CVPR 2009 Short Course on Light Field: Present and Future Slide 3/6

1. 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

2. 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

3. 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

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 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

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 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

136. 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

137. 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

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 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