1. Camera Model Calibration
Robot Vision Systems
Camera Model
Calibration

2. “perspectograph”
Alberti’s Grid

3. Pinhole Camera
scene
image plane
iris,
optical center

4. Pinhole Camera
“image coordinates”

5. Pinhole Camera
“Alberti’s Grid”

6. Pinhole Camera
Classical pin-hole x f r’ z

7. Classical Pinhole Camera
Similar Triangles x f c’ z

8. Classical Pinhole Camera
Similar Triangles x f c’ z
Image coordinates
Point expressed in the camera frame
Camera matrix (projection)
Projective coordinates

9. Classical Pinhole Camera
Similar Triangles x f
Convert pixels to mm: c’ z
world

10. Camera Calibration Used to determine the elements in the camera matrix
Use pairs of known world points and their corresponding image points
e.g. use calibration grid

11. Camera Model – Perspective Projection
Need to determine the parameters!

12. Camera Calibration
camera matrix

13. Camera Calibration
Projective Equivalence
Two equations in 12 unknowns

14. Camera Calibration Have 6 point pairs (c,r) and (x,y,z)
Correspondence known
World coordinates known accurately

15. Camera Calibration Method 1: assume a34 = 0 and solve for aij
Solve using pseudo-inverse

16. Camera Calibration
Method 2: make no assumption about a34 and solve for aij
The vector a = [a11, …, a34]T is in the nullspace of the design matrix

17. Camera Calibration Ba = 0,
Use SVD, then a is the column of V corresponding to the null singular value of B
One property of the SVD is that the columns of V corresponding to the zero singular value span the null space of B
The vector a = [a11, …, a34]T is in the nullspace of the design matrix

18. Camera Calibration
Given the camera matrix solved w.r.t. the robot base frame

19. Using the Camera Matrix
Projection ( is 3 x 4)– cannot invert!

20. Measurements from Images
Must have relationship between the image “pixels” and the world
2D imaging
the image plane and the “world” plane are in 1-1 correspondence

21. Using the Camera Matrix in 2D
If all world points are on a plane
Then z is a linear function of x & y

22. Using the Camera Matrix in 2D
Now the projection equations
Can be written

23. Using the Camera Matrix in 2D
Now the projection equations
Can be written

24. Using the Camera Matrix in 2D
Now the projection equations
Can be written
2 equations, 9 unknowns

25. Using the Camera Matrix in 2D
Four known points, i=1,..4
Linear equations -- can be solved
8 equations, 9 unknowns up to a scale (8 unknowns)

26. Using the Camera Matrix in 2D

27. Using the Camera Matrix in 2D
Now the projection equations are simpler
… and we can “invert” (map image points back to the world plane)

28. Measurements from Images

29. Robot to Plane
Homogeneous transformation from base (or end-effector) frame to the work-plane of the imaging system
Use “known” corresponding points to solve for the elements of T (6 unknowns)