1. Geometric Camera Calibration
EECS 274 Computer Vision
Geometric Camera Calibration

2. GEOMETRIC CAMERA CALIBRATION
Camera calibration problem
Least-squares techniques
Linear calibration from points
Analytical photogrammetry
Reading: Chapter 3 of FP, Chapters 2,6 of S

3. Calibration Determine the intrinsic and extrinsic parameters
Assume that the camera observe a set of features (points, or lines) with known positions
Calibration: modeled as an optimization to minimize the discrepancy between the observed image features and their theoretical projections (using the perspective projection equations)

4. Calibration Problem
Given n points, P1, …, Pn with known positions and their images points, p1, …, pn, find ξ

5. A x b = A x b = Linear Systems Square system: unique solution
Gaussian elimination =
Rectangular system ??
underconstrained:
infinity of solutions A x b =
overconstrained:
no solution
Minimize |Ax-b| 2

6. How do you solve overconstrained linear equations ?

7. In matrix form
Can be derived from the perspective projection matrix

8. A x = A x = Homogeneous Linear Systems Square system:
unique solution: 0
unless Det(A)=0 =
Rectangular system ??
0 is always a solution A x = 2
Minimize |Ax|
under the constraint |x| =1 2

9. How do you solve overconstrained homogeneous linear equations ?
The solution is e . 1

10. Example: Line Fitting
Problem: minimize
with respect to (a,b,d).
Minimize E with respect to d:
Minimize E with respect to a,b:
where
Solution is the unit eigenvector with minimum eigenvalue

11. Note:
Matrix of second moments of inertia
Axis of least inertia in mechanics

12. Linear Camera Calibration
min |Pm|2, |m|=1

13. Once M is known, you still got to recover the intrinsic and
extrinsic parameters !
This is a decomposition problem, not an estimation
problem. r
Intrinsic parameters
Extrinsic parameters

14. Decomposition of M As the recovered Orthonormal basis vector
θ is close to π/2 and has positive sine

15. Degenerate Point Configurations
Are there other solutions besides M ?
One solution: (l,m,n )=(m1, m2, m3)
Consider the points Pi all lie in some plane, s.t., P∙Pi=0 for some P
Coplanar points: choose (l,m,n )=(P,0,0) or (0,P,0) or (0,0,P ), or
any linear combination of these vectors yields a solution
Does not (usually) happen for 6 or more random points!

16. Radial distortion
Depends on the distance separating the optical axis from the point of interest, d
Barrel distortion
Corners are detected by fitting lines in each square
Using estimated distortion parameters

17. Correct radial distortion
Tsai’s algorithm (1987) exploits radial alignment constraints for estimating extrinsic parameters

18. Analytical Photogrammetry
Given n points, P1, …, Pn with known positions and
their images situations, p1, …, pn, find ξ
Non-Linear Least-Squares Methods
Newton
Gauss-Newton
Levenberg-Marquardt
Iterative, quadratically convergent in favorable situations

19. Mobile Robot Localization (Devy et al., 1997)

20. Calibration
Numerous ways that exploits properties of projective geometry
E.g. calibration using lines, calibration circular controlled points

21. Camera calibration toolbox
Excellent MATLAB toolbox by Jean-Yves Bouguet
Steps:
Generate calibration board
Collect images under different views
Select extreme points
Find corner points
Solve optimization problem

22. Calibration images

23. Extreme points

24. Guessed grid corners

25. Corner extraction

26. Repeat for all other images

27. Solving optimization problem

28. Reprojected corners

29. Camera centered view

30. World centered view

31. Applications Augmented reality Image registration Image stitching
Panoramic image

32. Panoramic image

33. Notes Camera pose estimation Multi-camera calibration
Auto/self calibration
Multi-camera self calibration
Projective geometry
Multi-view geometry
RANSAC (RANdom Sample Consensus)