1. GEOMETRIC CAMERA CALIBRATION The Calibration Problem Least-Squares Techniques Linear Calibration from Points Reading: Chapter 3

2. Calibration Problem

3. Linear Systems A A x xb b = = Square system: unique solution Gaussian elimination Rectangular system ?? underconstrained: infinity of solutions Minimize overconstrained: no solution

4. How do you solve overconstrained linear equations ??

5. Homogeneous Linear Systems A A x x0 0 = = Square system: unique solution: 0 unless Det(A)=0 Rectangular system ?? 0 is always a solution Minimize |Ax| under the constraint |x| =1 2 2

6. How do you solve overconstrained homogeneous linear equations ?? The vector that minimizes E is e 1

7. 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 !!

8. Note: Matrix of second moments of inertia Axis of least inertia

9. Linear Camera Calibration

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

11. Same equations for estimation of 3x3 projective transformation matrix M