1. 1 pb.  camera model  calibration  separation (int/ext)  pose Don’t get lost! What are we doing? Projective geometry Numerical tools Uncalibrated cameras Calibrated cameras Projective geometry Euclidean geometry Classical (Euclidean) tools

2. 2 First application: camera pose estimation 3-point algebraic method 4 coplanar points linear method Two most popular methods: In vision, robotics, virtual reality, … Pose estimation = extrinsic calibration = navigation by reference = where is the camera? = where am I in the scene?

3. 3 Pose estimation = calibration of only extrinsic parameters Given Estimate R and t

4. 4 3-point algebraic method First convert pixels u into normalized points x by knowing the intrinsic parameters Write down the fundamental equation: Solve this algebraic system to get the point distances first Compute a 3D transformation 3 reference points == 3 beacons

5. 5 X x O X’ x’ Fundamental euclidean geometric constraint:

6. 6 Solving the algebraic system by elimination: (using a symbolic computation software (Maple or Mathematica)) using … our hands a polynomial of degree m and a polynomial of degree n leads to a polynomial of degree m*n

7. 7 given 3 corresponding 3D points: 3D transformation estimation Compute the centroids as the origin Compute the scale (compute the rotation by quaternion) Compute the rotation axis Compute the rotation angle

8. 8 v v’ n Geometry of 3D rotation about an axis with angle theta

9. 9 2 equations for 3 unknowns, so two vectors are needed!

10. 10 Rotation axis is obtained, but not yet the angle …

11. 11 Linear pose estimation from 4 coplanar points (Projective method based on a homography) (Similar to plane-based calibration) Vector based (or affine geometry) method

12. 12 O A B C D x_a x_d

13. 13

14. 14 Now we get only the ratios of the unknown distances, to fix the ratio,