Tsai Camera Model
The Tsai model is based on a pinhole perspective projection model and the following eleven parameters are to estimate:
f - Focal length of camera,
k - Radial lens distortion coefficient,
Cx, Cy - Co-ordinates of centre of radial lens distortion,
Sx - Scale factor to account for any uncertainty due to imperfections in hardware timing for scanning and digitisation,
Rx, Ry, Rz - Rotation angles for the transformation between the world and camera co-ordinates,
Tx, Ty, Tz - Translation components for the transformation between the world and camera co-ordinates.
Figure 1: Tsai Camera re-projection model with perspective projection and radial distortion.
The transformation from world (Xw,Yw,Zw) to image (Xi,Yi,Zi) co-ordinates considers the extrinsic parameters of the camera (Translation T and Rotation R) within the equation:
where R and T characterize the 3D transformation from the world to the camera co-ordinate system and are defined as follows:
with
(Rx,Ry,Rz) the Euler angles of the rotation around the three axes.
(Tx,Ty,Tz) the 3D translation parameters from world to image co-ordinates.
The transformation from 3D position (in the image co-ordinate frame) to the image plane is then computed through the following steps (see Figure 1):
Transformation from 3D world co-ordinates (Xi,Yi) to undistorted image plane (Xu,Yu) co-ordinates
Transformation from undistorted (Xu,Yu) to distorted (Xd,Yd) image co-ordinates
where
, and k is the lens distortion coefficient.
Transformation from distorted co-ordinates in image plane (Xd,Yd) to the final image co-ordinates (Xf,Yf) are:
with
(dx,dy): distance between adjacent sensor elements in the X and Y direction. dx and dy are fixed parameters of the camera. They depend only on the CCD size and the image resolution, (Xf,Yf) are the final pixel position in the image.
References
[Debevec01]P. Debevec, Reconstructing and Augmenting Architecture with Image-Based Modelling, Rendering and Lighthing. Proceedings of the International Symposium on Virtual Architecture (VAA’01), pp. 1-10, Dublin 21-22 June 2001.[Faugeras93]O. Faugeras, Three-Dimensional Computer Vision. MIT Press, 1993.[Fitzgibbon98]A.W. Fitzgibbon, A. Zisserman, Automatic 3D Model Acquisition and Generation of New Images from Video Sequences. In Proceedings of European Signal Processing conference (EUSIPCO '98), Rhodes, Greece, pages 1261-1269, 1998.[Heikkila97]J. Heikkila, O. Silven, A Four-Step Camera Calibration Procedure with Implicit Image Correction. In Proc. of IEEE Computer Vision and Pattern Recognition, pp. 1106-1112, 1997.[Kumar94]R. Kumar and A. Hanson. Robust Methods for Estimating Pose and a Sensitivity Analysis. CVGIP-Image Understanding, Vol. 60, No. 3, pp. 313-342, 1994.[Kurazume02]R. Kurazume, K. Nishino, Z. Zhang, and K. Ikeuchi, Simultaneous 2D images and 3D geometric model registration for texture mapping utilizing reflectance attribute Proc. of Fifth Asian Conference on Computer Vision (ACCV), Vol. I, pp. 99-106, January 2002.[Pollefeys00]M. Pollefeys, 3D Modelling from Images, Tutorial notes, in conjunction with ECCV 2000, Dublin, Ireland, June 2000.[Tsai86]R.Y. Tsai, An Efficient and Accurate Camera Calibration Technique for 3D Machine Vision. Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, Miami Beach, FL, pp. 364-374, 1986.[Tsai87]R.Y. Tsai, Metrology Using Off-the-Shelf TV Cameras and Lenses IEEE Journal of Robotics and Automation, Vol. 3, No. 4, pp. 323-344, August 1987.[Wilson94]Reg G. Willson Modeling and Calibration of Automated Zoom Lenses Ph.D. thesis, Department of Electrical and Computer Engineering,Carnegie Mellon University, January 1994.[Zhang00]Z. Zhang. A flexible new technique for camera calibration. IEEE Transactionson Pattern Analysis and Machine Intelligence, Vol. 22, No. 11, pp. 1330-1334, 2000