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1 Calibrating a pan-tilt camera head P. M. Ngan and R. J. Valkenburg Machine Vision Team, Industrial Research Limited P. O. Bo 225, Auckland, New Zealand Tel: , Fa: fp.ngan, b.valkenburgg@irl.cri.n Abstract This paper describes a method for calibrating a pan-tilt camera head. The ke feature of the method is that the camera, the pan-tilt unit, and the relationship between the camera and the pan-tilt unit, are calibrated independentl. This paper develops a geometric model of the pan-tilt camera head and presents procedures to estimate the values of the unknown model parameters. Target image repositioning is discussed as an application of using a calibrated pan-tilt camera head. 1. Introduction In an active vision sstem, camera parameters are continuousl updated to improve the image representation of the target. It is often desirable to alter the pose of the camera and this can be achieved b mounting the camera on a pan-tilt unit. The utilit of a pan-tilt head is etended when it is calibrated. The method presented is an adaptation of that developed b Len and Tsai [1, 2], for the calibration of a Cartesian robot with mounted camera, to a pan-tilt camera head. A ke feature of the procedure is that the calibration is decoupled into three smaller calibration problems: namel, camera calibration, pan-tilt unit calibration, and camera to pan-tilt unit calibration. A further feature is that the latter two calibrations can be simplied b moving one rotar joint at a time. The pan-tilt unit has two rotational degrees of freedom and diers from the Cartesian robot which has three rotational and three translational degrees of freedom. Calibration of the pan-tilt camera head is not signicantl simpler than calibration of a Cartesian robot because Len and Tsai use the translational movements to keep the three dimensional calibration reference in the eld of view. The lack of translational movement in the pantilt camera head is compensated for b using a large coplanar reference which allows the camera pose to be estimated for a large range of angular movements. A geometric model for the pan-tilt camera head is described in Section 2. Section 3 describes the decoupling of the pan-tilt camera head model into three sub-models and describes a calibration procedure for each model. An application of the calibrated model is presented in Section 4.

2 C W T 7 C I T 2 C C T 1 C F T 3 T 8 γ C T T 4 T 6 β C P T 5 α C B Figure 1: Geometr of coordinate sstems 2. Geometric model Throughout this paper, all coordinate sstems are right handed Cartesian and all points are represented in homogeneous coordinates. The smbols C and P are used to represent a coordinate sstem and point respectivel, and a subscript is used to indicate the specic coordinate sstem. In addition, a coordinate transform between two coordinate sstems is denoted b an indeed smbol T k, while rigid bod motion within a coordinate sstem is denoted b an indeed smbol H k. A pan-tilt camera head consists of a pan-tilt unit together with a camera. The pan-tilt unit is a mechanical device that can move with two degrees of rotational freedom and the camera is ed rigidl onto its moving mount. The principal coordinate sstems of the pan-tilt camera head are shown in Figure 1. The world coordinate sstem C W is ed and denes the location of phsical entities such as the calibration reference. The -ais of the base coordinate sstem, C B, is dened to be collinear with the ais of pan, and C B is dened to be collinear with the ais of tilt when the pan step count,, is set to ero. Increasing step count is dened to give increasing rotational angles which in turn denes the direction of the aes of rotation b the right-hand rule. The pan coordinate sstem, C P, rotates about C B and is dened to be coincident with C B when = 0. The pan angle,, is the angle between C B and C P and hence = 0 when = 0. Similarl, the tilt coordinate sstem, C T, rotates about C P and is dened to be coincident with C P when the tilt step count,, is ero. The tilt angle,, is the angle between C P and C T and hence = 0 when = 0. The camera coordinate sstem, C C, moves with the camera and is dened b the optical ais of the camera. The image formed b the lens and the piel image formed b the framegrabber are represented b image, C I, and frame, C F, coordinate sstems.

3 A model of the pan-tilt camera head is given b the following sstem of equations. P F = T 3 T 2 P C (1) P C = T 1 P W (2) P F = T 7 P W (3) P T = T 8 P C (4) P B = T 5 T 4 P T (5) P B = T 6 P W (6) T 2 and T 3 model the intrinsic characteristics of the camera. T 2 models the projection from C C to C I, and T 3 models the scaling, oset and shear between C I and C F [3]. All other T i 's accommodate rotation and translation between coordinate sstems and are of the form; " # Ri t T i = i (7) 0 1 where R i is a 3 3 rotation matri and t i is a 3 1 translation vector. The coordinate sstems C T and C C are rigidl coupled and the coordinate transformation between them is denoted b T 8. This transformation is ed when the camera is bolted onto the pan-tilt unit mount. 3. Geometric calibration The task of nding the unknown parameters in the transforms T i of the model is commonl referred to as geometric calibration. The model described b Eqns. 1-6 can be decoupled into three smaller models and each of these models are calibrated separatel. The rst relates to the intrinsic camera parameters given in Eqn. 1 and the camera pose given in Eqn. 2. The second sub-model relates to the mechanical movements of the pan-tilt unit which is described b Eqn. 5. The third sub-model describes the relationship between C C and C T and is given b Eqn. 4. The calibration procedure for each sub-model is described in the following sections. The transformation, T 6, is ed and can be calculated once T 4 ; T 5 and T 8 have been determined. 3.1 Calibrating the camera Camera calibration involves nding the intrinsic parameters and pose. The intrinsic parameters are ed and calculated once at the start of calibration. The pose changes with and and must be calculated man times during the subsequent calibration steps. The intrinsic parameters are estimated using standard techniques [3] and make use of a non-coplanar calibration reference. Such techniques provide estimates for intrinsic and pose parameters, but the pose parameters are discarded since onl the intrinsic parameters are of interest at this stage. Subsequent estimates for pose parameters are found using a coplanar reference. Given the intrinsic parameters, the camera pose can be calculated using a small non-linear optimisation in the si pose variables. A good initial estimate can be obtained using an of the standard procedures. For eample, given four points, four combinations of three points are possible. To each of these combinations of points the three point spatial resection algorithm [4] can be applied to nd their position in C C. Each application of the three point algorithm gives four possible solutions, but selecting the common solution from each combination provides a unique solution. Given the position of the four references points in C C, T 1 can be estimated using a simple linear procedure based on singular value decomposition [4].

4 3.2 Calibrating pan-tilt unit The following section describes the procedure for estimating the elements of the transformations T 4 and T 5 which are associated with rotation about the tilt and pan aes respectivel. It is onl necessar to consider the estimation of T 5 because the procedure for estimating T 4 is identical. The transformation T 5 is of the form epressed in Eqn. 7. However, b the denition of C P it is clear that t 5 = 0. The rotation matri R 5 () is dependent onl on which can be epressed as; = s + (8) where is the step count (in controller register), s is the conversion factor from step count to radians, and is the angular oset. The goal of this calibration process is to nd the value of s and. B the denition of C P the value of is ero. The calibration procedure involves rotating the head about the pan ais over an angular range divided into i = 1 : : : N equal steps. At each location referred to as a station an image is taken, and an i is calculated as shown in Eqn. 11. When all i 's have been obtained, the value of s is given b s = 1 N NX i=1 i i (9) The essence of the calibration procedure is that C C and C T are ed together rigidl and undergo the same rotations. Let H i denote the rigid bod motion in C W which transforms the origin of C T from station 0 to station i. It is a simple matter to show that; H i = (T 1i )?1 T 10 (10) The rotation matri, R i, is the upper left 3 3 submatri of H i as described in Eqn. 7. The relative pan angle i can be recovered from R i b; i = tan?1 k i k (11) Tr(R i )? 1 where 2 6 = skew?1 (R) = 4 r 32? r 23 r 13? r 31 r 21? r (12) 3.3 Calibrating camera to pan-tilt unit transformation This section describes the procedure for estimating the elements of the transformation T 8 between C C and C T. The transformation T 8 can be epressed in terms of R 8 and t 8 as shown in Eqn. 7. The calibration procedure involves rotating the head about the pan and tilt ais to i = 1 : : : N stations and capturing an image at each station. Let T 9 = T 5 T 4, the coordinate transformation from C Ci to C Cj be denoted T 1ij and the coordinate transformation from C Ti to C Tj be denoted T 9ij. It is clear that; T 1ij = T 1j T?1 1 i and T 9ij = T?1 9 j T 9i (13) The values of T 1 and T 9, for the i th station, can be obtained b the methods described in Sections 3.1 and 3.2. From T 1ij and T 9ij the values of R 1ij ; R 9ij ; t 1ij and t 9ij can be etracted as suggested b Eqn. 7. If R is a rotation matri then dene the unit vector

5 P and angle to be the ais and angle of rotation with the orientation of P selected so 0. The ais of rotation, P, is given b the unit vector; where is dened b Eqn. 12. Dene P = kk P = 2 sin 2 P and P 0 = (14) 1 q4? kpk 2 P (15) It can be shown [2] that; where, for an vector v = (v ; v ; v ) T ; skew(p 1ij + P 9ij )P 0 8 = P 1ij? P 9ij (16) 2 6 skew(v) = 4 0?v v v 0?v?v v (17) A sstem of equations can be formed using Eqn. 16 for multiple stations and solved for P 0 8 using linear least squares. B Eqn. 15 it follows that; = 2 tan?1 P 0 (18) P = P0 kp 0 k (19) Hence the rotation matri, R 8, can then be obtained using Rodrigues formula [3]; R = I + sin H + 1? cos 2 H 2 where H = skew( P) (20) B considering the closed graph of coordinate transformations between C Ti ; C Tj ; C Cj and C Ci it follows; T 9ij = T 8 T 1ij T?1 8 (21) Multipling out Eqn. 21 gives; (R 9ij? I)t 8 = R 8 t 1ij? t 9ij (22) A sstem of equations can be formed using Eqn. 22 for multiple stations and solved for t 8 using linear least squares. 4. Target image repositioning A common application of a calibrated pan-tilt head is to orient the camera so that a known world point, P W, is repositioned to a predened frame position, p F (standard coordinates). Given P W and p F, a least squares solution for and can be obtained b solving the following non-linear least squares problem; where the projected image point is dened b; min kp F? ^p F (; )k 2 (23) ; ^p F (; ) = 1 s (u; v)t ; (u; v; s) T = T 3 T 2 (T 9 (; ) T 8 )?1 T 6 P W (24)

6 5. Discussion This paper has presented a model for the pan-tilt camera head and described a method of calibrating the sstem. The precision to which the unknown parameters are estimated is adequate for man applications. When more precision is required a more comple intrinsic camera model should be used. Aside from directl improving the camera parameters this will also improve the other calibration steps because pose estimation is central to calibrating the pan-tilt unit and pan-tilt unit camera relationship. Further precision can be obtained b using these parameter values as good initial estimates for a non-linear optimisation algorithm. The purpose of calibrating the pan-tilt head is so that active vision applications such as target repositioning can be performed ecientl. Acknowledgements This work was funded b the New Zealand Foundation for Research, Science and Technolog. References 1 R. K. Len and R. Y. Tsai. Calibrating a cartesian robot with ee-on-hand con- guration independent of ee-to-hand relationship. IEEE Transactions on Pattern Analsis and Machine Intelligence, 11(9), September R. Y. Tsai and R. K. Len. A new technique for full autonomous and ecient robotic hand/ee calibration. IEEE Transactions on Pattern Analsis and Machine Intelligence, 5(3):345{358, June Olivier Faugeras. Three-Dimensional Computer Vision. The MIT Press, R. M. Haralick and L. G. Shapiro. Computer and Robot Vision, volume 2. Addison-Wesle, 1993.