Tsai Camera Model&nbsp…

最近做AR涉及到了相机的自标定，又要开始跟线代做斗争了，网上找到Tsai最初的Calibrationmodel的文，记录下:http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/DIAS1/

Tsai CameraModel

TheTsai model is based on a pinhole perspective projection model andthe following eleven parameters are to estimate:

f - Focal length of camera,

k - Radial lens distortion coefficient,

C_x, C_y - Co-ordinates ofcentre of radial lens distortion,

S_x - Scale factor to account for anyuncertainty due to imperfections in hardware timing for scanningand digitisation,

R_x, R_y, R_z -Rotation angles for the transformation between the world and cameraco-ordinates,

T_x, T_y, T_z -Translation components for the transformation between the world andcamera co-ordinates.

 

Figure 1: Tsai Camera re-projection model with perspectiveprojection and radial distortion.

 

Thetransformation from world(X_w,Y_w,Z_w)to image(X_i,Y_i,Z_i)co-ordinates considers the extrinsic parameters of the camera(Translation T andRotation R) within the equation:

where R and T characterizethe 3D transformation from the world to the camera co-ordinatesystem and are defined as follows:

with

(R_x,R_y,R_z)the Euler angles of the rotation around the three axes.

(T_x,T_y,T_z)the 3D translation parameters from world to image co-ordinates.

Thetransformation from 3D position (in the image co-ordinate frame) tothe image plane is then computed through the following steps (seeFigure 1):

Transformation from 3D world co-ordinates(X_i,Y_i)to undistorted image plane(X_u,Y_u)co-ordinates

Transformation from undistorted(X_u,Y_u)to distorted(X_d,Y_d)image co-ordinates

where

, and k is thelens distortion coefficient.

Transformation from distorted co-ordinates in image plane(X_d,Y_d)to the final image co-ordinates(X_f,Y_f)are:

with

(d_x,d_y):distance between adjacent sensor elements in the X and Ydirection. d_x and d_y arefixed parameters of the camera. They depend only on the CCD sizeand the image resolution, (X_f,Y_f) are thefinal pixel position in the image.

References

[Debevec01]P. Debevec, 
Reconstructing and Augmenting Architecture with Image-BasedModelling, 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 Imagesfrom 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 ImageCorrection. 
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 SensitivityAnalysis. 
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 fortexture mapping utilizing reflectanceattribute 
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 3DMachine Vision.
Proceedings of IEEE Conference on Computer Vision and PatternRecognition, Miami Beach, FL, pp. 364-374, 1986.
[Tsai87]R.Y. Tsai, 
Metrology Using Off-the-Shelf TV Cameras andLenses 
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 ComputerEngineering,Carnegie Mellon University, January 1994.
[Zhang00]Z. Zhang. 
A flexible new technique for cameracalibration. 
IEEE Transactionson Pattern Analysis and Machine Intelligence, Vol.22, No. 11, pp. 1330-1334, 2000