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

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

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

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

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

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 O A B C D x_a x_d

13

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