Chong Chen Dan Schonfeld Department of Electrical and Computer Engineering

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Plenoptic Imaging

Chong Chen

Dan Schonfeld

Department of Electrical and Computer Engineering

University of Illinois at Chicago

May 7 2009

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Plenoptic Function (1)

• From plenus (complete or full) and optic.• An idealized function to express the image of a

scene from any possible viewing position at any viewing angle at any point in time.

( , , , , , , )x y zP V V V t

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Plenoptic Function (2)

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Plenoptic Camera (1)

R. Ng et al., Stanford University 2005

5R. Ng et al., Stanford University 2005

Plenoptic Camera (2)

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Image-Base Rendering

L. McMillan and G. Bishop, SIGGRAPH 1995

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LightfieldThe lightfield data is composed of six 4D functions, where the plane of the inner box is indexed with coordinate (u, v) and that of the outer box with coordinate (s, t).

M. Levoy and P. Hanrahan, SIGGRAPH 1996

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Sampling and Reconstruction• The lightfield reconstruction is computed as

• The sampled lightfield ls(u,v,s,t) is represented by

( , , , ) ( , , , )[ ( , , , ) ( , , , )]I u v s t R u v s t L u v s t P u v s t

( , , , ) ( , , , ) [ ( , , , ) ( , , , )]i u v s t r u v s t l u v s t p u v s t

1 2 1 2

1 2 1 2, , ,

( , , , ) ( , , , ) ( ) ( ) ( ) ( )sn n k k Z

l u v s t l u v s t u n u v n v s k s t k t

1 2 1 2

1 2 1 2

, , ,

2 2 2 2( , , , ) ( ) ( ) ( ) ( )s

n n k k Z

n n k kL u v s t L u L v L s L t

u v s t

H.Y. Shum et al., SIGGRAPH 2000

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Plenoptic Sampling (1)

( , ) ( ,0)( , , , )

ftl v t l v

z u v s t

Assumptions: Lambertian surfaces and no occlusion.

H.Y. Shum et al., SIGGRAPH 2000

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Plenoptic Sampling (2)( )

( ) ( )

0 0

( , , , ) ( , , , )

( , ,0,0)

u v s t

u v s t

j u v s tu v s t

j u v j s t

L l u v s t e dudvdsdt

fs ftl u v e dudv e dsdt

z z

2

0 0

4 '( , ) ( ) ( )u v s u t v

f fL

z z

The spectral support of a lightfield signal is bounded by the minimum and maximum depths only, no matter how complicated the spectral support might be because of depth variations in the scene.

where L’ is the 2D Fourier transform of l(x,y,0,0)

H.Y. Shum et al., SIGGRAPH 2000

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Unstructured Lumigraph

M. Cohen et al., SIGGRAPH 2001

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Parallel Cameras

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( , )

2 '( ) ( )

v t

v t v

L

d fLd d

Plenoptic signals taken by parallel cameras will be bandlimited, and their spectral support is bounded by the minimum and maximum depths.

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Unstructured Cameras (1)( )

2 2

21 1 1

( , ) ( , )

( 1) 1( ,0)

1

v tj v tv tL l v t e dvdt

d d f a fl td a d a dv

a

1 2[1,2 ]

[ ]

0

j vvv

BF e

0v

Plenoptic signals taken by unparallel cameras will not be bandlimited.

0v

( )v tj v te dvdt

1 1 1/2 '( ) [ ]t t

d qj jaf v a

t

d dL e F e

f f

[ , ]B n x is the Bessel Function.

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Unstructured Cameras (2)

Plenoptic signals taken by unparallel cameras with limited FOV and rotations can be approximated to be bandlimited

( )

1 1

( , ) ( , )

( ( tan ) ,0)

v tj v tv tL l v t e dvdt

d fl v f td d

tan tan( ) tan Assuming

( )v tj v te dvdt

1 tan1 12 '( ) ( )

vd f

jd

v t v

d d fe L

d d d

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Concentric Mosaic (1)

• for the constant depth concentric mosaic, the spectrum lies on a line with slope

After linearization

( )

( )

( , ) ( , )

( ,0)

2 '( ) ( )

v

v

j vv

j v

v v

L l v e dvd

rl v e dvd

r Rr

Lr R

C. Zhang and T. Chen, Carnegie Mellon University 2001

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Bernstein's inequality: if is a bounded function on with supported in the ball . Then for all multi-indices , there exist constants (depending only on and on the dimension n) such that

Before linearization

Concentric Mosaic (2)

( )

( )

sin

cos

( , ) ( , )

sin( ,0)

cos

2 '( ) [ ]

v

v

v

j vv

j v

rj

r Rv

L l v e dvd

rl v e dvd

r R

L F e

,nC

,b n b LLf C R f

bfnR

0,wB ˆbf

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Unoccluded Image Constraints

If any point (f(x), g(x)) on the surface of the scene is differentiable, there is no occlusion with the surface if and only if

'

'

( )( ) ( )

( )

f xf x g x L

g x

which can be proved by Cauchy's Mean-Value Theorem.

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Conclusions and Future Work

• Plenoptic signals taken by unparallel cameras will not be bandlimited.

• Plenoptic signals taken by unparallel cameras with limited FOV and rotations can be approximated to be bandlimited.

• Sampling (light conditions, surface luminance)

• Reconstruction (integral imaging)

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Thank you !