Branch and Bound in Rotation Space (ICCV 2007)

2. U

4. Direction (unit) vectors from cameras (blue) to points (black) are given : Find the positions of the cameras and points.

10. Branch and Bound in Rotation Space (ICCV 2007)

11. Essential Matrix Estimation • Encodes the relative displacement between two cameras. • Rotation • Translation • Needs at least 5 points X x1 x2 (R, t)

12. 2-view SfM with known rotations

14. Best current error We can eliminate all rotations within the ball of radius 0.3 about trial. Rotation Space

15. theta v

17. Isometry of Rotations and Quaternions Angle between two quaternions is half the angle between the corresponding rotations, defined by All rotations within a delta-neighbourhood of a reference rotation form a circle on the quaternion sphere.

18. Rotations are represented by a ball of radius pi in 3-Dimensional space. Angle-axis representation of Rotations Flatten out the meridians (longitude lines) Azimuthal Equidistant Projection

19. Subdividing and testing rotation space

20. Numbers of cubes left at each iteration (Log-10 scale) Remaining Volume at each iteration (Log-10 scale in cubic radians). Performance

21. Linear Programming, not SOCP X V’ V t C’ C Point correspondence in two views Coplanarity constraint with uncertainty

22. Multi-Camera Systems (Non-overlapping) – L inf Method Translation direction lies in a polyherdron (Green) from point correspondences

23. Multi-Camera Systems (Non-overlapping) – L inf Method

24. Each point correspondence gives two LP constraints on the direction t (epipolar direction).

25. Essential Matrix Calculated from 3 points (above) or 4 points (below) Possible rotations.

26. 360 degree camera Timing (in milliseconds) for E-matrix computation – 360 degree camera. Timing Examples 29 correspondences : 2.9 seconds 794 correspondences : 75 seconds. 6572 correspondeces : 3m 30 seconds

27. Further Application – 1D camera (e.g. robot moving in a plane) Joint work with Kalle Astrom, Fredrik Kahl, Carl Olsson and Olof Enquist Complete structure and motion problem for “planar motion” Optimal solution in L-infinity norm. Same idea of searching in rotation space.

28. Hockey Rink Data Reconstructed points and path Original and dual problems

29. Method works also for rigidly placed multi-camera systems. • Can be considered as a single “generalized” camera • One rotation, one translation to be estimated.

30. Robust 6DOF motion estimation from Non-overlapping images, Multi-camera systems 4 images from the right 4 images from the left (Images: Courtesy of UNC-Chapel Hill)

32. Generalized Cameras (Non-overlapping) Ladybug2 camera (The locally-central case) 5 cameras (horizontal) 1 camera (top)

33. Generalized Cameras (Non-overlapping) Experiment setup

34. Generalized Cameras (Non-overlapping) An Infinity-like path which the Ladybug2 camera follows (total 108 frames)

36. Robust 6DOF motion estimation from Non-overlapping images, Multi-camera systems Critical configuration

38. Generalized Cameras (Non-overlapping) – Linear Method Estimated path (Linear Method) vs. Ground truth

39. Generalized Cameras (Non-overlapping) – Linear Method

40. Generalized Cameras (Non-overlapping) – Linear Method Demo video : 16 sec (Click to play)

42. Multi-Camera Systems (Non-overlapping) – SOCP Method

43. Multi-Camera Systems (Non-overlapping) – L inf Method

45. Multi-Camera Systems (Non-overlapping) – L inf Method E+SOCP: Motion of multi-camera rigs using SOCP method BB+LP : Motion of multi-camera rigs using L inf method

46. Multi-Camera Systems (Non-overlapping) – L inf Method E+SOCP: Motion of multi-camera rigs using SOCP method BB+LP : Motion of multi-camera rigs using L inf method

47. Multi-Camera Systems (Non-overlapping) – L inf Method Estimated path (L inf Method) vs. Ground truth

48. Multi-Camera Systems (Non-overlapping) – L inf Method

49. Multi-Camera Systems (Non-overlapping) – L inf Method Demo video : 16 sec (Click to play)