1. Camera Calibration Sebastian Thrun, Gary Bradski, Daniel Russakoff Stanford CS223B Computer Vision http://robots.stanford.edu/cs223b (with material from David Forsyth, James Rehg and Allen Hanson)

2. Sebastian Thrun Stanford University CS223B Computer Vision A Quiz n How Many Flat Calibration Targets are Needed for Calibration? 1: 2: 3: 4: 5: 10: n How Many Corner Points do we need in Total? 1: 2: 3: 4: 10: 20:

3. Sebastian Thrun Stanford University CS223B Computer Vision Example Calibration Pattern

4. Sebastian Thrun Stanford University CS223B Computer Vision Perspective Camera Model Object Space

5. Sebastian Thrun Stanford University CS223B Computer Vision Calibration Model (extrinsic)

6. Sebastian Thrun Stanford University CS223B Computer Vision Experiment 1: Parallel Board

7. Sebastian Thrun Stanford University CS223B Computer Vision 30cm10cm20cm Projective Perspective of Parallel Board

8. Sebastian Thrun Stanford University CS223B Computer Vision Experiment 2: Tilted Board

9. Sebastian Thrun Stanford University CS223B Computer Vision 30cm10cm20cm 500cm50cm100cm Projective Perspective of Tilted Board

10. Sebastian Thrun Stanford University CS223B Computer Vision Calibration Model (extrinsic) rotation translation (3D)

11. Sebastian Thrun Stanford University CS223B Computer Vision Calibration Model (extrinsic) Homogeneous Coordinates

12. Sebastian Thrun Stanford University CS223B Computer Vision Homogeneous Coordinates n Idea: Most Operations Become Linear! n Extract Image Coordinates by Z-normalization

13. Sebastian Thrun Stanford University CS223B Computer Vision Advantage of Homogeneous C’s i-th data point

14. Sebastian Thrun Stanford University CS223B Computer Vision Calibration Model (intrinsic) Pixel size Focal length Image center

15. Sebastian Thrun Stanford University CS223B Computer Vision Intrinsic Transformation

16. Sebastian Thrun Stanford University CS223B Computer Vision Plugging the Model Together!

17. Sebastian Thrun Stanford University CS223B Computer Vision Summary Parameters n Extrinsic –Rotation –Translation n Intrinsic –Focal length –Pixel size –Image center coordinates –(Distortion coefficients - see JYB’s tutorial )

18. Sebastian Thrun Stanford University CS223B Computer Vision Q: Can We recover all Intrinsic Params? n No

19. Sebastian Thrun Stanford University CS223B Computer Vision Summary Parameters, Revisited n Extrinsic –Rotation –Translation n Intrinsic –Focal length –Pixel size –Image center coordinates –(Distortion coefficients - see JYB’s tutorial ) Focal length, in pixel units Aspect ratio

20. Sebastian Thrun Stanford University CS223B Computer Vision Calibration a la Trucco n Substitute n Advantage: Equations are linear in params n If over-constrained, minimize Least Mean Square fct n One possible solution: n Enforce constraint that R is rotation matrix n Lots of considerations to recover individual params…

21. Sebastian Thrun Stanford University CS223B Computer Vision Calibration a la Bouguet

22. Sebastian Thrun Stanford University CS223B Computer Vision Calibration a la Bouguet, cont’d n Calibration Examples: …

23. Sebastian Thrun Stanford University CS223B Computer Vision Calibration a la Bouguet, cont’d n Least Mean Square

24. Sebastian Thrun Stanford University CS223B Computer Vision Calibration a la Bouguet, cont’d n Least Mean Square n Gradient descent:

25. Sebastian Thrun Stanford University CS223B Computer Vision Trucco Versus Bouguet Trucco: n Mimization of Squared distance in parameter space Bouguet n Minimization of Squared distance in Image space

26. Sebastian Thrun Stanford University CS223B Computer Vision Q: How Many Images Do We Need? n Assumption: K images with M corners each n 4+6K parameters n 2KM constraints n 2KM  4+6K  M>3 and K  2/(M-3) n 2 images with 4 points, but will 1 images with 5 points work? n No, since points cannot be co-planar!

27. Sebastian Thrun Stanford University CS223B Computer Vision Nonlinear Distortions n Barrel and Pincushion n Tangential

28. Sebastian Thrun Stanford University CS223B Computer Vision Barrel and Pincushion Distortion telewideangle

29. Sebastian Thrun Stanford University CS223B Computer Vision Models of Radial Distortion distance from center

30. Sebastian Thrun Stanford University CS223B Computer Vision Tangential Distortion cheap glue cheap CMOS chip cheap lense image cheap camera

31. Sebastian Thrun Stanford University CS223B Computer Vision Image Rectification