Position Determination of a Robot End-Effector Using a 6D-Measurement System Based on the Two-View Vision

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1 Ope Joural of Applied Scieces, 203, 3, Published Olie November 203 ( Positio Determiatio of a Robot Ed-Effector Usig a 6D-Measuremet System Based o the wo-view Visio Aleej Jaz, Christia Pape, Eduard Reithmeier Istitute of Measuremet ad Automatic Cotrol, Leibiz Uiversität Haover, Haover, Germay aleej.jaz@imr.ui-haover.de Received September 6, 203; revised October, 203; accepted October 2, 203 Copyright 203 Aleej Jaz et al. his is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. ABSRAC A mechatroic system based o the micro-macro-kiematic cosists of a idustrial robot ad a piezoelectric stage mouted o the robot s ed-effector ad has to carry out operatios like micro-assembly or micro-millig. he piezoelectric stage has to compesate the positioig error of the robot. herefore, the positio of the robot s ed-effector has to be measured with high accuracy. his paper presets a high accuracy 6D-measuremet system, which is used to determie the positio ad orietatio of the robot s ed-effector. We start with the descriptio of the operatioal cocept ad compoets of the measuremet system. he we look at image processig methods, camera calibratio ad recostructio methods ad choose the most accurate oes. We apply the well-kow pi-hole camera model to calibrate sigle cameras. he we apply the epipolar geometry to describe the relatioship betwee two cameras ad calibrate them as a stereo visio system. A distortio model is also applied to ehace the accuracy of the system. he measuremet results are preseted i the ed of the paper. Keywords: 6D-Measuremet System; High Speed Camera; Image Processig; Recostructio; Essetial Matri; Fudametal Matri; Camera Projectio Matri. Itroductio he project Micro-Macro-Kiematics, which is curretly ruig at the Istitute of Measuremet ad Automatic Cotrol of the Leibiz Uiversität Haover, deals with the developmet ad aalysis of a robotic system, which is able to maipulate micro-objects with a accuracy of - 2 micrometers i a 3D-workspace of 0 cubic millimeters. For this purpose it is ecessary to determie the positio ad orietatio of the robot ed-effector with high accuracy ad i real time. At the begiig of the project we looked at various measurig systems, e.g., the Laser racker from Leica or the Laser RACER from Etalo AG. Such systems are able to carry out cotactless high accuracy measuremets i large workspaces. However, they are very epesive ad have several characteristics, for eample, the relatively high weight of the retro-reflector, which make them very difficult to use with the robot i this project the maimal load the robot ca carry is 700 grams. herefore, it was iitially decided to reduce the requiremets for the size of the workspace ad use a measuremet system based o two high-speed cameras. I order to preset this measuremet system, we will describe its cocept ad compoets i this paper. We will also take a look at the methods for image processig, cameras calibratio ad 3D-data recostructio, choose the most accurate oe ad eplai our choice. 2. Cocept ad Compoets of the System It is always ecessary to give clear requiremets of the measuremet system such as cotact-free measuremet of the 3D-positio ad orietatio of the robot ed-effector, measuremet accuracy of - 2 micrometers, workspace of at least 0 cubic cetimeters, real-time ability (oscillatios o the ed-effector of the robot are up to 50 Hz). he geeral cocept of the measuremet system is preseted i Figure. wo high speed moochrome cameras M20 Falco from DALSA are used i it. Both cameras have a resolutio of [h w] = piels with a piel size p = 7.4 micrometers. Both cameras are also coected to the frame grabber Xcelera-CL PX4 Dual, which has a trigger fuctio ad ca take up to 22 frames per secod simultaeously. Each camera has the

2 394 A. JANZ E AL. Figure. Geeral cocept of the measuremet system. bi-telecetric les C2336 from Opto-Egieerig with a magificatio k = hese bi-telecetric leses simplify image processig, because the dimesios of the objects remai early costat. he maimal image size of every camera ca be calculated based o the camera resolutio, piel size ad les magificatio: h p w p mm 3 mm. () k k I order to determie the positio ad orietatio of the robot ed effector, the 3D-mark a black plate with three white balls is fied o it. It is possible to determie the 3D-positio by usig oly a sigle ball. wo other balls are ecessary to determie the orietatio of the mark ad the ed-effector. I order to choose the size of the balls ad the offsets betwee them, oe has to cosider the image size, the desired workspace, the sub-piel accuracy ad agular accuracy. It is obvious that the sub-piel accuracy ca be icreased by makig the white balls bigger. However, the bigger the balls are, the smaller the real workspace becomes, because every ball must appear completely o the image for the successful image processig. he agular accuracy ca be improved by icreasig the offsets betwee white balls; however, oe should ot forget that the use of three balls reduces the real workspace. akig ito accout these properties, the diameter of the balls was set to 2.5 millimeters, ad the offset betwee the balls to 8 millimeters. It allows to measure the positio ad orietatio i the workspace of about 2 cubic cetimeters, with agular icliatio up to 60 as well, ad sub-piel accuracy of about 0.05 piels. I order to get more stable image processig, for eample, to elimiate the ifluece of the eteral illumiatio, the system has its ow light sources. At the very begiig of the project we have used rigs with white LED. Ufortuately, this solutio caused ihomogeeous illumiated image. For this reaso two rigs of red ad gree LEDs are used ow, see Figure 2. he red ad gree light filters are also mouted i the telecetric leses. his solutio elimiates ay ifluece of cameras o each other. 3. Image Processig wo cameras ca shoot two idepedet moochrome Figure 2. Illumiatio of the 3D-mark. images simultaeously with a frame frequecy of 22 Hz. herefore, the image processig must be fast ad simple. For that reaso images are ot processed completely. It is oly ecessary to process the regio of iterest (ROI) with size.5 times larger tha that of a white ball. However, the whole image area must be processed at first to idetify ad separate white balls from aother white object that could appear o the image. I order to fid ay white object o the image, all processed piels are grouped to white oes ad black oes with the help of a grey-level threshold which must be automatically foud for each sigle ROI or the first image. he method for global thresholdig, described i [], is used to fid this threshold. It cosists of: Defiitio of the iitial grey level threshold Divisio of the image usig threshold o two parts white ad black Computig of the average grey-level values i both parts Computig of the ew threshold: ew 2 0.5, (2) where μ, μ 2 are average grey-level values from the previous step Repeatig the last three steps util the differece betwee the ew ad the old threshold becomes smaller tha the predefied Δ. As we have already metioed above, the very first frame has to be processed completely. It meas that we have to fid the global grey level threshold first. he every sigle white object o this image must be aalyzed usig this threshold the roudess coefficiet must be computed: 4π S k 2, (3) P where S is the area of the object, ad P the perimeter or the legth of the object s cotour. If the roudess coefficiet is larger tha 0.85, the aalyzed object is treated as a white object. he ROI positio ad size are

3 A. JANZ E AL. 395 the computed for it. Furthermore, oly these ROIs will be processed. Figure 3 shows a eample of the first frame. White objects foud i ROIs will o loger be aalyzed with the roudess coefficiet. he fittig to circle method is used i order to fid the ceter of the circle o the image. First of all, a cotour of the circle must be foud. Oly eighbor piels are scaed to reduce the processig time to a miimum. he sca rus clockwise util the cotour is closed. his way the followig equatio is true for every poit of the cotour (see Figure 4): c c, y y R (4) where ad y are the coordiates of ay cotour poit, c ad y c the ukow coordiates of the circle ceter, R the ukow radius of the circle. he Equatio (4) ca also be epressed as: Figure 3. Fragmet of the first image. Figure 4. Fittig to the circle. or y 2 2 c y c yc yc R 0, A B C (5) AByC 2 y 2. (6) o use all poits of the cotour, Equatio (6) is epressed i the matri form: 2 2 ABy C y y A 2 2 A2 By2 C 2 y2 2 y2 B (7) C 2 2 A By C y y ABC XY or XY ABC Q (8) he ukow matri ABC ca be computed with the help of the followig equatio: ABC XY Q, (9) where (XY) is the pseudo iverse matri of the matri XY. he coordiates of the circle ceter c ad y c ca be easily foud from the two first elemets of the matri ABC. 4. Camera Calibratio o determie the positio of the robot s ed-effector i a 3D-workspace it is ot sufficiet to have the 2D image coordiates of the balls measured by the two cameras. he 2D-data from the two cameras has to be fused to get the 3D-data. o do so, the relatioship betwee the two cameras must be eactly determied. I order to fid this relatioship, it is ecessary to carry out a measuremet, i which the two cameras measure the image positios of a set of objects i the well-kow 3D-positio. Furthermore, parameters of each camera which cause systematic measuremet errors ca also be determied i this measuremet. his procedure is called camera matchig ad calibratio. Various camera calibratio methods were described i [2-4]. hey are mostly based o the pi-hole camera model, where the 3D-poits are projected to the image through a sigle poit called camera ceter. he accuracy of the camera calibratio ca be icreased if the distortio of the camera image, caused by leses mouted o the camera is cosidered. Most popular models of the les distortio ca be foud i [5-7]. he calibratio of a multiple view camera system is more comple, because the itrisic ad etrisic parameters of the sigle camera as well as the relatioship (relative positio ad orietatio) betwee two (or more) cameras must be foud. Q

4 396 A. JANZ E AL. Computatio of the multiple view geometry is described i [8]. he descriptio of the two view geometry ca be foud i [9-]. 4.. Camera Model he most ofte used ad well-kow camera model is the pi-hole camera model [8]. his model is described with the help of the projectio matri P of the camera: P K R RC, (0) where R is rotatio matri of the camera i the world coordiate system, C coordiates of the camera ceter i the absolute coordiate system, K the camera calibratio matri. he camera calibratio matri K icludes parameters such as focal legth f ad positio of the pricipal poit p, p y i the coordiate system of the image f K f p p y. 0 ω iwi yw i i P ωiwi 0 W i i P2 0, i i i i 0 yw W P3 () he projectio matri P is used for triagulatio ad computig of the 3D-data from the 2D image data, because it epresses the relatioship betwee the world poits W ad their correspodig image poits w: w PW, (2) where w, y, ad W X, Y, Z, Idepedet Calibratio As idepedet calibratio of two cameras or simply idepedet calibratio i this paper we have applied a calibratio method, i which a set of 3D poits with eact positios are shot with two cameras simultaeously. he projectio matri is the computed from the 3D world data ad the 2D image. he 3D data ca be recostructed with help of the computed projectio matri ad the image data of both cameras ad the geometric recostructio error ca be foud. If the recostructio accuracy does ot satisfy the requiremets of the measuremet system, the projectio matrices ca be refied by iteratively miimizig the geometric recostructio error (for eample, Leveberg-Marquardt algorithm). he computatio of iitial projectio matrices is based o the direct liear trasformatio (DL). Accordig to [8], for each correspodece W i w i the followig relatioship ca be derived: A P (3) where w, y, ω are ormalized image poits, W ormalized 3D data, P, P 2 ad P 3 rows of projectio matri P i a ormalized form ad i ide of the poit. he projectio matri P ca be foud after havig obtaied the sigular value decompositio (SVD) of the matri A for all the give poits. So if A = UDV, the the solutio is the last colum of the matri V. Whe projectio matrices of both cameras are computed, the 3D world poit ca be recostructed. For this purpose, the triagulatio [8] from the correspodig image poits w ad w' of both cameras ad projectio matrices P ad P' (also based o the DL) is used: where w PW 0 AW 0, w PW 0 P3 P yp3 P2 A P P yp P (4) (5) he 3D data ca be recostructed with the help of the SVD of the matri A by computig the projectio matrices, like i the solutio above. I order to miimize the geometric error, projectio matrices must be refied usig the Leveberg-Marquardt (LM) miimizatio method. For this purpose both projectio matrices are composed to a optimizatio matri ad the LM-algorithm rus util the geometric error reaches a miimum. Oce the geometric error is miimized, the projectio matrices stay costat for further measuremets util some mechaical parameters (for eample, camera orietatio) are chaged Fudametal ad Essetial Matrices I the previous chapter, the projectio matrices were matched to each other with the help of the LM-algorithm i order to obtai the miimal recostructio error. Aother method based o the computatio of the fudametal matri, i.e. a matri, which cotais the relatioship betwee two cameras, their etrisic ad itrisic parameters. he fudametal matri is defied by the equatio w Fw 0, (6) where F is fudametal matri, w ad w' correspodig image poits. From (6) it follows that the fudametal matri ca be computed whe havig oly the correspodig image poits. o do so, the (6) must be trasformed to

5 A. JANZ E AL. 397 y y y y Af yy y y f 0, (7) y y y y where [, y, ], [', y', ] are correspodig image coordiates of a sigle poit, vector f represets the fudametal matri F ad is the umber of correspodig poits. Usig the least squares solutio i (7), oe ca easily compute the fudametal matri. It is also importat to use the sigularity costrait after the computatio of the iitial fudametal matri [8]. Projectios matrices ca be directly etracted from the fudametal matri. Ufortuately, the use of the fudametal matri i this work did ot provide a stable recostructio of the 3D data, therefore it will ot be further cosidered. However, the solutio with the fudametal matri was ot a complete failure either. Istead of the fudametal matri we ca use its specificatio to the case of the ormalized image coordiates the essetial matri [8], which is defied as: ˆ ˆ 0, w Ew w K EK w (8) where ŵ' ad ŵ are ormalized image poits, w' ad w origial image poits, K' ad K camera calibratio matrices, E the essetial matri. he computatio of this matri differs from the computatio of the fudametal matri oly i the sigularity costrai ad ormalizatio of the image data with the help of calibratio matrices. hus, if the sigular values decompositio of the iitial essetial matri is E USV, (9) the the fial essetial matri usig the sigularity costrait is E Udiag,, 0 V. (20) o etract projectio matrices of the cameras from the essetial matri, the projectio matri of the first camera is assumed to be P = [I 0]. It meas that all the recostructed data is epressed i the coordiate system of the first camera. Four possible solutios eist for the secod camera. hese solutios are based o the two possible choices of the rotatio matri R ad the offset t betwee the ceters of the two cameras, E t R (2) where [t] is the skew-symmetric matri of the vector t defied by 0 t3 t2 t t3 0 t. (22) t2 t 0 hese four solutios are:, P UOV t (23) P UOV t, (24) P UO V t, (25) P UO V t. (26) Here U ad V are matrices from (20), t radius vector to the ceter of the ceter of the secod camera i the coordiate system of the first camera t U0,0, ad a orthogoal matri 0 0 O 0 0. (27) 0 0 he choice of the right solutio depeds o the 3D recostructio usig computed projectio matrices. First, the projectio matri, the recostructio data of which look correct, is chose. o refie the recostructio accuracy it is ecessary to apply the LM-algorithm. As i chapter 4.2, the re-projectio error should be miimized by usig the essetial matri as a optimizatio parameter. he essetial matri ad fudametal matri ca be very useful, if oe does ot have the eact 3D data. However, two camera calibratio matrices are ecessary for the computatio of the essetial matri, which ca be foud with the help of the decompositio of the projectio matrices from the chapter 4.. It meas that camera calibratio with the use of the eact 3D data is oly ecessary at first. Later, if itrisic camera parameters do ot chage, the matchig of the coordiate systems of both cameras ca be doe usig the computatio of the essetial matri Image Distortio he calibratio methods described above are based o the pi-hole camera model, which meas, that poits i the world frame are projected to the image plae through a straight lie. However the les surfaces i the real optical systems always have deviatios from the desiged form. hese deviatios cause geometric distortios of the image. Several ways to estimate the distortio are described i [2-4]. he most popular distortio model ca

6 398 A. JANZ E AL. be described with followig equatio p Δp kr kr kr p, d t k p k d p pr pr pr 2 ΔP2 P23 k 3 K 3 (28) where p d is the distorted image poit, Δp t vector of tagetial distortio, p ormalized distortio-free image poit, k, k 2 ad k 3 are coefficiets of radial distortio, r the legth of the radius vector from the pricipal poit to the image poit p. he tagetial distortio is commoly caused by the lack of aligmet of the leses. he deviatio of the leses surfaces from the desiged form causes the radial distortio. Simplifyig the (28) by igorig the tagetial distortio term ad epressig it i matri form for a set of poits we get or (29) Δ P PK, (30) where is the umber of image poits. Usig this epressio we ca compute the radial distortio coefficiets usig the pseudo iverse matri from P K P P Δ. (3) he re-projected image poits are used as ormalized distortio-free poits p to compute radial distortio coefficiets. Usig the computed values of the radial distortio as a iitial guess the Leveberg-Marquardt Algorithm should be used to refie the distortio coefficiets Ihomogeeous Distributio of the Light Itesity Not oly ca the radial ad tagetial distortio ifluece o the measuremet accuracy, but the ihomogeeous distributio of the light itesity ca also decrease the measuremet accuracy locally. For eample LEDs i our illumiatio system ca be aliged slightly asymmetric like i Figure 5. It will be fie for the measuremet mark positioed i oe certai poit, but the displacemet of this mark i the directio of the camera optical ais ca cause more light o oe side of the mark tha o aother. his will shift the measured positio of the white ball i directio of the more illumiated area. It is also importat to metio, that due to the spectral sesitivity of the camera the camera with gree illumiatio will be more affected by the ihomogeeous distributio of light. he Figure 6 illustrates the effect described above. A set of poits distributed i a cube i the world frame was Figure 5. Light distributio. shot by both cameras. Measured image poits are show i figure as gree poits. I the right part of Figure 6 the projectio of these poits to the sesor of the camera #2 is show. here is o sig of a radial distortio i this projectio. However lookig closely at poits o the lie, which is approimately parallel to the optical ais of the camera, it becomes obvious, that these poits sigificatly deviate from the straight lie. So i the left part of Figure 6 measured poits draw a bow as the 3D-mark moves i the world frame aloe a straight lie away from the camera. his behavior of the measuremet system is reproducible after several movemets i the world frame alog optical ais of the camera measured poits draw the same patter. I differet areas of the camera field of view measured poits are drawig differet patters. his ca sigificatly reduce the accuracy of the measuremet system ad ca oly be corrected usig look-up tables. Sice the coordiates of poits i the world frame ad their correspodig image poits are kow a look-up table ca be computed for each camera, where the correctio vector for image poits depeds o the correspodig world coordiates. For this purpose the correctio vector has to be foud for each image poit as a ormal vector from the actual image poit to the approimated lie. he approimated lie is show i the Figure 6 as the black lie. Correctio vectors are show as blue lies. Red poits describe corrected poits. o fid correctio vectors for poits i each aial lie the followig operatios have to be doe. First of all, the middle poit of the lie has to be foud with the help of the equatio N p i i M, (32) where M is the [2 ]-vector with coordiates of the midpoit, p is the [2 N]-matri with coordiates of poits i the lie, N umber of poits alog the lie. he vectors from the midpoit to poits o the lie have N

7 A. JANZ E AL. 399 Figure 6. Set of poits for the estimatio of the light distributio. to be computed, X p M (33) where X is the matri of size [N 2]. From this matri the sample covariace matri ca be computed N C, X X X X (34) jk ij j ik k N i where C is the matri of size [N N]. Usig sigular value decompositio of the covariace matri C USRl the orietatio of the approimated lie ca be computed as the rotatio matri R l. he average value of rotatio matri R has to be computed to force the same orietatio of approimated lies i the whole measurig volume. Usig this rotatio matri ad matri X the correctio vectors ca be computed ad d Δ RX Δy Δ, Δy R (35) (36) where d is the [2 N]-matri with correctio vectors for C Figure 7. riliear iterpolatio. 2 y,, z C, y, z C 2 C 2 2, y, z, 2 y,, z C, y, z C 2 2 2, y, z, each poit o the lie. 2 Cy,, z C, y, z After correctio vectors have bee computed for each measured poit the 3D-look-up table is created. o correct image poits durig a olie measuremet the 3Dpositio of the mark has to be recostructed. Usig this pre-computed 3D-positio ad look-up table the eight earest eighbor correctio vectors must be picked up from the table. o fid the correctio vector C(, y, z) for pre-computed positio [ y z] the triliear iterpolatio (Figure 7) is used. First of all, correctio vectors are iterpolated alog the -ais C or i matri form C 2 2 2, y, z, 2, y, z C, y, z C 2, y, z (37) (38) (39) (40)

8 400 A. JANZ E AL. C y,, z Cy, 2, z y,, z Cy,, z C C C 2 C y z y z, y, z C2, y, z,, C,, C 2 C y z y z, y2, z C2, y2, z,, C,, (4) After the iterpolatio alog -ais the iterpolatio alog y-ais usig computed matri C follows y2 y y2 y Cy C (42) y y y2 y Ad the the correctio vector for the pre-computed positio results the from the iterpolatio alog z-ais z2 z y2 y z2 z z z y y z z yz C C C 2 2 2,, y. zz y y zz z2 z y2 y z2 z (43) Usig iterpolated values of the correctio vector o image poits the 3D-positio will fially be recostructed. 5. Results of the 3D Data Recostructio I order to test the accuracy of the described measuremet system ad to compare the suggested camera calibratio methods, the 3D mark was mouted o the ed effector of the coordiate-measurig machie (CMM) Werth VCIP 3D. he tested measuremet system was placed o the table of the CMM. he measuremet ucertaity of this CMM ca be assumed i the workspace of 20 millimeters as.5 micrometers. he measuremet volume of the CMM is millimeters, which is more tha eough for the test. he complete test setup is preseted i Figure 8. he 3D mark o the ed effector of the CMM was moved i the 3D cubic space of millimeters ad shot by two cameras simultaeously. o elimiate the ifluece of the piel oise, the positio of the 3D mark was measured i test poits, where the mark had to stay still for two secods i order to get a average value of the measured positio. o get more precise camera calibratio, we measured 267 poits. he offset betwee the eighborig poits i X-, Y- ad Z-directios Figure 8. est of the measuremet system with the CMM. is millimeter. 5.. Recostructio with Projectio Matrices After the actual measuremet procedure had bee completed, the 3D poits from the CMM ad the correspodig 2D image poits were available for the aalysis. First, both cameras were idepedetly calibrated i order to get projectio ad calibratio matrices. he, usig the computed projectio matrices, the 3D-poits were recostructed from the image poits ad the recostructio error was determied for each poit as the geometric deviatio of the recostructed positio from the real oe. he results of the recostructio usig idepedet calibratio (chapter 4.2) without ad with the correctio of the light ihomogeeity are preseted i Figure 9 ad i Figure 0 respectively. It is obvious from these two cases, that whe usig the correctio of the light ihomogeeity described i chapter 4.5, the accuracy improvemet of up to - 3 µm ca be achieved. At the same time the estimatio of the radial distortio has show, that the distortio of the used leses is very small ad the correctio of the distortio does ot lead to ay accuracy improvemet. For that reaso it has ot bee used i further eperimets. he average value of the recostructio error for the idepedet calibratio with the light ihomogeeity correctio is 3.5 micrometers. he recostructio error gets less i the middle of the show poit cube, where the optical aes of the cameras cross each other. After the idepedet calibratio the camera calibratio matrices ca be etracted from the projectio matrices. he essetial matri ca be computed usig the calibratio matrices. his way, the calibratio of the whole stereo video measuremet system was carried out by oly usig the essetial matri. he 3D data was recostructed with the help of the projectio matrices etracted from the essetial matri. he recostructio error was

9 A. JANZ E AL. 40 Figure 9. Recostructio usig the idepedet calibratio. Figure. Recostructio usig the essetial matri. Figure 0. Recostructio usig the idepedet calibratio ad light ihomogeeity correctio. aalyzed i the way described above. he results of the recostructio usig calibratio with the essetial matri (chapter 4.3) are preseted i Figure without light ihomogeeity correctio ad i Figure 2 with light ihomogeeity correctio. Similar to the eperimet with the idepedet calibratio the light ihomogeeity correctio improves accuracy of the system i case of the essetial matri method up to µm. he average value of the recostructio error for this method is 0.05 micrometers. Like i the previous method the recostructio error gets less i the middle of the poits cube, but the distributio of this error i the workspace is much steadier. It is obvious that the recostructio results become much more accurate after the system has bee calibrated usig the essetial matri. It ca be eplaied by the fact that the essetial matri describes ot oly the itrisic parameters but also the positio ad orietatio of both cameras relatively to each other. Ulike it, a simple idepedet computatio of the projectio matrices oly describes the relatioship betwee the world poits ad the camera image. herefore, the calibratio with the Figure 2. Recostructio usig the essetial matri ad the light ihomogeeity correctio. essetial matri ca better match the coordiate systems of the two cameras tha the idepedet calibratio Affie Recostructio I respect to the fact that both cameras have telecetric leses, their camera models ca be approimated to the affie camera model all projectio lies are assumed to be parallel to each other ad the last row of the projectio matri has the form (0, 0, 0, ). Usig this assumptio, the affie recostructio ca be implemeted to get the 3D data from the correspodig image poits. he recostructio ca be carried out with the help of the factorizatio algorithm described i [8] (the algorithm 8.). First of all, the origi of every image must be traslated to the ceter of poits i this image. he traslatio vector t for the camera with the ide i is the i i t w, (44) j where w i are poits o the image i, j ide of the poit, umber of poits o the image. he traslatio is de- j

10 402 A. JANZ E AL. fied by the equatio i i i wˆ w mt, (45) where w i are origial image poits i the image i, ŵ i ew or traslated poits i the image i, m the vector of size [ ], all the elemets of which are equal. he the measuremet matri M of size [4 ] is composed from the ew image coordiates ŵ i : wˆ wˆ2 wˆ M (46) w ˆ ˆ ˆ w 2 w he from its SVD M = UDV the 3D ormalized poits are obtaied from the first three colums of V. As described above, the affie recostructio was tested usig the same measuremet data as used i chapter 5.; the geometric recostructio error was aalyzed as well. he results of the affie recostructio are preseted i Figure 3 without the light ihomogeeity correctio ad i Figure 4 with the light ihomogeeity correctio. he average value of the recostructio error for this method is approimately 3.25 micrometers. Ulike the recostructio usig the idepedet calibratio ad essetial matri, the affie recostructio does ot require the computatio of the camera projectio matrices. he recostructio occurs directly from the image data; at least four correspodig poits are required though. However, this recostructio method caot be used for a olie-measuremet of the 3D-positio due to the computatio effort caused by a large data volume. 6. Coclusios he recostructio results ad the measuremet accuracy ca be aalyzed more carefully. Figure 5 depicts a distributio of the recostructio error for all the three described methods. Here the whole error area is divided ito 50 itervals. he umber of poits with the recostructio error is foud i every iterval ad the preseted as poit-marks o the graphic. he average value of the error for all poit is preseted as a lie. It is obvious from the figure that the distributio of the recostructio error is similar to the affie recostructio ad the recostructio usig idepedet calibratio. he distributio of the error for the recostructio usig the essetial matri is very small i compariso to the other two methods. Furthermore, the average value of the recostructio error i case of the essetial matri is sigificatly smaller tha i the other two cases. herefore, the calibratio usig the essetial matri will be used for the calibratio of the system as the stadard method. he idepedet calibratio will be oly used to compute the iitial values of the calibratio matrices if the itrisic parameters have somehow chaged. Figure 3. Affie recostructio. Figure 4. Affie recostructio usig correctio of the light ihomogeeity. Figure 5. Distributio of the recostructio error. Note that i order to estimate the accuracy of the measuremet system, the image processig accuracy of.5 micrometers (the piel oise was strogly suppressed for the calibratio measuremet) must be added to the recostructio error. hus, if the average value of the recostructio error usig essetial matri is about 0.05 micrometers, the accuracy of the measuremet system is.55 micrometers.

11 A. JANZ E AL. 403 REFERENCES [] R. C. Gozalez ad R. E. Woods, Digital Image Processig, 2d Editio, Pretice Hall, Upper Saddle River, [2] J. Heikkilä, Geometric Camera Calibratio Usig Circular Cotrol Poits, IEEE rasactio o Patter Aalysis ad Machie Itelligece, Vol. 22, No. 0, 2000, pp [3] Z. Zhag, Fleible Camera Calibratio by Viewig a Plae from Ukow Orietatios, Proceedigs of the Iteratioal Coferece o Computer Visio ICCV, Kerkyra, September 999, pp [4] R. Y. sai, A Versatile Camera Calibratio echique for High-Accuracy 3D Machie Visio Metrology Usig Off-the-Shelf V Cameras ad Leses, IEEE Joural of Robotics ad Automatio, Vol. 3, No. 4, 987, pp [5] C. Ricolfe-Viala ad A.-J. Saches-Salmero, Les Distortio Models Evaluatio, Applied Optics, Vol. 49, No. 30, 202, pp [6] Z. Zhag, A Fleible New echique for Camera Cali- bratio, IEEE rasactio o Patter Aalysis ad Machie Itelligece, Vol. 22, No., 2000, pp [7] R. Hartley ad S. B. Kag, Parameter-free Radial Distortio Correctio with Cetre of Distortio Estimatio, Proceedig of the 0th IEEE Iteratioal Coferece o Computer Visio ICCV 05, Beijig, 7-2 October 2005, pp. -8. [8] R. Hartley ad A. Zisserma, Multiple View Geometry i Computer Visio, 2d Editio, Cambridge Uiversity Press, Cambridge, [9] R. Hartley ad C. Silpa-Aa, Recostructio from wo Views Usig Approimate Calibratio, Proceedigs of the 5th Asia Coferece o Computer Visio ACCV 02, Melboure, Jauary 2004, pp. -6. [0] B. Albouy, S. reuillet, Y. Lucas ad D. Birov, Fudametal Matri Estimatio Revisited hrough a Global 3D Recostructio Framework, Proceedigs of the Advaced Cocepts for Itelliget Visio Systems ACIVS 2004, Brussels, 3 August-3 September 2004, pp [] K. Kaatai ad Y. Sugaya, Compact Fudametal Matri Computatio, Proceedigs of the 3rd Pacific Rim Symposium of Image ad Video echology, okyo, 3-6 Jauary 2009, pp