ECE 549CS 543: COMPUTER VISON LECTURE 6

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ECE 549/CS 543 COMPUTER VISON LECTURE 6 CAMERA
CALIBRATION II

• Least-squares techniques
• MATLAB tutorial
• Linear calibration

Reading Chapter 3 Slides http//www-cvr.ai.uiuc.
edu/ponce/fall04/lect6.ppt Programming
assignment 1, due Tue. Sep. 21
http//www-cvr.ai.uiuc.edu/ponce/fall04/hw1/hw1.
pdf

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Calibration Problem
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Linear Systems
Square system
A
x
b

• unique solution
• Gaussian elimination

Rectangular system ??

• underconstrained
• infinity of solutions

A
x
b

• overconstrained
• no solution

Minimize Ax-b
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How do you solve overconstrained linear equations
??
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Homogeneous Linear Systems
Square system
A
x
0

• unique solution 0
• unless Det(A)0

Rectangular system ??

• 0 is always a solution

A
x
0

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Minimize Ax under the constraint x 1
2
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How do you solve overconstrained homogeneous
linear equations ??
The solution is e .
1
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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

• Done !!

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Note

• Matrix of second moments of inertia
• Axis of least inertia

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Linear Camera Calibration
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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

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Degenerate Point Configurations
Are there other solutions besides M ??

• Coplanar points (l,m,n )(P,0,0) or (0,P,0) or
  (0,0,P )

• Points lying on the intersection curve of two
  quadric
• surfaces

straight line twisted cubic
Does not happen for 6 or more random points!
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Analytical Photogrammetry
Non-Linear Least-Squares Methods

• Newton
• Gauss-Newton
• Levenberg-Marquardt

Iterative, quadratically convergent in favorable
situations
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Mobile Robot Localization (Devy et al., 1997)