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. 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. Isometry of Rotations and Quaternions

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

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

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

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

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. 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. 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. Demo video : 16 sec (Click to play)

53. Obtaining an initial region

65. 277,000 3D points triangulated. All but 281 proved by simple test to be minima. All except 153 proved to be global minima by more complex test.

66. Hardy: Pure mathematics is on the whole distinctly more useful than applied. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.