1. A Generic Concept for Camera Calibration Peter Sturm and Srikumar Ramaligam Sung Huh CPSC 643 Individual Presentation 4 April 15, 2009

2. Table of Content Introduction Calibration ◦ 2D Known and Unknown motion ◦ 3D Known and Unknown motion Discussion Conclusion Future Work

3. Introduction Develop a calibration method for any camera model ◦ Cameras w/o a single effective view point General model of camera adopted: ◦ Images consisting of pixels ◦ Each pixel captures light that travels along a ray in 3D Camera is fully described by: ◦ Coordinate Of rays ◦ Mapping b/w rays and pixels

4. Introduction – Related Works Existing Calibration methods R.I. Hartley, A. Zisserman. Multiple View Geometry in Computer Vision. Cambridge University Press, 2000 C.C. Slama. Manual of Photogrammetry. Fourth Edition, ASPRS, 1980 Calibration for the general imaging model M.D. Grossberg, S.K. Naar. A general imaging model and a method for finding its parameters. ICC, 2001 R. Swamminathan, M.D. Grossberg, and S.K. Nayar. Caustics of Catadioptric Cameras. ICCV, 2001

5. Introduction – Related Works Epipolar geometry estimation and modeling T. Pajdla. Stereo with oblique cameras. IJCV, 47(1), 2002. S. Seitz. The space of all stereo images. ICCV, 2001. Y. Wexler, A.W. Fitzgibbon, A. Zisserman. Learning epipolar geometryfrom image sequence. CVPR, 2003. Motion estimation for calibrated cameras J. Neumann, C. Fermuller, Y. Aloimonons. Polydioptric Camera design and 3D Motion Estimation. CVPR, 2003. R. Pless. Using Many Cameras as One. CVPR, 2003

6. Introduction – Related Works The special case of a linear calibration object P. Sturms, S. Ramalingam. A Generic Calibration Concept: Theory and Algorithms. Research Report 5058, INRIA, France, 2003

7. Camera Model Use infinite extensions of half-ray (Camera Rays) Non-central Camera ◦ Camera rays correspond to different pixel does not intersect Central Camera w/ optical center ◦ All camera rays intersect in a single point

8. Calibration Concept – 2D Known motion ◦ Object’s motion b/w image is known ◦ Two object points can be mapped to a single coord. Frame ◦ Joining two points to compute pixel’s camera ray ◦ Knowledge of point position relative to coord. frames of object and the motion b/w the two coord. frame

9. Calibration Concept – 2D Unknown Motion ◦ Estimate unknown motion ◦ Let Q, Q’, Q” be the points on the calibration object ◦ Common frame = coord. frame associated w/ object’s first position (relative motions are given by rotation matrix and translation vector) Collinear

10. Calibration Concept – 2D Coefficients of a trilinear matching tensor Depends on the calibration tensor Total 27 coefficients ( 8 are always zero, 6 pairs of identical ones) C i = Trilinear product of point coordinates V i = Associated coefficients of the tensor

11. Calibration Concept – 3D Known Motion ◦ Two views are sufficient ◦ Equivalent procedure as 2D camera Unknown Motion ◦ Start with Q, Q’, Q” ◦ Same coord. frame as in 2D ◦ Aligned points are collinear

12. Calibration Concept – 3D Rank of matrix must be less than 3 ◦ All sub-determinants of size 3 x 3 vanishes (4 of them) ◦ Corresponds to a trilinear equation in point coord. 34/64 coefficients are always zero

13. Calibration Concept – 3D Estimate tensors by solving linear equation system Let V i ’= Vi, W i ’ =  W i, i = 1…37 ◦ Estimate factors, ◦ Compute and, i = 1…37 ◦ Compute R’ and R” ◦ Compute t’ and t” by solving a straight forward linear least square problem

14. Experimental Result Result from Central camera Estimated motion parameters give rise to nearly perfectly collinear calibration points Radial distortion modeled correctly

15. Experimental Result Result from fish-eye lens Aligned calibration points are not always perfectly collinear Only the central image region has been calibrated

16. Discussion Algorithm for central cameras work fine Non-central catadioptric cameras give unsatisfying result ◦ Homography-based interpolation of calibration points General algorithm does not work for perspective cameras, but for multi-stereo systems consisting of sufficiently many cmeras

17. Conclusion Theory and algorithms for a highly general calibration concept proposed Different cameras will likely require different designs of calibration object

18. Future Works Developing bundle adjustment procedures to calibrate from multiple images Motion and pose estimation and triangulation from perspective to the general imaging model