Intitute o Tehnolog, 00. [3] L. Jong-Soo an J. Yu-Ho, "CCD amera alibration an projetion error anali", Pro. 000 IEEE Int. Con. Siene an Tehnolog, Korea-Ruia, Vol., 000, pp [4] R.Y. Tai, "A eratile amera alibration tehnique or high-aura 3D mahine iion metrolog uing o-the-hel TV amera an lene", IEEE J. Roboti an Automation, Vol. 3, No. 4, Aug. 987, pp [5] Z.Y. Zhang, "A leible ne tehnique or amera alibration", IEEE Tran. Pattern Anali an Mahine Intelligene, Vol., No., No. 000, pp [6] J. Heikkilä, "Geometri amera alibration uing irular ontrol point", IEEE Tran. Pattern Anali an Mahine Intelligene, Vol., No. 0, Ot. 000, pp [7] R.I. Hartle, "Sel-alibration rom multiple ie ith a rotating amera", Pro. 994 IEEE Int. Con. Computer Viion an Pattern Reognition, Ma 994, pp [8] D. Steart, "A platorm ith i egree o reeom", Pro. Int. Meh. Eng., Vol. 80, No. 5, 965, pp [9] J. Heikkilä, "A our-tep amera alibration proeure ith impliit image orretion", Pro. 997 IEEE Int. Con. Computer Viion an Pattern Reognition, June 997, pp [0] R.G. Wilon an S.A. Shaer, "What i the enter o the image", IEEE J. Optial Soiet o Ameria, Vol., No., No. 994, pp