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

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

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

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? Caustic Incoming light rays Reflector

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

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

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

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

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

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

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

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

Some Caustics. Caustic surface Reflector surface Caustic 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.

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.

Deriving Resolution Characteristics. Caustic f Reflector Z

Deriving Resolution Characteristics. Caustic Reflector

Resolution Characteristics. p Reflector z(t)

Resolution Characteristics. p Reflector z(t)

Calibration.

Calibration.

Calibration.

Calibration.

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