Plenoptic Imaging Chong Chen Dan Schonfeld Department of

Plenoptic Imaging Chong Chen Dan Schonfeld Department of Electrical and Computer Engineering University of Illinois at Chicago May 7 2009 1

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. 2

Plenoptic Function (2) 3

Plenoptic Camera (1) R. Ng et al. , Stanford University 2005 4

Plenoptic Camera (2) R. Ng et al. , Stanford University 2005 5

Image-Base Rendering L. Mc. Millan and G. Bishop, SIGGRAPH 1995 6

Lightfield The lightfield data is composed of six 4 D 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 7

Sampling and Reconstruction • The lightfield reconstruction is computed as • The sampled lightfield ls(u, v, s, t) is represented by H. Y. Shum et al. , SIGGRAPH 2000 8

Plenoptic Sampling (1) Assumptions: Lambertian surfaces and no occlusion. H. Y. Shum et al. , SIGGRAPH 2000 9

Plenoptic Sampling (2) where L’ is the 2 D Fourier transform of l(x, y, 0, 0) 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. H. Y. Shum et al. , SIGGRAPH 2000 10

Unstructured Lumigraph M. Cohen et al. , SIGGRAPH 2001 11

Parallel Cameras Plenoptic signals taken by parallel cameras will be bandlimited, and their spectral support is bounded by the minimum and maximum depths. 12

Unstructured Cameras (1) Plenoptic signals taken by unparallel cameras will not be bandlimited. is the Bessel Function. 13

Unstructured Cameras (2) Assuming Plenoptic signals taken by unparallel cameras with limited FOV and rotations can be approximated to be bandlimited 14

Concentric Mosaic (1) • for the constant depth concentric mosaic, the spectrum lies on a line with slope After linearization C. Zhang and T. Chen, Carnegie Mellon University 2001 15

Concentric Mosaic (2) Before linearization 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 16

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 which can be proved by Cauchy's Mean-Value Theorem. 17

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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) 20

Thank you ! 21