线结构光-标定和测距.pdf

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Complete calibration of a structured light stripe vision sensor through planar target of unknown orientations

Introduction

Mathematical model of the structured light stripe vision sensor

Camera model

Projector model

A flexible new calibration method

Camera calibration

Transformation from world coordinate frame to 3D camera coordinate frame

Computing 3D camera coordinates of control points

Projector calibration

Summary

Experiments

Conclusions

Acknowledgements

References

Image and Vision Computing 23 (2005) 59–67 www.elsevier.com/locate/imavis Complete calibration of a structured light stripe vision sensor through planar target of unknown orientations Fuqiang Zhou*, Guangjun Zhang School of Instrument Science and Opto-electronics Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, China Received 30 January 2004; accepted 7 July 2004 Abstract Structured light 3D vision inspection is a commonly used method for various 3D surface proﬁling techniques. In this paper, the mathematical model of the structured light stripe vision sensor is established. We propose a ﬂexible new approach to easily determine all primitive parameters of a structured light stripe vision sensor. It is well suited for use without specialized knowledge of 3D geometry. The technique only requires the sensor to observe a planar target shown at a few (at least two) different orientations. Either the sensor or the planar target can be freely moved. The motion need not be known. A novel approach is proposed to generate sufﬁcient non-collinear control points for structured light stripe vision sensor calibration. Real data has been used to test the proposed technique, and very good result has been obtained. Compared with classical techniques, which use expensive equipment such as two or three orthogonal planes, the proposed technique is easy to use and ﬂexible. It advances structured light vision one step from laboratory environments to real engineering 3D metrology applications. q 2004 Elsevier B.V. All rights reserved. Keywords: Structured light; Calibration; Control points; Planar target; Vision inspection 1. Introduction With the development of electronics, photo-electronics, image processing and computer vision technique, vision inspection has made giant strides in development. It is becoming increasingly relevant in industry for dimensional analysis, on-line inspection, component quality control and solid modeling, such as the 3D contouring and gauging of large-surface car parts and small-size microelectronic components, and the fast-dimensional analysis of object in relation to the recognition of targets by robots. And also it played an important role in reverse engineering. Recently, systems based on vision cameras allow automated and non-contact 3D measurements, which are based on two alternative techniques: stereovision and triangulation from structured light stripe vision. Structured light stripe vision inspection avoids the so-called corre- spondence problem of passive stereo vision and has gained * Corresponding author. Tel.: C86 10 82316930; fax: C86 10 82314804. E-mail address: fuqiang_zhou@yahoo.com (F. Zhou). 0262-8856/$ - see front matter q 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.imavis.2004.07.006 the widest acceptance [1–4] in industry inspections due to fast measuring speed, very simple optical arrangement, non- contact, moderate accuracy, low cost, and robust nature in the presence of ambient light source in situ. Since the early 1970s [5], research has been active on shape reconstruction and object recognition by projecting structured light stripes onto objects. A basic structured light stripe vision sensor consists of one camera and one laser projector, which together form an active stereo pair. The projector is displaced relative to the camera in space. For example, one of arrangements is shown in Fig. 1, the light stripe is modulated by the depth of the measured surface. Thus, the deformed light stripe contains rich 3D characteristic information of the object surface. The camera captures the image of the object with the deformed light stripe. The task of structured light vision inspection is to acquire the 3D characteristic information of the measured surface from the 2D deformed light stripe image, i.e. 3D reconstruction. A structured light stripe vision sensor generates dense world points by sampling image points on each light stripe in the image. The absence

60 F. Zhou, G. Zhang / Image and Vision Computing 23 (2005) 59–67 The approaches for estimating the structured light vision sensor’s model parameters are determined by the estab- lished mathematical model itself. Currently, there has been two major ways to establish the mathematical models of structured light stripe vision sensor. One is based on the simple triangular methodology, and another is based on the perspective, translation and rotation transforms in hom- ogenous coordinate system [4]. The perspective projection model is broadly used in the 3D vision inspection. Until now, the extraction of calibration world points for a structured light vision sensor has remained a difﬁcult problem. This is because usually the known world points on the calibration object do not normally fall on the light stripe plane illuminated from the projector, which poses difﬁculty for the sensor calibration. Different approaches for calibrating structured light vision sensor have been proposed in many literatures. In Refs. [12,13], several thin threads that are non- coplanar are strained in the space, and the illuminated light stripe plane from the projector of the structured light vision sensor will intersect the threads at several bright light dots, and these bright light dots are used as the control points. Then a 3D coordinate measuring system such as movable theodolite measurement system is used to measure the 3D coordinates of the bright light dots. However, the strained thread method has the following disadvantage: (1) another 3D calibration target is needed to calibrate camera; (2) the procedure to generate control points for projector cali- bration is too complex, and only a few points are obtained; (3) as the bright light dot is in fact a kind of distribution of light intensity, it is difﬁcult to match the world bright light dots on the thread strictly with the image bright light dots in the camera image plane, thus the accuracy of the obtained control points is usually poor. In the Ref. [14], a more sophisticated calibration target is used, which has a zigzag-like face. This method uses the intersection, which are generated by the emitted light stripe plane intersecting with the ridge of the zigzag, as the control points. In this method, the light stripe plane is perpendicular to the datum plane lying on the zigzag is required. Moreover, it has similar problems as the strained thread method. Chen and Kak [15] proposed the method of the known world line and image point correspondences to directly get the image-to-world transformation matrix. In this method, at least six world lines are required. Later Reid [16] extended the concept to the known world plane and image point correspondence. In Reid’s method, a number of known world planes are required. Xu et al. [17] and Huynh [18] proposed, respectively, a method that is based on the invariance of the cross-ratio to generate the world points on the light stripe plane. The control points can be got by a calibration target whose geometry in 3D space is known with very good precision. The 3D calibration target usually consists of two or three planes orthogonal to be to each other and is difﬁcult Fig. 1. Overview of the structured light vision sensor. of the difﬁcult matching problem makes active stereo an attractive method for many vision inspection tasks with the trade-off that the sensor must be fully calibrated prior to use [6]. In the camera calibration stage, The procedures to estimate the sensor model parameters are referred to sensor calibration. Sensor calibration is necessary step in vision inspection in order to extract metric information from 2D images. The traditional approach to calibrating a structured light stripe sensor incorporates two separate stages: camera calibration and projector calibration. the world-to-image perspective transformation matrix and the distorted coefﬁ- cient are estimated using at least six non-coplanar world points and their corresponding image projection points [7]. The transformation matrix can be decomposed into camera intrinsic parameters (e.g. effective focal length, principal point and skew coefﬁcient) and extrinsic parameters (including the 3D position and orientation of the camera frame relative to a certain world coordinate system). Camera calibration is performed by observing a calibration object whose geometry in 3D space is known with very good precision. Calibration can be done very efﬁciently [8, 9]. The calibration target usually consists of two or three planes orthogonal to each other. Sometimes, a plane undergoing a precisely known translation is also used [10]. These approaches require an expensive calibration apparatus, and an elaborate set-up. Zhang proposed a ﬂexible new technique for camera calibration by view a plane from the different unknown orientations, this approach is very easy to use and obtained very good calibration accuracy [11]. In the projector calibration stage, the coefﬁcients of the equation of light stripe plane relative to the same 3D coordination frame must be determined. The projector calibration is related to two principal procedures. One is constructing non-collinear control points. The 3D world coordinates of those control points and their corresponding 2D image coordinates can be estimated in this procedure. The other is establishing the mathematical model of the sensor and estimating the parameters of the model.

F. Zhou, G. Zhang / Image and Vision Computing 23 (2005) 59–67 61 2. Mathematical model of the structured light stripe vision sensor 2.1. Camera model manufactured accurately. It is also very difﬁcult to capture the good calibration image by simultaneously viewing the different planes of the 3D calibration object. These approaches require an expensive calibration apparatus. Besides the methods already mentioned, there are other methods in the literature, such as, standard ruler calibration target mounted on a step-motor-controlled stage with one linear translation axis [19–21], a photoelectrical aiming device combined with a 3D movable platform [22], and so on. But because all these methods are demanding for the precision of the moveable stage, inconvenient for operation, and time-consuming in calibration, none of them is suitable for or widely accepted on-line calibration. To sum up, the methods of calibrating a structured light vision sensor presented in the literature mainly have the following drawbacks: (1) it is difﬁcult to generate large number of highly accurate control points. (2) some methods require an expensive calibration appar- atus, and elaborate setup. (3) the method is well unsuitable for on line sensor calibration in situ. These factors obstruct the improvement of the calibration accuracy of structured light vision sensor and their engineering 3D metrology applications. In this paper, inspired by the work of Huynh and the work of Zhang, we propose a novel approach that employs the invariance of a cross-ratio to generate an arbitrary number of control points on the light stripe plane by viewing a plane from unknown orientations. The captured images of the planar target can be used for camera calibration and projector calibration simultaneously. The proposed techniques only requires the camera to observe a planar target shown at a few (at least two) different orientations. The target can be printed on a laser printer and attached to a ‘reasonable’ planar surface (e.g. a hard book cover). The planar target can be moved by hand. The motion need not be known. In many respects the approach taken here is similar to bundle adjustment procedures used in photogrammetry in that a nonlinear method is used to adjust the system parameters to best accommodate all measured data. The difference is that in our calibration approach, one of the components (the laser striper) is not camera, although it can be considered in some respects to be a special type of linear camera. The paper is organized as follows. Section 2 introduces the mathematical model of the structured light stripe vision sensor. The model includes the perspective projection model of the camera and the projector model. Section 3 describes the procedure of calibrating the sensor. This procedure includes camera calibration, constructing the control points lying onto the light stripe plane and projector calibration. Section 4 provides the experimental results. Real data is used to validate the proposed technique. ; yn Fig. 2 shows the perspective projection model of camera in the structured light vision sensor. Note that on is usually the center of the camera image plane pc, oc is the projection center of the camera and zc axis is the optical axis of the camera lens. The relative coordinate frames are deﬁned as follows: onxnyn is 2D normalized image coordinate frame, ouxuyu is 2D image plane coordinate frame, owxwywzw is 3D world coordinate frame and ocxcyczc is 3D camera coordinate frame. In addition, we deﬁne some known geometry relations in the camera model, such as ocxc//ou xu//onxn, ocyc//ouyu//onyn and oczc tpc. Given one point Pw in 3D space, its homogeneous Z coordinate in 3D world coordinate frame is denoted by ~pw ðxw ; 1ÞT and its homogeneous coordinate in 3D ; 1ÞT . camera coordinate frame is denoted by ~pc Pn is the ideal perspective projection of Pw in the image plane of camera, its 2D homogeneous coordinate in onxnyn is ; 1ÞT and its 2D homogeneous denoted by ~pn ; 1ÞT . Pd is coordinate in ouxuyu is denoted by ~pu the real projection of Pw in the image plane of camera, its ; 1ÞT . 2D real coordinate in ouXuYu is denoted by ~pd Based on the pinhole camera imaging theory and is ; yu Zðxd the camera model Zðxu Zðxn Zðxc ; yw ; zw ; yc ; zc ; yd perspective projection theory, described as follows: (1) Rigid body transformation from the 3D world coordi- nate to the 3D camera coordinate: ! (1) ~pc Z Hc w ~pw where Hc w Z Rc w Tc w 0T 1 Fig. 2. Perspective projection model of camera in the structured light vision sensor.

62 F. Zhou, G. Zhang / Image and Vision Computing 23 (2005) 59–67 is 4!4 transformation matrix, which relates the world coordinate frame to the camera coordinate frame. Tc w is 3!1 translation vector and Rc w is 3!3 orthogonal rotation matrix. (2) Transformation from the 3D camera coordinate to the 2D normalized image coordinate l1 ~pn ZðIj0Þ ~pc ; l1 s0 (2) where I is 3!3 identity matrix, l1 is an arbitrary scale factor. (3) Transformation from the 2D normalized image coordi- nate to the 2D undistorted image coordinate l2 ~pu Z A ~pn ; l2 s0 (3) Fig. 3. Projector model of the structured light stripe vision sensor. Given an arbitrary point ps lying on the light stripe plane ps. pn is ideal projection point in the image plane pc. ps is the intersection point of ocpn and ps. pn is the intersection point of ocpn and pc. Therefore, if the plane equation of ps and the line equation of ocpn are known in ocxcyczc, the 3D coordinate of the measured point ps can be determined completely in ocxcyczc. Z(xc, yc, zc)T be the coordinates of a point on the surface of object being proﬁled, in ocxcyczc relative to the object. The equation of the light stripe plane ps is described by Let pc axc C byc C czc C d Z 0 (6) Eqs. (4)–(6) constitute a complete real mathematical model of the structured light stripe vision sensor. Based on this model, the calibration of the structured light stripe vision sensor includes two separate steps: (1) camera calibration, i.e. estimating the intrinsic parameters and extrinsic parameters of the camera model, for which at least six non-coplanar world points and their corresponding image projection points are required; and (2) projector calibration, i.e. the calibration of the light stripe plane emitted from the projector. At least three non-collinear control points on the light stripe plane are required for this step. Often more points are necessary to improve the calibration accuracy. 3. A ﬂexible new calibration method 3.1. Camera calibration The camera intrinsic parameters including distortion coefﬁcients can be calibrated with multiple views of the planar target by applying the calibration algorithm described in Ref [11]. All the views of the planar target are acquired by one camera in different positions, thus each view has separate extrinsic parameters, but common intrinsic parameters. Due to the non-linear nature of Eq. (5), simultaneous estimation of the parameters involves using an iterative algorithm to minimize the residual between the model and N 0 B@ where A Z 1 CA fx 0 0 0 fy 0 u0 v0 1 is called the intrinsic parameters matrix, fx and fy are the effective focal length in pixels of the camera in the x and y direction, respectively, (u0,v0) is the principal point coordinates of the camera. l2 is an arbitrary scale factor. According to the Eqs. (1)–(3), the ideal camera model can be represented as follows: ; l Z l1l2 Z AðRc wÞ ~pw wjTc l ~pu s0 (4) ( We choose the following model to handle lens distortion effects. xd yd Z xu Z yu C xur2ðk1 C yur2ðk1 C k2r2Þ C 2p1xuyu C k2r2Þ C p1ðr2 C 2y2 C p2ðr2 C 2x2 uÞ uÞ C 2p2xuyu (5) C y2 where r2Z x2 u, k1 and k2 are the coefﬁcients of the u radial distortion, p1 and p2 are the coefﬁcients of tangential distortion. Eqs. (4) and (5) completely describe the real perspective projection model of camera in the structured light vision sensor. All the parameters in the camera model can be estimated by camera calibration. 2.2. Projector model Fig. 3 shows the Projector model of the structured light stripe vision sensor, which relatives the light stripe plane coordinate frame to the camera coordinate frame. It is useful to identify several coordinate frames as shown in Fig. 3. owxwywzw is 3D world coordinate frame. ocxcyczc is 3D camera coordinate frame. onxnyn is 2D normalized image coordinate frame. These three coordi- nate frames have the same deﬁnition as the frames in Section 2.1.

F. Zhou, G. Zhang / Image and Vision Computing 23 (2005) 59–67 63 observations. Typically, this procedure is performed with least squares ﬁtting, where the sum of squared residuals is minimized. The objective function is then expressed as F Z K xdiÞ2 Cðymi K ydiÞ2Þ (7) X ððxmi N iZ1 where (xdi, ydi) is the real image coordinate and (xmi, ymi) is the estimated image coordinate by 3D world point according to the camera projective model. Two coefﬁcients for both radial and tangential distortion are often enough [9]. Then, a total of eight intrinsic parameters are estimated. The more detailed algorithm is described in Ref. [11]. 3.2. Transformation from world coordinate frame to 3D camera coordinate frame Without loss of generality, we assume the calibration Z0 of the local world coordinate frame target plane is on zw Let’s denote the ith column of the rotation matrix by ri. From Eq. (4), we have Z Að r1 l ~pu r2 TÞ ~pw (8) by abuse of notation, we still use ~pw to denote the homogeneous coordinate of a point on the calibration ; 1T. Therefore, a cali- ~pw object’s plane, i.e. ~pw and its image ~pu is related by bration plane point homography H: Z½xw ; yw l ~pu Z H ~pw with 8>>>>>< >>>>>: (9) r2 TÞ Zð h1 h2 h3 Þ H Z Að r1 where hi denotes the ith column of the matrix H. Given four or more points ~pwi, where iZ1,.,n for nR4, in general position (i.e. no three points are collinear) on calibration plane and their perspective projection ~pui under a perspective center oc. If camera intrinsic parameters are known, the transformation from world coordinate frame to 3D camera coordinate frame can be obtained as follows [11]: r1 Z lA Z lA r2 Z r1 r3 T Z lA K1h1 K1h2 !r2 K1h3 where lZ 1=jjA matrix relates the world coordinate frame to the camera coordinate frame is expressed as: K1h2jj. Then the rotation K1h1jjZ 1=jjA (10) r2 r3 Þ ^R Zð r1 (11) The matrix ^R does not satisfy the orthonormality constraint of a standard rotation matrix, but we can normalize and orthonormalize it using the singular value decompo- sition (SVD): ^R Z UWVT The orthonormal version of ^R is given by R Z UW (12) (13) 0 VT 0Z diagð1; 1;jUVTjÞ. whereW In practice, the linear algorithm describes above is quite noisy because nine parameters for a system of equations with six degrees of freedom. However, the linear estimation is close to the true solution to sever as a good initialization for non-linear optimization technique. All the more accurate parameters of the transformation from the world frame to the camera frame can be estimated by Levenberg– Marquardt non-linear optimization of the cost function described in Eqs. (7) and (8). 3.3. Computing 3D camera coordinates of control points In Fig. 4, ocxcyczc is the 3D camera coordinate frame. oixiyizi is the ith 3D local world coordinate frame deﬁned on the planar calibration target for ziZ0, iZ1,.,m. oizi is the normal of the calibration plane. The plane pc is image plane, the plane ps is the light stripe plane and the plane pi is the calibration target plane. The plane ps intersects the plane pi at the straight line Ls. Lj, which consists of three or more known world points, is one column in the planar target. lj is the perspective projection of Lj. As shown in Section 2.2, to calibrate the projector, at least three non-collinear control points on the light stripe plane are required in the 3D camera coordinate frame. Often Fig. 4. Computing 3D camera coordinates based on the invariance of the cross-ratios. Using three collinear world points (dark circles {Aj, Bj, Cj j jZ 1,.,n}), open circles Qj’s world points lying on pi can be computed. oc and os are the perspective centers of the camera and projector.

64 F. Zhou, G. Zhang / Image and Vision Computing 23 (2005) 59–67 more control points are necessary to improve the calibration accuracy. Therefore, we must obtain the 3D camera coordinates of the control points lying on Ls prior to projector calibration. Under perspective projection, the length or two lengths’ ratio will change, but the ratio of two ratios, which is called the cross-ratios, remains unchanged. That is to say, the four collinear points (Aj,Qj,Bj,Cj) and their projected points in the images plane (aj,qj,bj,cj) have the same cross-ratios. The cross-ratio is deﬁned as: rðAj ; Qj ; Bj ; CjÞ Z AjBj QjBj : AjCj QjCj Z ajbj qjbj : ajcj qjcj (14) where jZ1,.,n. The image coordinates of the points aj, qj, bj, cj can be estimate with sub-pixel accuracy by image processing. Then the normalized image coordinates of the points aj, qj, bj, cj can be obtained from their corresponding image coordinates according to camera model. As shown in Fig. 4, the known collinear world points Aj, Bj, Cj do not fall onto ps, the world coordinate of the control point Qj that falls on ps can be computed according to Eq. (14). Then the camera coordinates of the control points Qj can be computed from their corresponding world coordinates based on Eq. (1). We only obtain the collinear control points if we use one view of the planar calibration target from an arbitrary orientation. Therefore, we must construct the calibration control points by using multiple views of the same planar calibration target from the different orientations. All views of the calibration plane are acquired by one camera in the different positions. Zð xi We use m views of the captured planar target image with 1ÞT n control points for each view. Let ~pi 1ÞT be the homogeneous coordi- j and ~pi cj nate of the jth control point in the ith local world coordinate frame and its camera coordinate, respectively, where iZ 1wm, jZ1wn. Zð xi yi cj zi cj yi j zi j cj j onto the light stripe plane, and their corresponding normal- ized image coordinates. Hc i and ~pi j are determined, ~pi cj for all the control points lying on the different views, respectively, can be computed according to (14). Theoretically speaking, as the scheme described above, many non-collinear control points lying on the light stripe plane can be constructed by viewing the same planar calibration target from multiple unknown different orientations. The camera coordinates of all the non-collinear control points lying on the light stripe plane and their corresponding normalized image coordinates can be denoted as follows. i ~pi ZgHc j Q ZgQi ~pi cj where iZ1wm, jZ1wn. Qi j is the normalized coordinate of images of the jth control points respective to the ith view. (16) j 3.4. Projector calibration An arbitrary number of control points (xci, yci, zci) (iZ 1,.,k) on the structured light stripe plane can easily be obtained as scheme described above. Use Eq. (6) to ﬁt the k control points with the nonlinear least squares. The objective function is the square sum of the distance from the control points to the ﬁtted plane: X k fðxÞ Z iZ1 d2 i ; i Z 1.k (17) xZ(a,b,c,d), where di Zjaxci C byci C czci C dj=ða2 C b2 C c2Þ1=2 After the camera and projector calibration are completed, we can reconstruct the metric 3D camera coordinates from the 2D real image points based on Eqs. (4)–(6). 3.5. Summary The 3D camera coordinates of the non-collinear control points can be computed from their local world coordinates frame as follows. In summary, for the light stripe plane illuminated from the projector, the procedure for the proposed calibration method is as follows: ! ~pi cj Z Hc i ~pi j where Z Hc i Rc i Tc i 0T 1 is 4!4 transformation matrix, which relates the ith local world coordinate frame to the 3D camera coordinate frame. is 3!3 orthogonal Tc i rotation matrix. is 3!1 translation vector and Rc i Hc i is estimated as the scheme described in Section 3.2 with additional at least six the known world points for each view, which fall onto the calibration target and don’t fall (15) (1) Extracting all the image coordinates of the light stripe (2) Correcting the distortion for those image coordinates according to Eq. (5) (3) Fitting the feature line with the image coordinates of undistorted image coordinates obtained in (2). (4) Fitting the straight lines with the undistorted image coordinates of the known world collinear points (at least three) with lying on the planar calibration target but not falling on the light stripe plane. (5) Computing the intersection points of the feature line and the ﬁtted lines in (4), respectively. (6) Computing the normalized image coordinates of the control points which are computed in (5) according to Eq. (3).

F. Zhou, G. Zhang / Image and Vision Computing 23 (2005) 59–67 65 Fig. 5. Designed structured light vision sensor. (7) Computing the cross-ratio as given by Eq. (14) for each image line lj and computing the local world coordinates. (8) Computing the ith transformation Hc i with another feature points lying on the ith calibration plane but not on the light stripe and their corresponding image coordinates. (9) Computing the 3D camera coordinates of the control points. (10) Estimating the equation of the light stripe plane in 3D camera coordinate frame as described in Section 3.4. Because our method uses unknown orientation planar calibration target, the proposed calibration method can be easily extended to handle an arbitrary large number of control points. The calibration is very easy and ﬂexible. 4. Experiments We designed a structured light vision sensor, which is shown in Fig. 5. The sensor consists of an off-the-shelf WATEC CCD camera (902H) with 16 mm lens and a stripe laser projector (wave length, 650 nm, line width %1 mm). The image resolution is 768!576. The distance between the camera and the laser projector is about 500 mm. The Fig. 6 shows the designed planar calibration target. The calibration target contains a pattern of 3!3 squares, so there are 36 corner points. The size of the pattern is 35!35 mm2. The distance between the near two squares is 35 mm in the horizontal and the vertical directions. It was printed with a high-quality printer and put on glass. Fig. 7. The normalized image coordinates of the control points lying on the light stripe plane. Five images of the plane under different orientations were taken. The 36 corner points were detected as the intersection of straight lines ﬁtted to each square. The camera intrinsic parameters including distortion coefﬁcients can be calibrated with those corner points by applying the calibration algorithm described in the reference [11]. The camera intrinsic parameters are fxZ1521.204 pixels, fyZ1515.462 pixels u0Z400.987 pixels, v0Z284.554 pixels k1ZK4.352!10 p1ZK1.789!10 K1, k2Z1.955 K3, p2ZK1.295!10 K3 Let the laser stripe plane fall onto the calibration squares as shown in Fig. 6, two images of the plane under different positions were taken. After correcting the distortion of the two images, we extract the feature light stripes and the corner points from the two undistorted images. The calibration control points used for calibrating the structured light vision sensor can be constructed with the scheme described above. The normalized image coordinates are shown in Fig. 7 and the 3D camera coordinates are shown in Fig. 8, respectively. We use 12 control points to calibrate the light stripe plane, and get the structured light stripe plane Fig. 6. The sketch of the planar calibration target and the feature light stripe lying on it. Fig. 8. The 3D camera coordinates of the calibration control points lying onto the light stripe plane.

66 F. Zhou, G. Zhang / Image and Vision Computing 23 (2005) 59–67 Table 1 Sensor calibration accuracy evaluation with camera coordinates of control points Number View 1 View 2 1 2 3 4 5 6 1 2 3 4 5 6 Camera coordinates of control points constructing by invariance of cross-ratio Camera coordinates of control points computing based on sensor model Camera coordinates errors x (mm) K85.434 K57.316 K29.197 K1.078 26.989 55.158 K78.418 K48.964 K19.606 9.800 39.194 68.612 y (mm) K50.255 K29.498 K8.742 12.013 32.843 53.526 K46.628 K28.285 K9.811 8.599 27.024 45.416 z (mm) 630.201 628.309 626.418 624.525 622.668 620.742 632.424 637.039 641.717 646.363 651.018 655.656 x (mm) K85.357 K57.242 K29.097 K1.140 27.030 55.234 K78.378 K48.961 K19.523 9.811 39.214 68.644 y (mm) K50.427 K29.625 K8.800 11.885 32.728 53.596 K46.777 K28.335 K9.880 8.510 26.943 45.393 z (mm) 630.411 628.508 626.602 624.709 622.802 620.892 632.921 637.474 642.031 646.571 651.122 655.677 RMS errors Dx (mm) K0.077 K0.074 K0.100 0.062 K0.041 K0.076 K0.040 K0.003 K0.083 K0.011 K0.020 K0.032 0.059 Dy (mm) 0.172 0.127 0.058 0.128 0.115 K0.070 0.149 0.050 0.069 0.089 0.081 0.023 0.103 Dz (mm) K0.210 K0.199 K0.184 K0.184 K0.134 K0.150 K0.497 K0.435 K0.314 K0.208 K0.104 K0.021 0.255 parameters as follows a Z K0:5324; b Z 0:7547; c Z 0:3832; d Z 248:998 If the local world coordinates and the normalized image coordinates of the control points are obtained with good accuracy, the camera coordinates of control points which are computed based on invariance of the cross-ratio as scheme described above should be accurate. We can use those camera coordinates as the approximate ground values of control points. In addition, we can estimate the camera coordinates by sensor model and we use those estimated coordinates as the measured values of control points. We evaluate the sensor calibration accuracy by comparing the approximate ground values to the measured values. Table 2 Sensor accuracy evaluation with distance between two control points (Point 1 point 2) View 1 dw (mm) drc (mm) (0 1) (0 2) (0 3) (0 4) (0 5) (1 2) (1 3) (1 4) (1 5) (2 3) (2 4) (2 5) (3 4) (3 5) (4 5) Dd RMS error (30) 35.001 70.001 105.002 140.003 175.003 35.001 70.001 105.002 140.002 35.001 70.002 105.002 35.001 70.001 35.000 35.026 70.090 104.918 140.013 175.149 35.064 69.892 104.987 140.123 34.829 69.924 105.060 35.095 70.231 35.136 Dd (mm) K0.025 K0.088 0.084 K0.010 K0.146 K0.063 0.109 0.015 K0.121 0.172 0.078 K0.058 K0.094 K0.230 K0.136 The results are listed Table 1. The RMS errors are as follows DxZ0.059 mm, DyZ0.103 mm, DzZ0.255 mm. The distance between any two points lying onto the same feature light stripe can be computed as follows q K xw;iC1Þ2 Cðyw;i ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðxw;i K zw;iC1Þ2 q ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ K zc;iC1Þ2 ðxc;i K yw;iC1Þ2 Cðzw;i K xc;iC1Þ2 Cðyc;i K yc;iC1Þ2 Cðzc;i (18) dw Z drc Z (19) where (xw,i, yw,i, zw,i) and (xc,i, yc,i, zc,i) (iZ0,.,5) are the local world coordinate of the ith point and the corresponding 3D camera coordinate which is obtained by measuring, respectively. Generally speaking, dw should be computed accurately based on invariance of the cross-ratios if the high-accuracy View 2 dw (mm) 35.004 70.005 105.008 140.010 175.013 35.002 70.004 105.007 140.009 35.003 70.005 105.008 35.002 70.005 35.003 0.085 mm drc (mm) 35.017 70.059 104.977 139.978 175.010 35.043 69.960 104.961 139.994 34.918 69.918 104.951 35.000 70.033 35.033 Dd (mm) K0.013 K0.054 0.031 0.032 0.003 K0.041 0.044 0.046 0.016 0.085 0.087 0.057 0.002 K0.028 K0.030

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