1. Ch. 3: Geometric Camera Calibration
Objective: Estimates the intrinsic and extrinsic
parameters of a camera.
Idea: Formulate camera calibration as an
optimization process, in which the
discrepancy between the theoretical and
observed image features is minimized
w.r.t. the camera’s parameters.
Steps: (1) Evaluate the perspective projection
matrix M of the camera,
(2) Estimate the intrinsic and extrinsic
parameters of the camera from M.

2. 。 Perspective Projection (Imaging Process)
where
ideally,
practically,

3. ○ Evaluate M
Let
Measure n pairs
of corresponding
image and scene points.

4. For each pair
we obtain
For all pairs,

5. In matrix form,
where

6. Solve for x using optimization techniques

7. 3.1 Least-Squares Parameter Estimation
Idea: Find the solution x by minimizing the squared
deviation ( ) from theoretical (Ux) to
observed (y) image features
3.1.1 Linear Least-Squares Methods
○ Consider a system of p linear equations in q
unknowns:

8. In vector-matrix form, , where

9. ○ Consider the over-constrained case (p > q)
Find x that minimizes the error
Let
The normal equations
: the pseudoinverse of U.

10. Homogenous systems:
Two issues:
(i) By equation
, we obtain trivial
solution
(ii) If x is a solution,
x is also a solution.
To resolve the issues, impose
。 The least squares error solution of
is the eigenvector of
corresponding
to the smallest eigenvalue.

11. 。 Find the least squares error solution by the
method of Lagrange multipliers
Error:
Constraint
Minimize
where
: Lagrange multiplier
Let
We obtain
The solution x is an eigenvector of
with eigenvalue

12. The associated error
The least squares error solution to
is the eigenvector of
corresponding
to the smallest eigenvalue.
Example:
Fit a line to a set
of data points in
the 2D space

13. Line equation:
Let

14. The perpendicular distance from point
to line is
Error measure:
Minimize E w.r.t. (a, b, d)
Let

15. is the unit eigenvector with the minimum eigenvalue of
where
。 Recall
, whose squared error
The solution of min
w.r.t. n under
constraint
is the unit eigenvector
with the minimum eigenvalue of

16. 3.1.2 Nonlinear Least-Squares Methods
where

17. f(x):
e.g.,
f(x) :
e.g.,

18. where
: the Jacobian of f

19. 。 Taylor expansion of
around point
f(x):
f(x) :

20. Newton’s Method (Gradient Descent) (i) Square Systems (p = q)
Idea: Given an initial x, find . .
s.t.
Since
When
: nonsingular,
Let
Repeat until f(x) stabilizes at some x
Drawbacks: i) Square system, ii) Nonsingular
iii) Locally optimal.

21. Finding x s.t.
Finding x s.t. F(x) = 0 (square system)
Since
: p by q matrix, f(x): p by 1,
: q by 1

22. where
: q by q matrix

31. (g)
Proof: From

33. From
and

35. 3.3. Shape Distortions Types of distortions:
○ Degenerated Point Configurations
e.g., points lie on a line or a plane, may
cause failure of camera calibration.
3.3. Shape Distortions
Types of distortions:
(a) Tangential distortion
(b) Radial distortion
Barrel distortion, Pincushion distortion

36. (a) Changes the distance between the image center and the image point
Radial distortion:
(a) Changes the distance between the image
center and the image point
(b) Does not affect the direction joining the
image center and the image point
d: actual distance
: distorted distance
: distortion function

37. Logarithmic model, Fisheye model,
Polynomial model:
where
: coefficients
FOV model:
: distortion coefficient
where
Logarithmic model, Fisheye model,
Radial model, Rational function model

38. 。 Consider Polynomial model
Given an image point (u,v), determine its actual d

40. Determine the distortion function
i.e., determine its coefficients

51. 3.5 Application: Mobile Robot Localization
-- Calibrate a static camera for monitoring a robot

52. 20 images of the planar rectangular grid
Image resolution: 576 by 768
Camera: height = 4m,
focal length = 4.5mm,
Skew = 0, precision = 0.1 pixel
3 radial distortion coefficients.
Experimental results:
Localization error: 2 cm in position and
1 degree in orientation
Maximum error: 5 cm in position and
5 degrees in orientation

53. 3.4. A Nonlinear Approach

55. and