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Augmenting Light Field

The document summarizes a presentation on light fields and the Wigner distribution function. It begins with introducing traditional light fields and their limitations in modeling diffraction and interference effects. It then describes how augmenting light fields can address these limitations by incorporating wave optics concepts like the Wigner distribution function. The presentation provides examples of how augmented light fields can model interference patterns from experiments like Young's double slit experiment. It concludes by discussing properties of the Wigner distribution function and how it relates to modeling light fields.

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Light Fields in Ray and Wave Optics Introduction to Light Fields: Ramesh Raskar Wigner Distribution Function to explain Light Fields: Zhengyun Zhang Break Augmenting LF to explain Wigner Distribution Function: Se Baek Oh Q&A Light Fields with Coherent Light: Anthony Accardi New Opportunities and Applications: Raskar and Oh Q&A: All
Space of LF representations Time-frequency representations Phase space representations Quasi light field Other LF representations Observable LF WDF Augmented LF Other LF Traditional representations light field incoherent Rihaczek Distribution Function coherent
Augmenting Light Fields explaining Wigner Distribution Function with LF Se Baek Oh Postdoctoral Associate 3D Optical Systems Group, Dept. of Mechanical Eng. Massachusetts Institute of Technology
Traditional Light Field 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 4
Motivation Traditional Light Field 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 5
Motivation light field direction position (θ, φ) radiance of ray Traditional (x, y) Light Field L(x, y, θ, φ) ref. plane 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 5
Motivation Traditional Light Field http://graphics.stanford.edu 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 5
Motivation Traditional Light Field ray optics based simple and powerful http://graphics.stanford.edu 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 5
Motivation Wigner Distribution Function Traditional Light Field ray optics based simple and powerful 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 6
Motivation rigorous but cumbersome wave optics based Wigner Distribution Function Traditional Light Field ray optics based simple and powerful 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 6
Motivation rigorous but cumbersome wave optics based Wigner Distribution Function holograms beam shaping Traditional Light Field 1µm 1µm ray optics based simple and powerful rotational PSF limited in diffraction & interference 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 6
Augmented LF rigorous but cumbersome wave optics based Wigner Distribution Function Traditional Light Field ray optics based simple and powerful limited in diffraction & interference 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 7
Augmented LF rigorous but cumbersome wave optics based Wigner WDF Distribution Function Augmented LF Traditional Traditional Light Field Light Field ray optics based simple and powerful limited in diffraction & interference 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 7
Augmented LF rigorous but cumbersome wave optics based Wigner WDF Distribution Function Augmented LF Traditional Traditional Light Field Light Field ray optics based simple and powerful Interference & Diffraction limited in diffraction & interference Interaction w/ optical elements 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 7
Augmented LF rigorous but cumbersome wave optics based Wigner WDF Distribution Function Augmented LF Traditional Traditional Light Field Light Field ray optics based simple and powerful Interference & Diffraction limited in diffraction & interference Interaction w/ optical elements Non-paraxial propagation 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 7
Augmented LF • not a new light field • a new methodology/framework to create, modulate, and propagate light fields • stay purely in position-angle space • wave optics phenomena can be understood with the light field 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 8
Augmented LF framework LF (diffractive) optical element 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 9
Augmented LF framework LF LF (diffractive) optical element LF propagation 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 9
Augmented LF framework light field transformer LF LF LF negative radiance (diffractive) optical element LF propagation 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 9
Augmented LF framework light field transformer LF LF LF LF negative radiance (diffractive) optical element LF propagation LF propagation 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 9
Augmented LF framework light field transformer LF LF LF LF negative radiance (diffractive) optical element LF propagation LF propagation 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 9
Augmented LF framework light field transformer LF LF LF LF negative radiance (diffractive) optical element LF propagation LF propagation Diffraction can be included in the light field framework! Tech report, S. B. Oh et al. http://web.media.mit.edu/~raskar/RayWavefront/ 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 9
outline 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 10
outline • Limitations of Light Field analysis 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 10
outline • Limitations of Light Field analysis • Augmented Light Field • free-space propagation u u x x 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 10
outline • Limitations of Light Field analysis • Augmented Light Field • free-space propagation • virtual light projector in the ALF • coherence 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 10
outline • Limitations of Light Field analysis • Augmented Light Field • free-space propagation • virtual light projector in the ALF • coherence (x1 , θ1 ) (x2 , θ2 ) • light field transformer L1 (x1 , θ1 ) L2 (x2 , θ2 ) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 10
Assumptions • monochromaticinto polychromatic coherent) (= temporally •can be extended • flatland extendedobservation plane) (= 1D • can be to the real world • scalarbefield and into polarized lightone polarization) diffraction (= • can extended • no non-linear effect (two-photon, SHG, loss, absorption, etc) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 11
Young’s experiment screen light from double a laser slit x d I(x) z 2π d I(x) = 1 + cos x λ z 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 12
Young’s experiment screen light from double a laser slit x d I(x) z 2π d I(x) = 1 + cos x λ z 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 12
Young’s experiment screen light from double a laser slit x d |r1 − r2 | = mλ constructive interference I(x) z 2π d I(x) = 1 + cos x λ z 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 12
Young’s experiment screen light from double a laser slit x destructive interference |r1 − r2 | = (m + 1/2)λ d |r1 − r2 | = mλ constructive interference I(x) z 2π d I(x) = 1 + cos x λ z 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 12
Young’s experiment x θ+ θ u (= θ/λ) A B A B x A B x θ− Light Field WDF z ref. plane 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 13
Young’s experiment θ+ x θ u (= θ/λ) A B A B x A B x θ− Light Field WDF z ref. plane 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 14
Young’s experiment projection projection θ+ x θ u (= θ/λ) A B A B x A B x θ− Light Field WDF z I(x) I(x) ref. plane 3D Optical x x Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 14
Wigner Distribution Function x x Wg (x, u) = g x+ g ∗ x− e−j2πx u dx 2 2 space local spatial frequency (u = θ/λ) (= fξ in Zhengyun’s slide) • local spatial frequency spectrum (similar as wavelet) • ex) global vs. local frequency in a song global freq. local freq. • complex input g(x), WDF is always real • intensity = projection of WDF along u • WDF can be defined for light (E-field) as well as optical elements (e.g., gratings or apertures) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 15
Wigner Distribution Function x x Wg (x, u) = g x+ g ∗ x− e−j2πx u dx 2 2 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 16
Wigner Distribution Function x x Wg (x, u) = g x+ g ∗ x− e−j2πx u dx jαx2 2 2 g(x) = e x 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 16
Wigner Distribution Function x x Wg (x, u) = g x+ g ∗ x− e−j2πx u dx jαx2 2 2 g(x) = e “ ”2 x −jα x− x g ∗ x− =e 2 x 2 “ ”2 x x jα x+ 2 g x+ =e jα(2xx ) 2 e x x /2 x /2 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 16
Wigner Distribution Function x x Wg (x, u) = g x+ g ∗ x− e−j2πx u dx jαx2 2 2 g(x) = e “ ”2 x −jα x− x g ∗ x− =e 2 x 2 “ ”2 x x jα x+ 2 g x+ =e jα(2xx ) 2 e x x x /2 x /2 x 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 16
Wigner Distribution Function x x Wg (x, u) = g x+ g ∗ x− e−j2πx u dx jαx2 2 2 g(x) = e “ ”2 x −jα x− x g ∗ x− =e 2 x 2 “ ”2 x x jα x+ 2 g x+ =e jα(2xx ) 2 e x x . . x /2 x /2 . . . . x 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 16
Wigner Distribution Function x x Wg (x, u) = g x+ g ∗ x− e−j2πx u dx jαx2 2 2 g(x) = e “ ”2 x −jα x− x g ∗ x− =e 2 x 2 “ ”2 x x jα x+ 2 g x+ =e jα(2xx ) 2 e x F x . . x /2 x /2 . . . . x 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 16
Wigner Distribution Function x x Wg (x, u) = g x+ g ∗ x− e−j2πx u dx jαx2 2 2 g(x) = e “ ”2 x −jα x− x g ∗ x− =e 2 x 2 “ ”2 x x jα x+ 2 g x+ =e jα(2xx ) 2 e x F x . . x /2 x /2 . . u Wg (x, u) . . x x 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 16
Wigner Distribution Function plane wave spherical wave point source incoherent light 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 17
Augmented Light Field 1. free-space propagation 2. virtual light projector with negative radiance 3. light field transformer
Free-space propagation • In homogeneous medium and the paraxial region, • LF = ALF = WDF WDF Augmented LF Traditional Light Field 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 19
Free-space propagation • two plane parameterization equivalent to θ x x x x d 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 20
Free-space propagation • two plane parameterization equivalent to θ x x x x d 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 21
Free space propagation • wave optics: Huygen’s principle • point sources on the wavefront • coherent superposition of wavelets 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 22
Free space propagation • wave optics: Huygen’s principle • point sources on the wavefront • coherent superposition of wavelets 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 22
Free space propagation • Mathematical description j 2π r e λ r E-field jλr point source j 2π r e λ (x, y) E(x , y ) = E(x, y) ⊗ (x , y ) jλr r= (x − x)2 + (y − y)2 + z 2 z E(x, y) E(x , y ) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 23
Free space propagation • Mathematical description j 2π r e λ r E-field jλr point source j 2π r e λ (x, y) E(x , y ) = E(x, y) ⊗ (x , y ) jλr r= (x − x)2 + (y − y)2 + z 2 z E(x, y) E(x , y ) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 23
Free space propagation • with the paraxial approximation spherical wavefront 1 quadratic (x − x)2 + (y − y)2 + z2 ≈z+ (x − x)2 + (y − y)2 wavefront 2z z point source exp j 2π λ (x − x)2 + (y − y)2 + z 2 E(x , y ) = E(x, y) dxdy jλ (x − x)2 + (y − y)2 + z 2 j 2π z e λ π ≈ E(x, y) exp j [(x − x)2 + (y − y)2 ] dxdy jλz λz Fresnel diffraction formula 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 24
Free space propagation • with the paraxial approximation spherical wavefront 1 quadratic (x − x)2 + (y − y)2 + z2 ≈z+ (x − x)2 + (y − y)2 wavefront 2z source & aperture size << z z point source exp j 2π λ (x − x)2 + (y − y)2 + z 2 E(x , y ) = E(x, y) dxdy jλ (x − x)2 + (y − y)2 + z 2 j 2π z e λ π ≈ E(x, y) exp j [(x − x)2 + (y − y)2 ] dxdy jλz λz Fresnel diffraction formula 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 24
Fresnel propagation • w/ WDF x x E1 (x) E2 (x ) W1 (x, u) W2 (x , u ) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 25
Fresnel propagation • w/ WDF x x E1 (x) E2 (x ) W1 (x, u) W2 (x , u ) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 25
Fresnel propagation • w/ WDF x x E1 (x) E2 (x ) W1 (x, u) W2 (x , u ) W2 (x , u ) = W1 (x − λzu , u ) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 25
Fresnel propagation • w/ WDF x x E1 (x) E2 (x ) W1 (x, u) W2 (x , u ) W2 (x , u ) = W1 (x − λzu , u ) u x 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 25
Fresnel propagation • w/ WDF x x E1 (x) E2 (x ) W1 (x, u) W2 (x , u ) W2 (x , u ) = W1 (x − λzu , u ) u u x-shear transform 1/(λz) x x 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 25
diffraction vs. distance single slit a = 64λ laser from Zhengyun’s slide z 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 26
diffraction vs. distance Position and Direction in Wave Optics single slit a = 64λ laser from Zhengyun’s slide z 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 26
diffraction vs. distance Position and Direction in Wave Optics near zone: few λ (evanescent wave) Fresnel regime Fraunhofer regime (paraxial region) (Far-field) single slit a = 64λ laser from Zhengyun’s slide z 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 26
diffraction vs. distance Position and Direction in Wave Optics near zone: few λ (evanescent wave) 1 FN Fresnel regime FN 1 Fraunhofer regime (paraxial region) (Far-field) non-paraxial single slit region a = 64λ laser from Zhengyun’s slide a2 z rule of thumb: Fresnel number FN = λz 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 26
Augmented Light Field 1. free-space propagation 2. virtual light projector with negative radiance 3. light field transformer
Virtual light projector WDF Augmented LF Traditional Light Field Diffraction and Interference With simple modifications in Light Field - virtual light projector (negative radiance) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 28
Young’s experiment x θ u A B A B x A B x Light Field WDF ref. plane 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 29
Young’s experiment projection projection x θ u A B A B x A B x Light Field WDF I(x) I(x) ref. plane x x 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 29
Young’s experiment projection projection x θ u A B A B x A B x Light Field WDF I(x) I(x) ref. plane x x 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 29
Virtual light projector projection real projector θ x real projector Augmented LF intensity=0 Not conflict with physics 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 30
Virtual light projector projection 2π real projector cos λ [a − b]θ θ negative virtual light projector positive at the mid point x real projector Augmented LF intensity=0 Not conflict with physics 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 30
Virtual light projector first null real projector (OPD = λ/2) real projector 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 31
Virtual light projector first null real projector (OPD = λ/2) virtual light projector real projector 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 31
Virtual light projector hyperbola first null (OPD = λ/2) asymptote of λ/2 hyperbola valid in Fresnel regime (or paraxial) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 32
Virtual light projector destructive interference in high school physics class, (need negative radiance from virtual light projector) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 33
Virtual light projector destructive interference in high school physics class, (need negative radiance from virtual light projector) m = λ/2 m = 3λ/2 m = 5λ/2 m = 7λ/2 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 33
Question • Does a virtual light projector also work for incoherent light? 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 34
Question • Does a virtual light projector also work for incoherent light? • Yes! 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 34
Coherence • Degree of making interference • coherent partially coherent incoherent • Correlation of two)points on wavefront • E(p , t )E (p , t ∗ 1 1 (≈phase difference) 2 2 p1 Coherent: deterministic phase relation Incoherent: uncorrelated phase relation p2 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 35
Coherence • throwing stones...... 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 36
Coherence • throwing stones...... single point source 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 36
Coherence • throwing stones...... single point source coherent 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 36
Coherence • throwing stones...... single point source many point sources coherent 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 36
Coherence • throwing stones...... single point source many point sources coherent if thrown identically, still coherent! 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 36
Coherence • throwing stones...... single point source many point sources coherent if thrown identically, still coherent! if thrown randomly, then incoherent! 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 36
Coherence • Temporal coherence: E(p, t )E (p, t ) 1 ∗ 2 • spectral bandwidth • monochromatic: temporally coherent • broadband (white light): temporally incoherent • Spatial coherence: E(p1 , t)E ∗ (p2 , t) • spatial bandwidth (angular span) • point source: spatially coherent • extended source: spatially incoherent 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 37
Example 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 38
Example Temporally incoherent; spatially coherent 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 38
Example Temporally incoherent; spatially coherent 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 38
Example Temporally incoherent; Temporally & spatially coherent spatially coherent 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 38
Example Temporally incoherent; Temporally & spatially coherent spatially coherent 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 38
Example Temporally incoherent; Temporally & spatially coherent spatially coherent Temporally & spatially incoherent 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 38
Example Temporally incoherent; Temporally & spatially coherent spatially coherent Temporally & spatially incoherent Temporally coherent; spatially incoherent ? 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 38
Example Temporally incoherent; Temporally & spatially coherent spatially coherent Temporally & spatially incoherent Temporally coherent; spatially incoherent rotating diffuser laser 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 38
Temporal coherence • Broadband light is incoherent • ALF (also LF and WDF) can be defined for different wavelength and treated independently 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 39
Young’s Exp. w/ white light x I(x) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 40
Young’s Exp. w/ white light u Red x x u Green x u Blue x 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 41
Young’s Exp. w/ white light Red I(x) x x Green I(x) x Blue I(x) x 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 42
Young’s Exp. w/ white light Red x Green I(x) x Blue 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 42
Spatial coherence • ALF w/ virtual light projectors is defined for spatially coherent light • For partially coherent/incoherent light, adding the defined ALF still gives valid results! 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 43
Young’s Exp. w/ spatially incoherent light x I(x) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 44
Young’s Exp. w/ spatially incoherent light x I(x) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 45
Young’s Exp. w/ spatially incoherent light x I(x) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 46
Young’s Exp. w/ spatially incoherent light x w/ random phase (uncorrelated) I(x) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 47
Young’s Exp. w/ spatially incoherent light x w/ random phase (uncorrelated) I(x) spatially incoherent light: infinite number of waves propagating along all the direction with random phase delay 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 47
Young’s Exp. w/ spatially incoherent light u x w/ random phase (uncorrelated) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 48
Young’s Exp. w/ spatially incoherent light u x w/ random phase (uncorrelated) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 48
Young’s Exp. w/ spatially incoherent light u x w/ random phase (uncorrelated) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 48
Young’s Exp. w/ spatially incoherent light u x w/ random phase (uncorrelated) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 48
Young’s Exp. w/ spatially incoherent light u x w/ random phase (uncorrelated) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 48
Young’s Exp. w/ spatially incoherent light u x w/ random phase (uncorrelated) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 48
Young’s Exp. w/ spatially incoherent light u x w/ random phase (uncorrelated) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 48
Young’s Exp. w/ spatially incoherent light u x w/ random phase Addition (uncorrelated) u x 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 48
Young’s Exp. w/ spatially incoherent light u x x w/ random phase (uncorrelated) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 49
Young’s Exp. w/ spatially incoherent light u x x w/ random phase Addition (uncorrelated) u x 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 49
Virtual light projectors • Very simple modification to the LF • interference and diffraction within light field (geometry based) representation 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 50
Augmented Light Field 1. free-space propagation 2. virtual light projector with negative radiance 3. light field transformer
Light Field Transformer WDF Augmented LF Light Field Interaction at the optical elements 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 52
Light Field Transformer light field transformer WDF LF LF LF LF negative radiance Augmented LF (diffractive) optical element Light Field LF propagation LF propagation Interaction at the optical elements 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 52
Light Field Transformer • Q.Virtual light projector for a big aperture? • put the virtual light projectors for all the possible pairs of two points • WDF of optical elements 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 53
Light Field Transformer • Q.Virtual light projector for a big aperture? • put the virtual light projectors for all the possible pairs of two points • WDF of optical elements 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 53
Light Field Transformer • Q.Virtual light projector for a big aperture? • put the virtual light projectors for all the possible pairs of two points equivalent to compute the WDF mathematically.... 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 53
Light Field Transformer • Q.Virtual light projector for a big aperture? • put the virtual light projectors for all the possible pairs of two points equivalent to compute the WDF mathematically.... • WDF of optical elements 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 53
Light Field Transformer • Q.Virtual light projector for a big aperture? • put the virtual light projectors for all the possible pairs of two points equivalent to compute the WDF mathematically.... • WDF of optical elements representing properties of optical elements 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 53
Light Field Transformer Tech report: S. B. Oh et. al 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 54
Light Field Transformer • light field interactions w/ optical elements (x1 , θ1 ) (x2 , θ2 ) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 55
Light Field Transformer • light field interactions w/ optical elements (x1 , θ1 ) (x2 , θ2 ) L1 (x1 , θ1 ) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 55
Light Field Transformer • light field interactions w/ optical elements (x1 , θ1 ) (x2 , θ2 ) L1 (x1 , θ1 ) L2 (x2 , θ2 ) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 55
Light Field Transformer • light field interactions w/ optical elements (x1 , θ1 ) (x2 , θ2 ) L1 (x1 , θ1 ) L2 (x2 , θ2 ) Light field transformer T (x2 , x1 , θ1 , θ2 ) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 55
Light Field Transformer Dimension Property Note 8D(4D) thick, shift variant, 8D reflectance field, T (x2 , x1 , θ1 , θ2 ) angular variant volume hologram 6D(3D) thin, shift variant, 6D display, T (x, θ1 , θ2 ) angular variant BTF 4D(2D) thin, shift variant, many optical elements T (x, θ) angular invariant 2D(1D) attenuation shield field T (x) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 56
8D LF Transformer • the most generalized case (x1 , θ1 ) (x2 , θ2 ) L2 (x2 , θ2 ) L1 (x1 , θ1 ) L2 (x2 , θ2 ) = T (x2 , θ2 , x1 , θ1 )L1 (x1 , θ1 )dx1 dθ1 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 57
6D LF Transformer • For thin optical elements x 6D Display L2 (x, θ2 ) L1 (x, θ1 ) Courtesy of Martin Fuchs Bidirectional L2 (x, θ2 ) = T (x, θ2 , θ1 )L1 (x, θ1 )dθ1 Texture Function Courtesy of Paul Debevec 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 58
er of times with ference terms are tood with the in- the two pinholes 4D LF Transformer • Figure 7: Concept of virtual light sources for coherent light. w/ anglethe LF representation, no interference is predicted. By In shift invariant elements (in the paraxial region) virtual light sources, the LF propagation still including the n for diffraction • can be used. ould be included. e.g. aperture, lens, thin grating, etc oducing the con- have negative ra- es at a and b as al light source is π[a − b] λ along θ by integrating the l light sources do ne, which agrees Once the virtual L2 (x, θ) = T (x, θ − θ)L1 (x, θ )dθ propagation still Figure 8: Angle shift invariance in a thin transparency. In erly modeled3Dby Group (a) and (b), the output rays rotate in the same fashion 59 Se Baek Oh Optical Systems CVPR 2009 - Light Fields: Present and Future as
Light field transformer • only amplitude variation (occluders) x shield fields for occluders L2 (x, θ) L1 (x, θ) L2 (x, θ) = T (x, θ)L1 (x, θ) Courtesy of D. Lanman 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 60
Applications On wavefront coding holography 315 rendering the screen was very large. As expected, we see (Fig. 9) th Fraunhofer diffraction pattern. 1.1. Double-helix point spread function (DH-PSF) A DH-PSF system can be implemented by introducing a phase mask in the Fourier plane of an otherwise standard imaging system. The phase mask is designed such that its transmittance function generates a rotating pattern in the focal region of a Fourier transform lens [15-18]. Specifically, the DH-PSF exhibits two lobes that spin around the opticalaperture. An animate Figure 9: Diffraction from a square axis as shown in Fig. 1(a). Note that DH-PSF displays this experiment with of orientation with defocusappears in of a significant change varying the aperture size over an gaussian beam rotating PSF extended depth. In contrast, the standard PSF presents a slowly changing and expanding plementary material as a video. The distance from the ap symmetrical pattern throughout the same region [Fig. 1(b)]. the screen is 1 m. 316 317 Double rectangular apertures: Next we created two r lar apertures and probe them with the AMP. Note that we 3D Optical Fig. 1. Comparison of the (a) DH-PSF and the (b) standard PSF at different axial planes for a Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future system with 0.45 numerical aperture (NA) and 633nm wavelength. 61
Space of LF representations Time-frequency representations Phase space representations Quasi light field Other LF representations Observable LF WDF Augmented LF Other LF Traditional representations light field incoherent Rihaczek Distribution Function coherent
Property of the Representation Constant along Interference Non-negativity Coherence Wavelength rays Cross term Traditional LF always always only zero no constant positive incoherent nearly always any Observable LF constant positive coherence any yes state only in the positive and Augmented LF paraxial region negative any any yes only in the positive and WDF paraxial region negative any any yes Rihaczek DF no; linear drift complex any any reduced 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 63
Benefits & Limitations of the Representation Adaptability Ability to Modeling Simplicity of to current Near Field Far Field propagate wave optics computation pipe line Traditional Light Fields x-shear no very simple high no yes Observable not x- yes modest low yes yes Light Fields shear Augmented Light Fields x-shear yes modest high no yes WDF x-shear yes modest low yes yes better than Rihaczek DF x-shear yes WDF, not as low no yes simple as LF 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 64
Conclusion • WDFoptics generalized version of the LF in wave is the • Augmented Light Field • identicalregion) propagation (in the paraxial free-space • virtual light projectors • light field transformers • Wave opticswith geometrical be based understood phenomena can ray representations 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 65
Light Fields in Ray and Wave Optics Introduction to Light Fields: Ramesh Raskar Wigner Distribution Function to explain Light Fields: Zhengyun Zhang Augmenting LF to explain Wigner Distribution Function: Se Baek Oh Q&A Break Light Fields with Coherent Light: Anthony Accardi New Opportunities and Applications: Raskar and Oh Q&A: All

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Augmenting Light Field

  • 1.
    Light Fields inRay and Wave Optics Introduction to Light Fields: Ramesh Raskar Wigner Distribution Function to explain Light Fields: Zhengyun Zhang Break Augmenting LF to explain Wigner Distribution Function: Se Baek Oh Q&A Light Fields with Coherent Light: Anthony Accardi New Opportunities and Applications: Raskar and Oh Q&A: All
  • 2.
    Space of LFrepresentations Time-frequency representations Phase space representations Quasi light field Other LF representations Observable LF WDF Augmented LF Other LF Traditional representations light field incoherent Rihaczek Distribution Function coherent
  • 3.
    Augmenting Light Fields explainingWigner Distribution Function with LF Se Baek Oh Postdoctoral Associate 3D Optical Systems Group, Dept. of Mechanical Eng. Massachusetts Institute of Technology
  • 4.
    Traditional Light Field 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 4
  • 5.
    Motivation Traditional Light Field 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 5
  • 6.
    Motivation light field direction position (θ, φ) radiance of ray Traditional (x, y) Light Field L(x, y, θ, φ) ref. plane 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 5
  • 7.
    Motivation Traditional Light Field http://graphics.stanford.edu 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 5
  • 8.
    Motivation Traditional Light Field ray optics based simple and powerful http://graphics.stanford.edu 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 5
  • 9.
    Motivation Wigner Distribution Function Traditional Light Field ray optics based simple and powerful 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 6
  • 10.
    Motivation rigorous but cumbersome wave optics based Wigner Distribution Function Traditional Light Field ray optics based simple and powerful 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 6
  • 11.
    Motivation rigorous but cumbersome wave optics based Wigner Distribution Function holograms beam shaping Traditional Light Field 1µm 1µm ray optics based simple and powerful rotational PSF limited in diffraction & interference 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 6
  • 12.
    Augmented LF rigorous but cumbersome wave optics based Wigner Distribution Function Traditional Light Field ray optics based simple and powerful limited in diffraction & interference 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 7
  • 13.
    Augmented LF rigorous but cumbersome wave optics based Wigner WDF Distribution Function Augmented LF Traditional Traditional Light Field Light Field ray optics based simple and powerful limited in diffraction & interference 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 7
  • 14.
    Augmented LF rigorous but cumbersome wave optics based Wigner WDF Distribution Function Augmented LF Traditional Traditional Light Field Light Field ray optics based simple and powerful Interference & Diffraction limited in diffraction & interference Interaction w/ optical elements 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 7
  • 15.
    Augmented LF rigorous but cumbersome wave optics based Wigner WDF Distribution Function Augmented LF Traditional Traditional Light Field Light Field ray optics based simple and powerful Interference & Diffraction limited in diffraction & interference Interaction w/ optical elements Non-paraxial propagation 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 7
  • 16.
    Augmented LF • not a new light field • a new methodology/framework to create, modulate, and propagate light fields • stay purely in position-angle space • wave optics phenomena can be understood with the light field 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 8
  • 17.
    Augmented LF framework LF (diffractive) optical element 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 9
  • 18.
    Augmented LF framework LF LF (diffractive) optical element LF propagation 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 9
  • 19.
    Augmented LF framework light field transformer LF LF LF negative radiance (diffractive) optical element LF propagation 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 9
  • 20.
    Augmented LF framework light field transformer LF LF LF LF negative radiance (diffractive) optical element LF propagation LF propagation 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 9
  • 21.
    Augmented LF framework light field transformer LF LF LF LF negative radiance (diffractive) optical element LF propagation LF propagation 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 9
  • 22.
    Augmented LF framework light field transformer LF LF LF LF negative radiance (diffractive) optical element LF propagation LF propagation Diffraction can be included in the light field framework! Tech report, S. B. Oh et al. http://web.media.mit.edu/~raskar/RayWavefront/ 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 9
  • 23.
    outline 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 10
  • 24.
    outline • Limitations of Light Field analysis 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 10
  • 25.
    outline • Limitations of Light Field analysis • Augmented Light Field • free-space propagation u u x x 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 10
  • 26.
    outline • Limitations of Light Field analysis • Augmented Light Field • free-space propagation • virtual light projector in the ALF • coherence 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 10
  • 27.
    outline • Limitations of Light Field analysis • Augmented Light Field • free-space propagation • virtual light projector in the ALF • coherence (x1 , θ1 ) (x2 , θ2 ) • light field transformer L1 (x1 , θ1 ) L2 (x2 , θ2 ) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 10
  • 28.
    Assumptions • monochromaticinto polychromatic coherent) (= temporally •can be extended • flatland extendedobservation plane) (= 1D • can be to the real world • scalarbefield and into polarized lightone polarization) diffraction (= • can extended • no non-linear effect (two-photon, SHG, loss, absorption, etc) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 11
  • 29.
    Young’s experiment screen light from double a laser slit x d I(x) z 2π d I(x) = 1 + cos x λ z 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 12
  • 30.
    Young’s experiment screen light from double a laser slit x d I(x) z 2π d I(x) = 1 + cos x λ z 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 12
  • 31.
    Young’s experiment screen light from double a laser slit x d |r1 − r2 | = mλ constructive interference I(x) z 2π d I(x) = 1 + cos x λ z 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 12
  • 32.
    Young’s experiment screen light from double a laser slit x destructive interference |r1 − r2 | = (m + 1/2)λ d |r1 − r2 | = mλ constructive interference I(x) z 2π d I(x) = 1 + cos x λ z 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 12
  • 33.
    Young’s experiment x θ+ θ u (= θ/λ) A B A B x A B x θ− Light Field WDF z ref. plane 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 13
  • 34.
    Young’s experiment θ+ x θ u (= θ/λ) A B A B x A B x θ− Light Field WDF z ref. plane 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 14
  • 35.
    Young’s experiment projection projection θ+ x θ u (= θ/λ) A B A B x A B x θ− Light Field WDF z I(x) I(x) ref. plane 3D Optical x x Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 14
  • 36.
    Wigner Distribution Function x x Wg (x, u) = g x+ g ∗ x− e−j2πx u dx 2 2 space local spatial frequency (u = θ/λ) (= fξ in Zhengyun’s slide) • local spatial frequency spectrum (similar as wavelet) • ex) global vs. local frequency in a song global freq. local freq. • complex input g(x), WDF is always real • intensity = projection of WDF along u • WDF can be defined for light (E-field) as well as optical elements (e.g., gratings or apertures) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 15
  • 37.
    Wigner Distribution Function x x Wg (x, u) = g x+ g ∗ x− e−j2πx u dx 2 2 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 16
  • 38.
    Wigner Distribution Function x x Wg (x, u) = g x+ g ∗ x− e−j2πx u dx jαx2 2 2 g(x) = e x 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 16
  • 39.
    Wigner Distribution Function x x Wg (x, u) = g x+ g ∗ x− e−j2πx u dx jαx2 2 2 g(x) = e “ ”2 x −jα x− x g ∗ x− =e 2 x 2 “ ”2 x x jα x+ 2 g x+ =e jα(2xx ) 2 e x x /2 x /2 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 16
  • 40.
    Wigner Distribution Function x x Wg (x, u) = g x+ g ∗ x− e−j2πx u dx jαx2 2 2 g(x) = e “ ”2 x −jα x− x g ∗ x− =e 2 x 2 “ ”2 x x jα x+ 2 g x+ =e jα(2xx ) 2 e x x x /2 x /2 x 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 16
  • 41.
    Wigner Distribution Function x x Wg (x, u) = g x+ g ∗ x− e−j2πx u dx jαx2 2 2 g(x) = e “ ”2 x −jα x− x g ∗ x− =e 2 x 2 “ ”2 x x jα x+ 2 g x+ =e jα(2xx ) 2 e x x . . x /2 x /2 . . . . x 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 16
  • 42.
    Wigner Distribution Function x x Wg (x, u) = g x+ g ∗ x− e−j2πx u dx jαx2 2 2 g(x) = e “ ”2 x −jα x− x g ∗ x− =e 2 x 2 “ ”2 x x jα x+ 2 g x+ =e jα(2xx ) 2 e x F x . . x /2 x /2 . . . . x 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 16
  • 43.
    Wigner Distribution Function x x Wg (x, u) = g x+ g ∗ x− e−j2πx u dx jαx2 2 2 g(x) = e “ ”2 x −jα x− x g ∗ x− =e 2 x 2 “ ”2 x x jα x+ 2 g x+ =e jα(2xx ) 2 e x F x . . x /2 x /2 . . u Wg (x, u) . . x x 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 16
  • 44.
    Wigner Distribution Function plane wave spherical wave point source incoherent light 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 17
  • 45.
    Augmented Light Field 1. free-space propagation 2. virtual light projector with negative radiance 3. light field transformer
  • 46.
    Free-space propagation • In homogeneous medium and the paraxial region, • LF = ALF = WDF WDF Augmented LF Traditional Light Field 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 19
  • 47.
    Free-space propagation • two plane parameterization equivalent to θ x x x x d 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 20
  • 48.
    Free-space propagation • two plane parameterization equivalent to θ x x x x d 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 21
  • 49.
    Free space propagation • wave optics: Huygen’s principle • point sources on the wavefront • coherent superposition of wavelets 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 22
  • 50.
    Free space propagation • wave optics: Huygen’s principle • point sources on the wavefront • coherent superposition of wavelets 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 22
  • 51.
    Free space propagation • Mathematical description j 2π r e λ r E-field jλr point source j 2π r e λ (x, y) E(x , y ) = E(x, y) ⊗ (x , y ) jλr r= (x − x)2 + (y − y)2 + z 2 z E(x, y) E(x , y ) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 23
  • 52.
    Free space propagation • Mathematical description j 2π r e λ r E-field jλr point source j 2π r e λ (x, y) E(x , y ) = E(x, y) ⊗ (x , y ) jλr r= (x − x)2 + (y − y)2 + z 2 z E(x, y) E(x , y ) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 23
  • 53.
    Free space propagation • with the paraxial approximation spherical wavefront 1 quadratic (x − x)2 + (y − y)2 + z2 ≈z+ (x − x)2 + (y − y)2 wavefront 2z z point source exp j 2π λ (x − x)2 + (y − y)2 + z 2 E(x , y ) = E(x, y) dxdy jλ (x − x)2 + (y − y)2 + z 2 j 2π z e λ π ≈ E(x, y) exp j [(x − x)2 + (y − y)2 ] dxdy jλz λz Fresnel diffraction formula 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 24
  • 54.
    Free space propagation • with the paraxial approximation spherical wavefront 1 quadratic (x − x)2 + (y − y)2 + z2 ≈z+ (x − x)2 + (y − y)2 wavefront 2z source & aperture size << z z point source exp j 2π λ (x − x)2 + (y − y)2 + z 2 E(x , y ) = E(x, y) dxdy jλ (x − x)2 + (y − y)2 + z 2 j 2π z e λ π ≈ E(x, y) exp j [(x − x)2 + (y − y)2 ] dxdy jλz λz Fresnel diffraction formula 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 24
  • 55.
    Fresnel propagation • w/ WDF x x E1 (x) E2 (x ) W1 (x, u) W2 (x , u ) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 25
  • 56.
    Fresnel propagation • w/ WDF x x E1 (x) E2 (x ) W1 (x, u) W2 (x , u ) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 25
  • 57.
    Fresnel propagation • w/ WDF x x E1 (x) E2 (x ) W1 (x, u) W2 (x , u ) W2 (x , u ) = W1 (x − λzu , u ) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 25
  • 58.
    Fresnel propagation • w/ WDF x x E1 (x) E2 (x ) W1 (x, u) W2 (x , u ) W2 (x , u ) = W1 (x − λzu , u ) u x 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 25
  • 59.
    Fresnel propagation • w/ WDF x x E1 (x) E2 (x ) W1 (x, u) W2 (x , u ) W2 (x , u ) = W1 (x − λzu , u ) u u x-shear transform 1/(λz) x x 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 25
  • 60.
    diffraction vs. distance single slit a = 64λ laser from Zhengyun’s slide z 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 26
  • 61.
    diffraction vs. distance Position and Direction in Wave Optics single slit a = 64λ laser from Zhengyun’s slide z 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 26
  • 62.
    diffraction vs. distance Position and Direction in Wave Optics near zone: few λ (evanescent wave) Fresnel regime Fraunhofer regime (paraxial region) (Far-field) single slit a = 64λ laser from Zhengyun’s slide z 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 26
  • 63.
    diffraction vs. distance Position and Direction in Wave Optics near zone: few λ (evanescent wave) 1 FN Fresnel regime FN 1 Fraunhofer regime (paraxial region) (Far-field) non-paraxial single slit region a = 64λ laser from Zhengyun’s slide a2 z rule of thumb: Fresnel number FN = λz 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 26
  • 64.
    Augmented Light Field 1. free-space propagation 2. virtual light projector with negative radiance 3. light field transformer
  • 65.
    Virtual light projector WDF Augmented LF Traditional Light Field Diffraction and Interference With simple modifications in Light Field - virtual light projector (negative radiance) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 28
  • 66.
    Young’s experiment x θ u A B A B x A B x Light Field WDF ref. plane 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 29
  • 67.
    Young’s experiment projection projection x θ u A B A B x A B x Light Field WDF I(x) I(x) ref. plane x x 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 29
  • 68.
    Young’s experiment projection projection x θ u A B A B x A B x Light Field WDF I(x) I(x) ref. plane x x 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 29
  • 69.
    Virtual light projector projection real projector θ x real projector Augmented LF intensity=0 Not conflict with physics 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 30
  • 70.
    Virtual light projector projection 2π real projector cos λ [a − b]θ θ negative virtual light projector positive at the mid point x real projector Augmented LF intensity=0 Not conflict with physics 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 30
  • 71.
    Virtual light projector first null real projector (OPD = λ/2) real projector 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 31
  • 72.
    Virtual light projector first null real projector (OPD = λ/2) virtual light projector real projector 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 31
  • 73.
    Virtual light projector hyperbola first null (OPD = λ/2) asymptote of λ/2 hyperbola valid in Fresnel regime (or paraxial) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 32
  • 74.
    Virtual light projector destructive interference in high school physics class, (need negative radiance from virtual light projector) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 33
  • 75.
    Virtual light projector destructive interference in high school physics class, (need negative radiance from virtual light projector) m = λ/2 m = 3λ/2 m = 5λ/2 m = 7λ/2 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 33
  • 76.
    Question • Does a virtual light projector also work for incoherent light? 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 34
  • 77.
    Question • Does a virtual light projector also work for incoherent light? • Yes! 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 34
  • 78.
    Coherence • Degree of making interference • coherent partially coherent incoherent • Correlation of two)points on wavefront • E(p , t )E (p , t ∗ 1 1 (≈phase difference) 2 2 p1 Coherent: deterministic phase relation Incoherent: uncorrelated phase relation p2 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 35
  • 79.
    Coherence • throwing stones...... 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 36
  • 80.
    Coherence • throwing stones...... single point source 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 36
  • 81.
    Coherence • throwing stones...... single point source coherent 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 36
  • 82.
    Coherence • throwing stones...... single point source many point sources coherent 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 36
  • 83.
    Coherence • throwing stones...... single point source many point sources coherent if thrown identically, still coherent! 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 36
  • 84.
    Coherence • throwing stones...... single point source many point sources coherent if thrown identically, still coherent! if thrown randomly, then incoherent! 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 36
  • 85.
    Coherence • Temporal coherence: E(p, t )E (p, t ) 1 ∗ 2 • spectral bandwidth • monochromatic: temporally coherent • broadband (white light): temporally incoherent • Spatial coherence: E(p1 , t)E ∗ (p2 , t) • spatial bandwidth (angular span) • point source: spatially coherent • extended source: spatially incoherent 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 37
  • 86.
    Example 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 38
  • 87.
    Example Temporally incoherent; spatially coherent 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 38
  • 88.
    Example Temporally incoherent; spatially coherent 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 38
  • 89.
    Example Temporally incoherent; Temporally & spatially coherent spatially coherent 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 38
  • 90.
    Example Temporally incoherent; Temporally & spatially coherent spatially coherent 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 38
  • 91.
    Example Temporally incoherent; Temporally & spatially coherent spatially coherent Temporally & spatially incoherent 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 38
  • 92.
    Example Temporally incoherent; Temporally & spatially coherent spatially coherent Temporally & spatially incoherent Temporally coherent; spatially incoherent ? 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 38
  • 93.
    Example Temporally incoherent; Temporally & spatially coherent spatially coherent Temporally & spatially incoherent Temporally coherent; spatially incoherent rotating diffuser laser 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 38
  • 94.
    Temporal coherence • Broadband light is incoherent • ALF (also LF and WDF) can be defined for different wavelength and treated independently 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 39
  • 95.
    Young’s Exp. w/white light x I(x) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 40
  • 96.
    Young’s Exp. w/white light u Red x x u Green x u Blue x 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 41
  • 97.
    Young’s Exp. w/white light Red I(x) x x Green I(x) x Blue I(x) x 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 42
  • 98.
    Young’s Exp. w/white light Red x Green I(x) x Blue 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 42
  • 99.
    Spatial coherence • ALF w/ virtual light projectors is defined for spatially coherent light • For partially coherent/incoherent light, adding the defined ALF still gives valid results! 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 43
  • 100.
    Young’s Exp. w/spatially incoherent light x I(x) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 44
  • 101.
    Young’s Exp. w/spatially incoherent light x I(x) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 45
  • 102.
    Young’s Exp. w/spatially incoherent light x I(x) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 46
  • 103.
    Young’s Exp. w/spatially incoherent light x w/ random phase (uncorrelated) I(x) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 47
  • 104.
    Young’s Exp. w/spatially incoherent light x w/ random phase (uncorrelated) I(x) spatially incoherent light: infinite number of waves propagating along all the direction with random phase delay 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 47
  • 105.
    Young’s Exp. w/spatially incoherent light u x w/ random phase (uncorrelated) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 48
  • 106.
    Young’s Exp. w/spatially incoherent light u x w/ random phase (uncorrelated) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 48
  • 107.
    Young’s Exp. w/spatially incoherent light u x w/ random phase (uncorrelated) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 48
  • 108.
    Young’s Exp. w/spatially incoherent light u x w/ random phase (uncorrelated) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 48
  • 109.
    Young’s Exp. w/spatially incoherent light u x w/ random phase (uncorrelated) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 48
  • 110.
    Young’s Exp. w/spatially incoherent light u x w/ random phase (uncorrelated) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 48
  • 111.
    Young’s Exp. w/spatially incoherent light u x w/ random phase (uncorrelated) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 48
  • 112.
    Young’s Exp. w/spatially incoherent light u x w/ random phase Addition (uncorrelated) u x 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 48
  • 113.
    Young’s Exp. w/spatially incoherent light u x x w/ random phase (uncorrelated) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 49
  • 114.
    Young’s Exp. w/spatially incoherent light u x x w/ random phase Addition (uncorrelated) u x 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 49
  • 115.
    Virtual light projectors • Very simple modification to the LF • interference and diffraction within light field (geometry based) representation 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 50
  • 116.
    Augmented Light Field 1. free-space propagation 2. virtual light projector with negative radiance 3. light field transformer
  • 117.
    Light Field Transformer WDF Augmented LF Light Field Interaction at the optical elements 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 52
  • 118.
    Light Field Transformer light field transformer WDF LF LF LF LF negative radiance Augmented LF (diffractive) optical element Light Field LF propagation LF propagation Interaction at the optical elements 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 52
  • 119.
    Light Field Transformer • Q.Virtual light projector for a big aperture? • put the virtual light projectors for all the possible pairs of two points • WDF of optical elements 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 53
  • 120.
    Light Field Transformer • Q.Virtual light projector for a big aperture? • put the virtual light projectors for all the possible pairs of two points • WDF of optical elements 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 53
  • 121.
    Light Field Transformer • Q.Virtual light projector for a big aperture? • put the virtual light projectors for all the possible pairs of two points equivalent to compute the WDF mathematically.... 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 53
  • 122.
    Light Field Transformer • Q.Virtual light projector for a big aperture? • put the virtual light projectors for all the possible pairs of two points equivalent to compute the WDF mathematically.... • WDF of optical elements 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 53
  • 123.
    Light Field Transformer • Q.Virtual light projector for a big aperture? • put the virtual light projectors for all the possible pairs of two points equivalent to compute the WDF mathematically.... • WDF of optical elements representing properties of optical elements 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 53
  • 124.
    Light Field Transformer Tech report: S. B. Oh et. al 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 54
  • 125.
    Light Field Transformer • light field interactions w/ optical elements (x1 , θ1 ) (x2 , θ2 ) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 55
  • 126.
    Light Field Transformer • light field interactions w/ optical elements (x1 , θ1 ) (x2 , θ2 ) L1 (x1 , θ1 ) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 55
  • 127.
    Light Field Transformer • light field interactions w/ optical elements (x1 , θ1 ) (x2 , θ2 ) L1 (x1 , θ1 ) L2 (x2 , θ2 ) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 55
  • 128.
    Light Field Transformer • light field interactions w/ optical elements (x1 , θ1 ) (x2 , θ2 ) L1 (x1 , θ1 ) L2 (x2 , θ2 ) Light field transformer T (x2 , x1 , θ1 , θ2 ) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 55
  • 129.
    Light Field Transformer Dimension Property Note 8D(4D) thick, shift variant, 8D reflectance field, T (x2 , x1 , θ1 , θ2 ) angular variant volume hologram 6D(3D) thin, shift variant, 6D display, T (x, θ1 , θ2 ) angular variant BTF 4D(2D) thin, shift variant, many optical elements T (x, θ) angular invariant 2D(1D) attenuation shield field T (x) 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 56
  • 130.
    8D LF Transformer • the most generalized case (x1 , θ1 ) (x2 , θ2 ) L2 (x2 , θ2 ) L1 (x1 , θ1 ) L2 (x2 , θ2 ) = T (x2 , θ2 , x1 , θ1 )L1 (x1 , θ1 )dx1 dθ1 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 57
  • 131.
    6D LF Transformer • For thin optical elements x 6D Display L2 (x, θ2 ) L1 (x, θ1 ) Courtesy of Martin Fuchs Bidirectional L2 (x, θ2 ) = T (x, θ2 , θ1 )L1 (x, θ1 )dθ1 Texture Function Courtesy of Paul Debevec 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 58
  • 132.
    er of timeswith ference terms are tood with the in- the two pinholes 4D LF Transformer • Figure 7: Concept of virtual light sources for coherent light. w/ anglethe LF representation, no interference is predicted. By In shift invariant elements (in the paraxial region) virtual light sources, the LF propagation still including the n for diffraction • can be used. ould be included. e.g. aperture, lens, thin grating, etc oducing the con- have negative ra- es at a and b as al light source is π[a − b] λ along θ by integrating the l light sources do ne, which agrees Once the virtual L2 (x, θ) = T (x, θ − θ)L1 (x, θ )dθ propagation still Figure 8: Angle shift invariance in a thin transparency. In erly modeled3Dby Group (a) and (b), the output rays rotate in the same fashion 59 Se Baek Oh Optical Systems CVPR 2009 - Light Fields: Present and Future as
  • 133.
    Light field transformer • only amplitude variation (occluders) x shield fields for occluders L2 (x, θ) L1 (x, θ) L2 (x, θ) = T (x, θ)L1 (x, θ) Courtesy of D. Lanman 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 60
  • 134.
    Applications On wavefront coding holography 315 rendering the screen was very large. As expected, we see (Fig. 9) th Fraunhofer diffraction pattern. 1.1. Double-helix point spread function (DH-PSF) A DH-PSF system can be implemented by introducing a phase mask in the Fourier plane of an otherwise standard imaging system. The phase mask is designed such that its transmittance function generates a rotating pattern in the focal region of a Fourier transform lens [15-18]. Specifically, the DH-PSF exhibits two lobes that spin around the opticalaperture. An animate Figure 9: Diffraction from a square axis as shown in Fig. 1(a). Note that DH-PSF displays this experiment with of orientation with defocusappears in of a significant change varying the aperture size over an gaussian beam rotating PSF extended depth. In contrast, the standard PSF presents a slowly changing and expanding plementary material as a video. The distance from the ap symmetrical pattern throughout the same region [Fig. 1(b)]. the screen is 1 m. 316 317 Double rectangular apertures: Next we created two r lar apertures and probe them with the AMP. Note that we 3D Optical Fig. 1. Comparison of the (a) DH-PSF and the (b) standard PSF at different axial planes for a Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future system with 0.45 numerical aperture (NA) and 633nm wavelength. 61
  • 135.
    Space of LFrepresentations Time-frequency representations Phase space representations Quasi light field Other LF representations Observable LF WDF Augmented LF Other LF Traditional representations light field incoherent Rihaczek Distribution Function coherent
  • 136.
    Property of theRepresentation Constant along Interference Non-negativity Coherence Wavelength rays Cross term Traditional LF always always only zero no constant positive incoherent nearly always any Observable LF constant positive coherence any yes state only in the positive and Augmented LF paraxial region negative any any yes only in the positive and WDF paraxial region negative any any yes Rihaczek DF no; linear drift complex any any reduced 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 63
  • 137.
    Benefits & Limitationsof the Representation Adaptability Ability to Modeling Simplicity of to current Near Field Far Field propagate wave optics computation pipe line Traditional Light Fields x-shear no very simple high no yes Observable not x- yes modest low yes yes Light Fields shear Augmented Light Fields x-shear yes modest high no yes WDF x-shear yes modest low yes yes better than Rihaczek DF x-shear yes WDF, not as low no yes simple as LF 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 64
  • 138.
    Conclusion • WDFoptics generalized version of the LF in wave is the • Augmented Light Field • identicalregion) propagation (in the paraxial free-space • virtual light projectors • light field transformers • Wave opticswith geometrical be based understood phenomena can ray representations 3D Optical Se Baek Oh Systems Group CVPR 2009 - Light Fields: Present and Future 65
  • 139.
    Light Fields inRay and Wave Optics Introduction to Light Fields: Ramesh Raskar Wigner Distribution Function to explain Light Fields: Zhengyun Zhang Augmenting LF to explain Wigner Distribution Function: Se Baek Oh Q&A Break Light Fields with Coherent Light: Anthony Accardi New Opportunities and Applications: Raskar and Oh Q&A: All

Editor's Notes

  • #4 my background
  • #5 what is the light field?
  • #6 4D parameterization of plenoptic function, radiance,
  • #7 4D parameterization of plenoptic function, radiance,
  • #8 4D parameterization of plenoptic function, radiance,
  • #9 4D parameterization of plenoptic function, radiance,
  • #10 4D parameterization of plenoptic function, radiance,
  • #11 in wave optics, WDF exhibit similar property, compare the two,
  • #12 in wave optics, WDF exhibit similar property, compare the two,
  • #13 in wave optics, WDF exhibit similar property, compare the two,
  • #14 in wave optics, WDF exhibit similar property, compare the two,
  • #15 the motivation, to augment lf, model diffraction in light field formulation
  • #16 the motivation, to augment lf, model diffraction in light field formulation
  • #17 the motivation, to augment lf, model diffraction in light field formulation
  • #18 the motivation, to augment lf, model diffraction in light field formulation
  • #19 the motivation, to augment lf, model diffraction in light field formulation
  • #20 the motivation, to augment lf, model diffraction in light field formulation
  • #21 the motivation, to augment lf, model diffraction in light field formulation
  • #22 the motivation, to augment lf, model diffraction in light field formulation
  • #23 the motivation, to augment lf, model diffraction in light field formulation
  • #24 the motivation, to augment lf, model diffraction in light field formulation
  • #25 more specifically, same lf propagation,
  • #26 more specifically, same lf propagation,
  • #27 more specifically, same lf propagation,
  • #28 more specifically, same lf propagation,
  • #29 more specifically, same lf propagation,
  • #30 more specifically, same lf propagation,
  • #31 more specifically, same lf propagation,
  • #32 more specifically, same lf propagation,
  • #33 more specifically, same lf propagation,
  • #34 more specifically, same lf propagation,
  • #35 more specifically, same lf propagation,
  • #36 more specifically, same lf propagation,
  • #37 more specifically, same lf propagation,
  • #38 more specifically, same lf propagation,
  • #39 more specifically, same lf propagation,
  • #40 more specifically, same lf propagation,
  • #41 more specifically, same lf propagation,
  • #42 more specifically, same lf propagation,
  • #43 more specifically, same lf propagation,
  • #44 more specifically, same lf propagation,
  • #45 more specifically, same lf propagation,
  • #46 more specifically, same lf propagation,
  • #47 more specifically, same lf propagation,
  • #48 more specifically, same lf propagation,
  • #49 more specifically, same lf propagation,
  • #50 more specifically, same lf propagation,
  • #51 more specifically, same lf propagation,
  • #52 more specifically, same lf propagation,
  • #53 more specifically, same lf propagation,
  • #54 more specifically, same lf propagation,
  • #55 more specifically, same lf propagation,
  • #56 more specifically, same lf propagation,
  • #57 more specifically, same lf propagation,
  • #58 more specifically, same lf propagation,
  • #59 more specifically, same lf propagation,
  • #60 more specifically, same lf propagation,
  • #61 more specifically, same lf propagation,
  • #62 more specifically, same lf propagation,
  • #63 more specifically, same lf propagation,
  • #64 more specifically, same lf propagation,
  • #79 to demonstrate the limitation of LF,
  • #80 to demonstrate the limitation of LF,
  • #81 to demonstrate the limitation of LF,
  • #82 to demonstrate the limitation of LF,
  • #83 to demonstrate the limitation of LF,
  • #84 to demonstrate the limitation of LF,
  • #85 to demonstrate the limitation of LF,
  • #86 to demonstrate the limitation of LF,
  • #87 to demonstrate the limitation of LF,
  • #88 to demonstrate the limitation of LF,
  • #89 to demonstrate the limitation of LF,
  • #90 to demonstrate the limitation of LF,
  • #91 to demonstrate the limitation of LF,
  • #92 to demonstrate the limitation of LF,
  • #93 to demonstrate the limitation of LF,
  • #94 in terms of lf,
  • #99 so what is wigner?
  • #173 recall young&amp;#x2019;s, to make the light field model, we can bring the interference term
  • #174 recall young&amp;#x2019;s, to make the light field model, we can bring the interference term
  • #175 recall young&amp;#x2019;s, to make the light field model, we can bring the interference term