http://scripts.mit.edu/~raskar/lightfields/index.php?title=An_Introduction_to_The_Wigner_Distribution_in_Geometric_Optics

1. Wave Phenomenon in
Geometric Optics
Tom Cuypers
Se Baek Oh
Roarke Horstmeyer
Ramesh Raskar

2. Part 1
Introduction

3. Overview
1. Introduction and Welcome
2. Relating wave propagation to Light Fields
3. Augmented Light Fields
4. Applications in Imaging

4. Motivation
• Dual representation of light:
– Photons travelling in a straight line
Computational Photography Computer Graphics
http://graphics.stanford.edu/projects/lightfield
http://graphics.ucsd.edu/~henrik/images

5. Motivation
• Dual representation of light:
– Photons travelling in a straight line
– Waves traveling in all directions
Optics Holography
http://www.humanproductivitylab.com/images

6. Motivation
• Dual representation of light:
– Photons travelling in a straight line
– Waves traveling in all directions
• Goal of the course:
Provide a gentle introduction of wave
phenomenon using ray-based representations

7. Wave phenomena in the real world
• Fluid surfaces
http://4.bp.blogspot.com/_NpINLHeo8rM/Rsl52vjOKII/AAAAAAAAFMM/WnESejvzq5Y/s400/s
plash-water-waves-4559.JPG

8. Wave phenomena in the real world
• Fluid surfaces
• Sound waves
http://fetch1.com/wp-content/uploads/2009/11/hd-800_detail_sound-waves1.jpg

9. Wave phenomena in the real world
• Fluid surfaces
• Sound waves
• Electromagnetic waves
– Microscopic scale
http://upload.wikimedia.org/wikipedia/commons/archive/1/1f/20090127195426!Ggb_in_soap_bubble_1.jpg

10. Coherence
• Degree of making interference
– coherent ⇐ partially coherent ⇒ incoherent
• Correlation of two points on wavefront
– (≈phase difference)
Coherent: deterministic phase relation
Incoherent: uncorrelated phase
relation

11. Coherence
• throwing stones......
single point source many point sources
⇒ coherent ⇒ if thrown identically, still coherent!
⇒ if thrown randomly, then incoherent!

12. Coherence
• Temporal coherence:
– spectral bandwidth
• monochromatic: temporally coherent
• broadband (white light): temporally incoherent
• Spatial coherence:
– spatial bandwidth (angular span)
• point source: spatially coherent
• extended source: spatially incoherent

13. Example
Temporally incoherent; Temporally &
spatially coherent spatially coherent
Temporally & Temporally coherent;
spatially incoherent spatially incoherent
rotating
diffuser
laser

14. What is a wave?
• Types
– Electromagnetic waves
– Mechanical Waves
http://en.wikipedia.org/wiki/File:EM_spectrum.svg

15. What is a wave?
• Types
– Electromagnetic waves
– Mechanical Waves
http://www.gi.alaska.edu/chaparral/acousticspectrum.jpg

16. What is a wave?
• Types λ
• Properties A
– Wavelength: λ
– Frequency : p=0
p=π/2
– Phase: p p=π
p=3π/2
– Amplitude: A
– Polarization
http://www.ccrs.nrcan.gc.ca/glossary/images/3104.gif

17. What are wave phenomena?
• Huygens principle

18. What are wave phenomena?
• Huygens principle

19. What are wave phenomena?
• Huygens principle

20. What are wave phenomena?
• Huygens principle

21. What are wave phenomena?
• Huygens principle
• Diffraction

22. What are wave phenomena?
• Huygens principle
• Diffraction

23. What are wave phenomena?
• Huygens principle
• Diffraction

24. What are wave phenomena?
• Huygens principle
• Diffraction

25. What are wave phenomena?
• Huygens principle
• Diffraction

26. What are wave phenomena?
• Huygens principle Wave A
• Diffraction Wave B
• Interference
Constructive
interference

27. What are wave phenomena?
• Huygens principle Wave A
• Diffraction Wave B
• Interference
Destructive
interference

28. What are wave phenomena?
• Huygens principle
• Diffraction
• Interference
• Example N
Reflection
Ray-based

29. What are wave phenomena?
• Huygens principle
• Diffraction
• Interference
• Example
Reflection
Huygens Principle

30. What are wave phenomena?
• Huygens principle
• Diffraction
• Interference
• Example
Reflection
Huygens Principle

31. What are wave phenomena?
• Huygens principle
• Diffraction
• Interference
• Example
Reflection
Huygens Principle

32. What are wave phenomena?
• Huygens principle
• Diffraction
• Interference
• Example
Reflection
Huygens Principle

33. What are wave phenomena?
• Huygens principle
• Diffraction
• Interference
• Example
Reflection
Huygens Principle

34. What are wave phenomena?
• Huygens principle
• Diffraction
• Interference
• Example
Reflection
Huygens Principle

35. Part 2
Relating Wave Phenomena
to Light Fields

36. Introduction
• Review of Light Fields
• Review of Waves using Fourier optics
principles ? (intro)
• Introduction to the Wigner Distribution
Function
• Augmented Light Fields to represent wave
phenomena

37. Plenoptic Function
• Q: What is the set of all things that we can ever see?
• A: The Plenoptic Function (Adelson & Bergen)
Let’s start with a stationary person and try to parameterize
everything that he can see…

38. Gray Snapshot
• P(θ,φ) is intensity of light
– Seen from a single view point
– At a single time
– Averaged over the wavelengths of the visible spectrum
• (can also do P(x,y), but spherical coordinate are nicer)

39. Color Snapshot
P(θ,φ,λ) is intensity of light
– Seen from a single view point
– At a single time
– As a function of wavelength

40. Movie
P(θ,φ,λ,t) is intensity of light
– Seen from a single view point
– Over time
– As a function of wavelength

41. Holographic Movie
P(θ,φ,λ,t,Vx, Vy, Vz) is intensity of light
• – Seen from ANY single view point
• – Over time
• – As a function of wavelength

42. Plenoptic Function
P(θ,φ,λ,t,Vx, Vy, Vz)
• Can reconstruct every possible view, at every moment, from every position,
at every wavelength
• Contains every photograph, every movie, everything that anyone has ever
seen.

43. Sampling Plenoptic Function
(top view)

44. Ray
Let’s not worry about time and color:
5D : P(θ,φ,VX,VY,VZ)
• – 3D position
• – 2D direction

45. Ray
• No Occluding Objects
P(θ,φ,VX,VY,VZ)
• 4D
2D position
– 2D direction
• The space of all lines in 3-D space is 4D.

46. Representation
(θ,φ)
(u,v)
(x,y) (x,y)
Position-angle 2 plane
representation representation

47. Light Field Camera
Point Grey
Mark levoy

48. Why Study Light Fields Using Wave Optics?
z=z0 θ
x
Light
z=0 Field
Macro
Micro f
z=z0
x
Wigner
z=0 Distribution

49. Wave Optics
• Waves instead of rays Parallel rays Plane waves
• Interference & diffraction
• Plane of point emitters
(Huygen’s principle)
• Each emitter has amplitude
and phase

50. Position and direction in wave optics
• Spatial frequency: f
1
f

51. Position and direction in wave optics
• Spatial frequency: f
• Direction of wave: θ
λ
Small θ assumption: θ 1
f

52. Position and direction in wave optics
Complex wavefront = parallel wavefronts

53. Wigner Distribution Function
Auto correlation of complex wavefront
• Input: one-dimensional function of position
• Output: two-dimensional function of position
and spatial frequency
• (some) information about spectrum at each
position

54. Wigner Distribution Function
.
.
.
.
.
.

55. 2D Wigner Distribution
• Projection along
frequency yields power
• Projection along position
yield spectral power
f W(x,f)
x

56. 2D Wigner Distribution
|h(x)|² • Projection along
frequency yields power
x • Projection along position
yield spectral power
f W(x,f)
x

57. 2D Wigner Distribution
|h(x)|² • Projection along
frequency yields power
x • Projection along position
yield spectral power
f W(x,f) f
|f(x)|²
x

58. 2D Wigner Distribution
|h(x)|² • Projection along
frequency yields power
x • Projection along position
yield spectral power
f W(x,f) f
|f(x)|²
x

59. 2D Wigner Distribution
Remarks:
• Possible negative values
• Uncertainty principle
f W(x,f)
x

60. Relationship with Light Fields:
Observable Light Fields
• Move aperture
across plane
• Look at direction
spread
• Continuous form
of plenoptic Scene
camera

61. Relationship with Light Fields:
Observable Light Fields
• Move aperture
across plane
• Look at direction
spread
• Continuous form
of plenoptic Scene
camera

62. Relationship with Light Fields:
Observable Light Fields
• Move aperture
across plane
• Look at direction
spread
• Continuous form
of plenoptic Scene
camera

63. Relationship with Light Fields:
Observable Light Fields
• Move aperture
across plane
• Look at direction
spread
• Continuous form
of plenoptic Scene
camera

64. Relationship with Light Fields:
Observable Light Fields
• Move aperture
across plane
• Look at direction
spread
• Continuous form
of plenoptic Scene
camera

65. Relationship with Light Fields:
Observable Light Fields
• Move aperture
across plane
• Look at direction
spread
• Continuous form
of plenoptic Scene
camera

66. Relationship with Light Fields:
Observable Light Fields
• Move aperture
across plane
• Look at direction
spread
• Continuous form
of plenoptic Scene
θ
camera
Aperture
Position x

67. Relationship with Light Fields:
Observable Light Fields

68. Relationship with Light Fields:
Observable Light Fields
Aperture Window Power
Wave Fourier Transform

69. Relationship with Light Fields:
Observable Light Fields
Aperture Window Power
Wave Fourier Transform

70. Relationship with Light Fields:
Observable Light Fields
Aperture Window Power
Wave Fourier Transform
Wigner Distribution Wigner Distribution
of wave function of aperture window

71. Relationship with Light Fields:
Observable Light Fields
Blur trades off
resolution in position
with direction
Wigner Distribution Wigner Distribution
of wave function of aperture window

72. Relationship with Light Fields:
Observable Light Fields
At zero wavelength limit
(regime of ray optics)
Wigner Distribution
of wave function

73. Relationship with Light Fields:
Observable Light Fields
At zero wavelength limit
(regime of ray optics)
Observable light field and Wigner equivalent!

74. Observable Light Field
• Observable light field is a blurred Wigner
distribution with a modified coordinate
system
• Blur trades off resolution in position with
direction
• Wigner distribution and observable light field
equivalent at zero wavelength limit

75. Light Fields and Wigner
• Observable Light Fields = special case of
Wigner
• Ignores wave phenomena
• Can we also introduce wave phenomena in
light fields?
– -> Augmented Light Fields

76. Part 3
Augmenting Light Fields

77. Introduction
light field
position radiance of ray
Traditional
Light Field
ray optics based
simple and powerful ref. plane

78. Introduction
light field
direction
position radiance of ray
Traditional
Light Field
ray optics based
simple and powerful ref. plane

79. Introduction
Traditional
Light Field
ray optics based
simple and powerful

80. Introduction
rigorous but cumbersome
wave optics based
Wigner
Distribution
Function
Traditional
Light Field
ray optics based
simple and powerful
limited in diffraction & interference

81. Introduction
rigorous but cumbersome
wave optics based
Wigner
Distribution
Function
holograms beam shaping
Traditional
Light Field
ray optics based
rotational PSF
simple and powerful
limited in diffraction & interference

82. 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

83. 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

84. Augmented LF framework
LF
(diffractive)
optical
element

85. Augmented LF framework
LF LF
(diffractive)
optical
element
LF propagation

86. Augmented LF framework
light field
transformer
LF LF LF
negative
radiance
(diffractive)
optical
element
LF propagation

87. Augmented LF framework
light field
transformer
LF LF LF LF
negative
radiance
(diffractive)
optical
element
LF propagation LF propagation
Tech report, S. B. Oh et al.

88. Outline
• Limitations of Light Field analysis
– Ignore wave phenomena
– Only positive ray -> no interference

89. Outline
• Limitations of Light Field analysis
• Augmented Light Field
– free-space propagation

90. Outline
• Limitations of Light Field analysis
• Augmented Light Field
– free-space propagation
– virtual light projector in the ALF
• Possible negative
• Coherence

91. Outline
• Limitations of Light Field analysis
• Augmented Light Field
– free-space propagation
– virtual light projector in the ALF
• Possible negative
• Coherence
– light field transformer

92. Assumptions
• Monochromatic (= temporally coherent)
– can be extended into polychromatic
• Flatland (= 1D observation plane)
– can be extended to the real world
• Scalar field and diffraction (= one polarization)
– can be extended into polarized light
• No non-linear effect (two-photon, SHG, loss,
absorption, etc)

93. Young’s experiment
screen
light from double
a laser slit
constructive
interference

94. Young’s experiment
screen
light from double
a laser slit
destructive
interference

95. Young’s experiment
Light Field WDF
ref. plane

96. Young’s experiment
projection projection
Light Field WDF
ref. plane

97. Virtual light projector
projection
real projector
negative
virtual light projector positive
at the mid point
real projector
Augmented
LF
intensity=0
Not conflict with physics

98. Virtual light projector
first null
real projector (OPD = λ/2)
virtual light projector
real projector

99. Virtual light projector
hyperbola first null
(OPD = λ/2)
asymptote of
λ/2
hyperbola
valid in Fresnel regime
(or paraxial)

100. Virtual light projector
in high school physics destructive interference
(need negative radiance from
class, virtual light projector)
Video
waves

101. Question
• Does a virtual light projector also work for
incoherent light?
• Yes!

102. Temporal coherence
• Broadband light is incoherent
• ALF (also LF and WDF) can be defined
for different wavelength and treated
independently

103. Young’s Exp. w/ white light

104. Young’s Exp. w/ white light
Red
Green
Blue

105. Young’s Exp. w/ white light
Red
Green
Blue

106. 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!

107. Young’s Exp. w/ spatially
incoherent light

108. Young’s Exp. w/ spatially
incoherent light

109. Young’s Exp. w/ spatially
incoherent light

110. Young’s Exp. w/ spatially
incoherent light
w/ random
phase
(uncorrelated)
spatially incoherent light:
infinite number of waves propagating along all
the direction with random phase delay

111. Young’s Exp. w/ spatially
incoherent light
w/ random
phase Addition
(uncorrelated)

112. Young’s Exp. w/ spatially
incoherent light
w/ random
phase Addition
(uncorrelated)

113. Light Field Transformer
• light field interactions w/ optical elements
Light field transformer

114. Light Field Transformer
Dimension Property Note
8D(4D) thick, shift variant, 8D reflectance field,
angular variant volume hologram
6D(3D) thin, shift variant, 6D display,
angular variant BTF
4D(2D) thin, shift variant,
many optical elements
angular invariant
2D(1D) attenuation shield field

115. 8D LF Transformer
• the most generalized case

116. 6D LF Transformer
• For thin optical elements
6D Display
Courtesy of Martin Fuchs
Bidirectional
Texture Function
Courtesy of Paul Debevec

117. 4D LF Transformer
• w/ angle shift invariant elements (in the
paraxial region)
– e.g. aperture, lens, thin grating, etc

118. Part 4
Applications in Imaging

119. Message
• LF is a very powerful tool to understand
wave-related phenomena
– and potentially design and develop new systems
and applications

120. Augmented LF
light field
transformer
WDF LF LF LF LF
negative
radiance
Augmented LF (diffractive)
optical
element
Light
Field
LF LF
propagation propagation

121. Outline
gaussian beam wavefront coding
rotating PSF holography

122. Gaussian Beam
(from a laser pointer)
• Beam from a laser
– a solution of paraxial wave equation
20 mm beam
width
20 m distance

123. Gaussian Beam
• ALF (and WDF) of the Gaussian Beam is also Gaussian in x-θ
space

124. Gaussian Beam
x-θ space z-x space
20 mm beam 20 m
width distance

125. Wavefront coding
• ALF of a phase mask(slowly varying ϕ(x))
conventional wavefront coding
extended DOF
(w/ deconvolution)

126. Unusual PSF for depth from
defocus
standard PSF DH PSF
Defocus circle with distance
Prof. Rafael Piestun’s group
Courtesy of S. R. P. Pavani
Univ. of Colorado@Boulder U. of Colorado@Boulder

127. Rotating PSF
• Rotating beams
– Superposition along a straight line
– Rotation rate related to slope of
line
– Both intensity and phase rotate
– Maximum rotation rate in
Rayleigh range
intensity
Courtesy of S. R. P. Pavani

128. Rotating PSF
Courtesy of S. R. P. Pavani

129. Conceptually...

130. Conceptually...
other modes need to be balanced...

131. WDF (ALF) of (1,1) order
intensity
R. Simon and G. S. Agarwal, "Wigner
representation of Laguerre-Gaussian beams", Opt.
Lett., 25(18), (2000)

132. WDF in θx- θy
θy intensity in x- WDF in θx- θy
y
θx
y
θy
x θx
WDF in θx- θy
θy
θx

133. Holography
Recording Reconstruction
laser virtual
object
image
object wave real image
reference reference
wave hologram wave
hologram
observer

134. Holography
• For a point object
recording
reconstruction

135. Future direction
• Tomography & Inverse problems
• Beam shaping/phase mask design by ray-
based optimization
• New processing w/ virtual light source

136. 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

137. Property of the Representation
Constant Non- Interference
Coherence Wavelength
along rays negativity Cross term
always always only
Traditional LF constant positive incoherent zero no
Observable nearly always any
constant positive coherence any yes
LF state
Augmented only in the positive and
paraxial negative any any yes
LF region
only in the positive and
WDF paraxial negative any any yes
region
no; linear
Rihaczek DF complex any any reduced
drift

138. Benefits & Limitations of the Representation
Simplicity of Adaptability
Ability to Modeling
computatio to current Near Field Far Field
propagate wave optics
n pipe line
Traditional very
Light Fields
x-shear no 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 WDF, not
DF
x-shear yes as simple low no yes
as LF

139. Conclusions
• Wave optics phenomena can be understood with
geometrical ray based representation
• There are many different phase-space
representations
• We hope to inspire researchers in computer
vision/graphics as well as in optics graphics to
develop new tools and algorithms based on joint
exploration of geometric and wave optics concepts