1. Zhengyou Zhang Vision Technology Group Microsoft Research
Flexible Camera Calibration by Viewing a Plane from Unknown Orientations
Zhengyou Zhang
Vision Technology Group
Microsoft Research
Please remind the audience that this is all completely confidential. We are in the midst of the patent process and certainly do not want to jeopardize that effort in any way. 1

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)
Examples include:
virtual world navigation. We are familiar with Euclidean world. If we don’t know the camera internal parameters, we can at best obtain projective structure. It would need a lot of training for human being to navigate successfully in a projective world.
Vision-based user interface. How far is the person from the screen? Where is s/he looking at?
Product advertisement. An on-line customer would prefer to check the exact Euclidean shape, rather than projective distorted one.

4. Classical Approach (Photogrammetry)
Use precisely known 3D points
Decision theory group focuses on solving problems with probability and decision theory. Focus is on making application of these methods practical. Related methods, approximations, heuristics from AI, OR, Statistics, Economics are applied as needed. Groups’s work typically motivated by real-world problems, e.g., those posed by MS product and systems groups.
Known displacement
Shortcomings: Not flexible
very expensive to make such a calibration apparatus. 3

5. Futuristic Approach (Self-calibration)
Move the camera in a static environment
match feature points across images
make use of rigidity constraint
Decision theory group focuses on solving problems with probability and decision theory. Focus is on making application of these methods practical. Related methods, approximations, heuristics from AI, OR, Statistics, Economics are applied as needed. Groups’s work typically motivated by real-world problems, e.g., those posed by MS product and systems groups.
Shortcoming: Not always reliable
too many parameters to estimate 3

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. Camera Model C  m

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

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

10. Geometric interpretation
Plane at infinity
Absolute conic C

11. Linear Equations Let symmetric 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. Correction of Radial Distortion
Original image
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.