Flexible Camera Calibration by Viewing a Plane from Unknown Orientations

1. Flexible Camera Calibration by Viewing a Plane from Unknown Orientations Zhengyou Zhang Vision Technology Group Microsoft Research

2. Problem Statement • Determine the characteristics of a camera (focal length, aspect ratio, principal point) from visual information (images)

3. Motivations • Recovery of 3D Euclidean structure from images is essential for many applications. • This requires camera calibration. • Look for a flexible and robust technique, suitable for desktop vision systems. (such that it can be used by the general public)

4. Classical Approach(Photogrammetry) • Use precisely known 3D points Known displacement • Shortcomings:Not flexible • very expensive to make such a calibration apparatus.

5. Futuristic Approach(Self-calibration) • Move the camera in a static environment • match feature points across images • make use of rigidity constraint • Shortcoming:Not always reliable • too many parameters to estimate

6. Realistic Approach(my new method) • Use only one plane • Print a pattern on a paper • Attach the paper on a planar surface • Show the plane freely a few times to the camera • Advantages: • Flexible! • Robust? Yes. See RESULTS

7.  m C Camera Model

8.  C m with Plane projection • For convenience, assume the plane at z = 0. • The relation between image points and model points is then given by:

9. Given H, which is defined up to a scale factor, And let , we have What do we get from one image? • We can obtain two equations in 6 intermediate homogeneous parameters. This yields

10. Absolute conic Geometric interpretation Plane at infinity C

11. Linear Equations • Let • Define up to a scale factor • Rewrite as linear equations: symmetric

12. What do we get from 2 images? • If we impose  = 0, which is usually the case with modern cameras, we can solve all the other camera intrinsic parameters. How about more images? Better! More constraints than unknowns.

13. Solution • Show the plane under n different orientations (n > 1) • Estimate the n homography matrices (analytic solution followed by MLE) • Solve analytically the 6 intermediate parameters (defined up to a scale factor) • Extract the five intrinsic parameters • Compute the extrinsic parameters • Refine all parameters with MLE

14. Experimental results

15. Extracted corner points

16. Result (1)

17. Result (2)

18. Original image Correction of Radial Distortion Corrected image

19. Errors vs. Noise Levels in data

20. Errors vs. Number of Planes

21. Errors vs. Angle of the plane

22. Errors vs. Noise in model points

23. Errors vs. Spherical non-planarity

24. Errors vs. Cylindrical non-planarity

25. Application to object modeling

26. Reconstructed VRML Model

27. Conclusion • We have developed a flexible and robust technique for camera calibration. • Analytical solution exists. • MLE improves the analytical solution. • We need at least two images if c = 0. • We can use as many images of the plane as possible to improve the accuracy.

28. It really works! • Currently used routinely in both Vision and Graphics Groups. • Binary executable will be distributed on the Web to the public soon. • Source code will also be made available.