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.