1. Omnidirectional epipolar geometry
Kostas Daniilidis and Chris Geyer
University of Pennsylvania

2. Inside-out immersive…
Geyer, Daniilidis
ICRA 2003

3. And recently…
Geyer, Daniilidis
ICRA 2003

4. Omnidirectional input
Porta del Sol, Tiwanaku Bolivia
Geyer, Daniilidis
ICRA 2003

5. Central Catadioptric Projection
is a double projection:First on the mirror, then on the image plane.

6. Corollary: Conventional cameras are a just a singularity.
Unifying Theorem:
All central catadioptric projections are equivalent to double projection through the sphere.
Corollary: Conventional cameras are a just a singularity.

7. Equivalence with the sphere
Image of object obtained on image plane identical to catadioptric projection

8. Two facts
1. Parabolic projection = central projection to the sphere then stereo-graphic projection to a plane
2. Perspective projection = central projection to the sphere followed by central projection to a plane from the same center ! Our model covers all conventional perspective cameras!!
Geyer, Daniilidis
ICRA 2003

9. Inside-Out-Inside: Motion estimation
Geyer, Daniilidis
ICRA 2003

10. The projection of a line in space is a conic section and in parabolic mirrors it is a circle.
Geyer, Daniilidis
ICRA 2003

11. A new representation of image features
While the projective plane captures both points and lines, we do not have a space suitable for points and circles. We need a
CIRCLE SPACE!
Geyer, Daniilidis
ICRA 2003

12. Lift a circle line projection in parabolic omnicameras
Geyer, Daniilidis
ICRA 2003

13. Take inverse stereographic image
Geyer, Daniilidis
ICRA 2003

14. Construct cone tangent to locus P is the circle representation
Geyer, Daniilidis
ICRA 2003

15. By varying the radius we model points, circles, and imaginary circles!
Geyer, Daniilidis
ICRA 2003

16. Not every circle is a line projection (it has to be projection of a great circle). All these feasible lines lie on a plane circle space.
Geyer, Daniilidis
ICRA 2003

17. Image of the absolute conic
calibrating conic
Geyer, Daniilidis
ICRA 2003

18. Transformations of circle space
Motivation: In the perspective case the group of transformations is the set of collineations, i.e. non-singular matrices in PGL(3)
Goal: find the natural transformation group of circle space.
Geyer, Daniilidis
ICRA 2003

19. A translation in the plane….
If the sphere has projective quadratic form
Then for A to preserve the sphere we must have
(Note similarity with )
Geyer, Daniilidis
ICRA 2003

20. The Lorentz group What is the general set of 4×4 matrices satisfying
Actually since Q is projective we only need
Lorentz Group O(3,1) It is a six dimensional Lie group
Geyer, Daniilidis
ICRA 2003

21. The Lorentz group
rotation about x-axis
Geyer, Daniilidis
ICRA 2003

22. The Lorentz group
rotation about y-axis
Geyer, Daniilidis
ICRA 2003

23. The Lorentz group
rotation about z-axis
Geyer, Daniilidis
ICRA 2003

24. The Lorentz group
x translation
Geyer, Daniilidis
ICRA 2003

25. The Lorentz group
y translation
Geyer, Daniilidis
ICRA 2003

26. The Lorentz group
scale
Geyer, Daniilidis
ICRA 2003

27. Linear transformation from uncalibrated pixels to calibrated rays.
Such a linear transformation exists and its kernel contains the parameters of this mapping.
Geyer, Daniilidis
ICRA 2003

28. Scene reconstruction and ego-motion using omnidirectional cameras:
Geyer, Daniilidis
ICRA 2003

29. Two view perspective: the essential matrix
Recall that two images p1, p2 of the same space point X satisfy the bilinear constraint
where E is a 3×3 rank 2 matrix independent of X, (Tsai-Huang and Longuet-Higgins)
Geyer, Daniilidis
ICRA 2003

30. Assume p1 and p2 are the catadioptric projections of X
Two views
Assume p1 and p2 are the catadioptric projections of X
Geyer, Daniilidis
ICRA 2003

31. There exists an essential matrix E such that ________
Two views
There exists an essential matrix E such that ________
Geyer, Daniilidis
ICRA 2003

32. However there exist Lorentz group elements K1 & K2 such that
Two views
However there exist Lorentz group elements K1 & K2 such that
Geyer, Daniilidis
ICRA 2003

33. Catadioptric fundamental matrix
i.e. the lifted image points satisfy a bilinear epipolar constraint!!!
F is the 4×4 catadioptric fundamental matrix
The kernel of F is the kernel of K.
Geyer, Daniilidis
ICRA 2003

34. Reconstruction algorithm much simpler than in perspective !
Recover camera parameters with kernel computation and intersection
2. Recover rotation and translation
3. Reconstruct environment or produce novel views.
Geyer, Daniilidis
ICRA 2003

35. Epipolar circles
Geyer, Daniilidis
ICRA 2003