caustics of catadioptric cameras rahul swaminathan, michael d. grossberg and shree k. nayar cave...
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Caustics of Catadioptric Cameras
Rahul Swaminathan,Michael D. Grossberg and Shree K. Nayar
CAVE Lab, Columbia University
ICCV Conference
Vancouver, July 2001
Imaging Systems.
Camera
All incoming rays of light meet at a point.
Detector
Effective Viewpoint
Camera
Incoming light rays
The Single Viewpoint
Catadioptric Systems: Single Viewpoint.
Only combinations of conic reflectors produce single viewpoint systems.
(Nayar , Baker Peri )
Parashot 360 Acquired image.
Parabolicmirror
Telecentriclens.
What about such systems?
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?
Locus of Viewpoints: Caustics.
Adapted from [ Jensen, 1997 ]
How is a Caustic formed?
Reflector
Incoming light rays
Caustic
Computing the Caustic.
Reflector
)(tVr
Entry Pupil
)(tVi
Caustic
)(tSr
)()(),( tVrtSrtS rrc
r
At the caustic, there exists a singularity :
0))(( rr VrSJDet
Burkhard & Shealy, 1973
Conic Catadioptrics
(t)
z(t)
)(tVr
)(tVi
Imaging plane
Reflector
d
Op
222 2)1()(
)(
ppttet
ttz
Ellipse : e < 1Parabola : e = 1
Hyperbola : e > 1
Conic Catadioptric Cameras
Imaging plane
Reflector
Op
Reflector
d
Op
Perspective lens based Telecentric lens based
Imaging plane
d
Some Caustics.
Elliptical Reflector : e =.75 , p = 1.0 , d = 0.5
p
d
Pinhole
Reflector
Caustic
Some Caustics.
Parabolic Reflector : e =1.0 , p = 1.0 , d = 1.0
p
d
Pinhole
Reflector
Caustic
Some Caustics.
Elliptical Reflector :
e =.75 , p = 1.0 , d =
Reflector
Caustic
Some Caustics.
Elliptical Reflector :
e =.75 , p = 1.0 , d =
Reflector
Caustic
Caustic surface
Reflector surface
Some Caustics.
Caustic surface
Reflector surface
Lens pinhole
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.
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.
Deriving Resolution Characteristics.
Reflector
2
22
))()(( tzdCos
rfA
c
c
A
Caustic
S
f
Z
Resolution Characteristics.
(t)
p
z(t)Reflector
O
Resolution Characteristics.
(t)
p
z(t)Reflector
O
Calibration.
Calibration.
Calibration.
Calibration.
Caustics formed by the refracting water surface,lead to such exotic light patterns.
Realistic reproduction, requires modeling of caustics
Importance of Caustics to Graphics.