1. Caustics of Catadioptric Cameras
Rahul Swaminathan,
Michael D. Grossberg and Shree K. Nayar
CAVE Lab, Columbia University
ICCV Conference
Vancouver, July 2001

2. Imaging Systems.
Camera

3. The Single Viewpoint All incoming rays of light meet at a point.
Camera
Detector
Incoming light rays
Effective Viewpoint

4. Catadioptric Systems: Single Viewpoint.
(Nayar , Baker Peri )
Only combinations of conic reflectors produce single viewpoint systems.
Parabolic
mirror
Telecentric
lens.
Parashot 360
Acquired image.

5. What about such systems?

6. Non-Single Viewpoint Cameras.
No single viewpoint. – Locus of Viewpoints formed.
Can we elegantly describe the geometry of viewpoints?
Effect of new geometry on imaging characteristics.
Is there a simple technique to calibrate such systems?

7. Locus of Viewpoints: Caustics.
Adapted from [ Jensen, 1997 ]

8. How is a Caustic formed?
Caustic
Incoming light rays
Reflector

9. Computing the Caustic. r Entry Pupil
Burkhard & Shealy, 1973
Entry Pupil
At the caustic, there exists a singularity : r
Caustic
Reflector

10. Jacobian Method. At the caustic, moving along Vr is the
Burkhard, 1973
Jacobian Method.
At the caustic, moving along Vr is the
same as moving along the reflector.
Entry pupil
Reflector

11. Conic Catadioptrics d (t) O p z(t) Imaging plane Ellipse : e < 1
Parabola : e = 1
Hyperbola : e > 1
Reflector
z(t)

12. Conic Catadioptric Cameras
Imaging plane d
Imaging plane
d  O O p p
Reflector
Reflector
Perspective lens based
Telecentric lens based

13. Some Caustics. d p Pinhole Elliptical Reflector :
e =.75 , p = 1.0 , d = 0.5 d p
Caustic
Reflector

14. Some Caustics. d p Pinhole Parabolic Reflector :
e =1.0 , p = 1.0 , d = 1.0 d
Reflector p
Caustic

15. Some Caustics. Elliptical Reflector : e =.75 , p = 1.0 , d =  Caustic

16. Some Caustics. Elliptical Reflector : e =.75 , p = 1.0 , d =  Caustic

17. Some Caustics. Caustic surface Reflector surface Caustic surface
Lens pinhole

18. 3D Caustic Surfaces.
Caustic for a parabola, source outside the reflector and along the axis of symmetry.
Caustic surface for a source placed off-axis with respect to the reflector.
All caustics are computed in closed form, then plotted.

19. 3D Caustic Surfaces.
Caustic for a parabola, source outside the reflector and along the axis of symmetry.
Caustic surface for a source placed off-axis with respect to the reflector.
All caustics are computed in closed form, then plotted.

20. Geometric Properties. Cusp: Lies along optical axis. Reflector Caustic
Provides bound on optimal global approximation for a single viewpoint.
Reflector
Caustic
Extent: Determines extent of the caustic surface.
Caustic touches reflector at this point.

21. Deriving Resolution Characteristics.
Caustic f
Reflector Z

22. Deriving Resolution Characteristics.
Caustic
Reflector

23. Resolution Characteristics.p
Reflector
z(t)

24. Resolution Characteristics.p
Reflector
z(t)

25. Calibration.

26. Calibration.

27. Calibration.

28. Calibration.

29. Realistic reproduction, requires modeling of caustics
Importance of Caustics to Graphics.
Caustics formed by the refracting water surface,lead to such exotic light patterns.
Realistic reproduction, requires modeling of caustics