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Wigner Distributions and How They Relate to the Light Field Zhengyun Zhang, Marc Levoy Stanford University IEEE International Conference on Computational Photography 2009 Light Fields and Wave Optics Zhengyun Zhang, Marc Levoy Stanford

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SLIDE 1

Wigner Distributions and How They Relate to the Light Field

Zhengyun Zhang, Marc Levoy Stanford University

IEEE International Conference on Computational Photography 2009

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SLIDE 2

Light Fields and Wave Optics

Zhengyun Zhang, Marc Levoy Stanford University

IEEE International Conference on Computational Photography 2009

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SLIDE 3

Why Study Light Fields Using Wave Optics?

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SLIDE 4

Why Study Light Fields Using Wave Optics?

macro micro

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SLIDE 5

Why Study Light Fields Using Wave Optics?

s u

z=0 z=z0

x f

z=0 z=z0

light field Wigner distribution macro micro

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SLIDE 6

Outline

  • review light fields and wave optics
  • observable light field and

the Wigner distribution

  • applications
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SLIDE 7

Light Fields

  • radiance per ray
  • ray parametrization:
  • position (s)
  • direction (u)
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SLIDE 8

Light Fields

  • radiance per ray
  • ray parametrization:
  • position (s)
  • direction (u)
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SLIDE 9

Light Fields

  • radiance per ray
  • ray parametrization:
  • position (s)
  • direction (u)

reference plane

position

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SLIDE 10

Light Fields

  • radiance per ray
  • ray parametrization:
  • position (s)
  • direction (u)

reference plane

position direction

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SLIDE 11

Wave Optics

  • waves instead of rays
  • interference, diffraction
  • plane of point emitters

(Huygen’s principle)

  • each emitter has

amplitude and phase parallel rays plane waves

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SLIDE 12

Wave Optics

  • waves instead of rays
  • interference, diffraction
  • plane of point emitters

(Huygen’s principle)

  • each emitter has

amplitude and phase

(coherent and flatland)

parallel rays plane waves

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SLIDE 13

Wave Optics

(coherent and flatland)

  • waves instead of rays
  • interference, diffraction
  • plane of point emitters

(Huygen’s principle)

  • each emitter has

amplitude and phase

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SLIDE 14

Wave Optics

(coherent and flatland)

  • waves instead of rays
  • interference, diffraction
  • plane of point emitters

(Huygen’s principle)

  • each emitter has

amplitude and phase

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SLIDE 15

Wave Optics

(coherent and flatland)

  • waves instead of rays
  • interference, diffraction
  • plane of point emitters

(Huygen’s principle)

  • each emitter has

amplitude and phase

U(x) = A(x)ejφ(x)

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SLIDE 16

Position and Direction in Wave Optics

  • recall: light field

describes how power is spread over position and direction

  • point emitters on

plane have amplitude and phase

  • positional spread is

amplitude squared

U(x) = A(x)ejφ(x)

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SLIDE 17

Position and Direction in Wave Optics

  • recall: light field

describes how power is spread over position and direction

  • point emitters on

plane have amplitude and phase

  • positional spread is

amplitude squared

U(x) = A(x)ejφ(x) I(x) =

  • A(x)ejφ(x)
  • 2
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SLIDE 18

Position and Direction in Wave Optics

  • recall: light field

describes how power is spread over position and direction

  • point emitters on

plane have amplitude and phase

  • positional spread is

amplitude squared

U(x) = A(x)ejφ(x) I(x) = A2(x)

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SLIDE 19
  • direction
  • axial
  • oblique
  • more oblique

Position and Direction in Wave Optics

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SLIDE 20
  • direction
  • axial
  • oblique
  • more oblique

Position and Direction in Wave Optics

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SLIDE 21
  • direction
  • axial
  • oblique
  • more oblique

Position and Direction in Wave Optics

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SLIDE 22
  • direction
  • axial
  • oblique
  • more oblique

Position and Direction in Wave Optics

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SLIDE 23
  • direction
  • axial
  • oblique
  • more oblique

Position and Direction in Wave Optics

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SLIDE 24
  • direction
  • axial
  • oblique
  • more oblique

Position and Direction in Wave Optics

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SLIDE 25

Position and Direction in Wave Optics

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SLIDE 26

Position and Direction in Wave Optics

axial

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SLIDE 27

Position and Direction in Wave Optics

zero spatial frequency axial

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SLIDE 28

Position and Direction in Wave Optics

zero spatial frequency axial

  • blique
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SLIDE 29

Position and Direction in Wave Optics

zero spatial frequency axial low spatial frequency

  • blique
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SLIDE 30

Position and Direction in Wave Optics

zero spatial frequency axial low spatial frequency

  • blique

more oblique

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SLIDE 31

Position and Direction in Wave Optics

zero spatial frequency axial low spatial frequency

  • blique

higher spatial frequency more oblique

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SLIDE 32

Position and Direction in Wave Optics

plane waves

?

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SLIDE 33

Position and Direction in Wave Optics

plane waves

?

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SLIDE 34

Position and Direction in Wave Optics

aperture = 128 wavelengths

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SLIDE 35

Position and Direction in Wave Optics

aperture = 64 wavelengths

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SLIDE 36

Position and Direction in Wave Optics

aperture = 32 wavelengths

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SLIDE 37

Position and Direction in Wave Optics

aperture = 16 wavelengths

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SLIDE 38

Position and Direction in Wave Optics

aperture = 8 wavelengths

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SLIDE 39

Position and Direction in Wave Optics

aperture = 4 wavelengths

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SLIDE 40

Position and Direction in Wave Optics

aperture = 2 wavelengths

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SLIDE 41

Position and Direction in Wave Optics

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SLIDE 42

Recap

  • to determine both position and

spatial frequency, need to look at a window

  • f finite (nonzero) width

ray optics position direction wave optics position spatial frequency

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SLIDE 43

2D Wigner Distribution

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SLIDE 44

2D Wigner Distribution

x h(x)

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SLIDE 45

2D Wigner Distribution

Fourier

  • h (x) e−j2πfxxdx

x h(x)

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SLIDE 46

2D Wigner Distribution

Fourier

  • h (x) e−j2πfxxdx

x h(x) fx H(fx)

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SLIDE 47

2D Wigner Distribution

Fourier

  • h (x) e−j2πfxxdx

x h(x) x h(x) fx H(fx)

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SLIDE 48

2D Wigner Distribution

Fourier Wigner

  • h
  • x + ξ

2

  • h∗

x − ξ

2

  • e−j2πfξξdξ
  • h (x) e−j2πfxxdx

x h(x) x h(x) fx H(fx)

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SLIDE 49

2D Wigner Distribution

Fourier Wigner

  • h
  • x + ξ

2

  • h∗

x − ξ

2

  • e−j2πfξξdξ
  • h (x) e−j2πfxxdx

x h(x) x fξ Wh (x, fξ) x h(x) fx H(fx)

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SLIDE 50

2D Wigner Distribution

  • input: one-dimensional function of position
  • output: two-dimensional function of

position and frequency

  • (some) information about spectrum at each

position

Wh(x, fξ) =

  • h
  • x + ξ

2

  • h∗

x − ξ

2

  • e−j2πfξξdξ
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SLIDE 51

x fξ Wh (x, fξ)

2D Wigner Distribution

  • projection along frequency

yields power

  • projection along position

yields spectral power

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SLIDE 52

x fξ Wh (x, fξ)

2D Wigner Distribution

x |h(x)|2 • projection along frequency

yields power

  • projection along position

yields spectral power

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SLIDE 53

x fξ Wh (x, fξ)

2D Wigner Distribution

x |h(x)|2 |H (fξ)|2 fξ

  • projection along frequency

yields power

  • projection along position

yields spectral power

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SLIDE 54

x fξ Wh (x, fξ)

2D Wigner Distribution

x |h(x)|2 |H (fξ)|2 fξ

  • tradeoff between

width and height (fixed “area” or space-bandwidth product)

  • uncertainty principle
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SLIDE 55

x fξ Wh (x, fξ)

2D Wigner Distribution

x |h(x)|2 |H (fξ)|2 fξ

  • tradeoff between

width and height (fixed “area” or space-bandwidth product)

  • uncertainty principle
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SLIDE 56

x fξ Wh (x, fξ)

2D Wigner Distribution

x |h(x)|2 |H (fξ)|2 fξ

  • tradeoff between

width and height (fixed “area” or space-bandwidth product)

  • uncertainty principle
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SLIDE 57

2D Wigner Distribution

  • information about both

position and frequency

  • fixed space-bandwidth product

Wh(x, fξ) =

  • h
  • x + ξ

2

  • h∗

x − ξ

2

  • e−j2πfξξdξ
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SLIDE 58

Observable Light Field

  • move aperture

across plane

  • look at

directional spread

  • continuous

form of plenoptic camera

scene

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SLIDE 59

Observable Light Field

  • move aperture

across plane

  • look at

directional spread

  • continuous

form of plenoptic camera

scene

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SLIDE 60

Observable Light Field

  • move aperture

across plane

  • look at

directional spread

  • continuous

form of plenoptic camera

scene

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SLIDE 61

Observable Light Field

  • move aperture

across plane

  • look at

directional spread

  • continuous

form of plenoptic camera

scene

aperture position s direction u

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SLIDE 62

Observable Light Field

l(T )

  • bs (s, u) =
  • U(x)T(x − s)e−j2π u

λ xdx

  • 2
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SLIDE 63

Observable Light Field

Fourier transform

l(T )

  • bs (s, u) =
  • U(x)T(x − s)e−j2π u

λ xdx

  • 2
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SLIDE 64

Observable Light Field

wave Fourier transform

l(T )

  • bs (s, u) =
  • U(x)T(x − s)e−j2π u

λ xdx

  • 2
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SLIDE 65

Observable Light Field

wave Fourier transform aperture window

l(T )

  • bs (s, u) =
  • U(x)T(x − s)e−j2π u

λ xdx

  • 2
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SLIDE 66

Observable Light Field

wave Fourier transform aperture window power

l(T )

  • bs (s, u) =
  • U(x)T(x − s)e−j2π u

λ xdx

  • 2
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SLIDE 67

Observable Light Field

l(T )

  • bs (s, u) =
  • U(x)T(x − s)e−j2π u

λ xdx

  • 2

l(T )

  • bs (s, u) = WU
  • s, u

λ

  • ⊗ WT
  • −s, u

λ

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SLIDE 68

Observable Light Field

l(T )

  • bs (s, u) =
  • U(x)T(x − s)e−j2π u

λ xdx

  • 2

l(T )

  • bs (s, u) = WU
  • s, u

λ

  • ⊗ WT
  • −s, u

λ

  • Wigner distribution
  • f wave function
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SLIDE 69

Observable Light Field

l(T )

  • bs (s, u) =
  • U(x)T(x − s)e−j2π u

λ xdx

  • 2

l(T )

  • bs (s, u) = WU
  • s, u

λ

  • ⊗ WT
  • −s, u

λ

  • Wigner distribution
  • f wave function

Wigner distribution

  • f aperture window
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SLIDE 70

Observable Light Field

l(T )

  • bs (s, u) =
  • U(x)T(x − s)e−j2π u

λ xdx

  • 2

l(T )

  • bs (s, u) = WU
  • s, u

λ

  • ⊗ WT
  • −s, u

λ

  • Wigner distribution
  • f wave function

Wigner distribution

  • f aperture window

blur trades off resolution in position with direction

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SLIDE 71

l(T )

  • bs (s, u) = WU
  • s, u

λ

  • WT
  • −s, u

λ

  • ⊗

Observable Light Field

Wigner distribution

  • f wave function

Wigner distribution

  • f aperture window

at zero wavelength limit (regime of ray optics)

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SLIDE 72

l(T )

  • bs (s, u) = WU
  • s, u

λ

  • ⊗ δ(−s, u)

Observable Light Field

Wigner distribution

  • f wave function

at zero wavelength limit (regime of ray optics)

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SLIDE 73

l(T )

  • bs (s, u) = WU
  • s, u

λ

  • Observable Light Field

at zero wavelength limit (regime of ray optics)

  • bservable light field and Wigner equivalent!
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SLIDE 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

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SLIDE 75

Application - Refocusing

s u

light field

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SLIDE 76

Application - Refocusing

s u Isaksen

  • et. al

2000

light field

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SLIDE 77

Application - Refocusing

s u image at z=0 Isaksen

  • et. al

2000

light field

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SLIDE 78

Application - Refocusing

s u image at z=z0 Isaksen

  • et. al

2000

light field

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SLIDE 79

Application - Refocusing

s u

Fourier

fs fu Isaksen

  • et. al

2000

light field light field spectrum

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SLIDE 80

Application - Refocusing

s u

Fourier

fs fu Isaksen

  • et. al

2000 Ng 2005

light field light field spectrum

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SLIDE 81

Application - Refocusing

s u

Fourier

fs fu image at z=0 Isaksen

  • et. al

2000 Ng 2005

light field light field spectrum

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SLIDE 82

Application - Refocusing

s u

Fourier

fs fu image at z=z0 Isaksen

  • et. al

2000 Ng 2005

light field light field spectrum

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SLIDE 83

fξ x

Application - Refocusing

s u

Fourier

fs fu Isaksen

  • et. al

2000 Ng 2005

fx ξ Fourier

light field light field spectrum Wigner distribution ambiguity function

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SLIDE 84

fξ x

Application - Refocusing

s u

Fourier

fs fu Isaksen

  • et. al

2000 Ng 2005 image at z=0

fx ξ Fourier

light field light field spectrum Wigner distribution ambiguity function

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SLIDE 85

fξ x

Application - Refocusing

s u

Fourier

fs fu Isaksen

  • et. al

2000 Ng 2005 image at z=z0

fx ξ Fourier

light field light field spectrum Wigner distribution ambiguity function

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SLIDE 86

fξ x

Application - Refocusing

s u

Fourier

fs fu Isaksen

  • et. al

2000 Ng 2005

fx ξ Fourier

image at z=0

light field light field spectrum Wigner distribution ambiguity function

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SLIDE 87

fξ x

Application - Refocusing

s u

Fourier

fs fu Isaksen

  • et. al

2000 Ng 2005

fx ξ Fourier

image at z=z0

light field light field spectrum Wigner distribution ambiguity function

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SLIDE 88

fξ x

Application - Refocusing

s u

Fourier

fs fu Isaksen

  • et. al

2000 Ng 2005

fx ξ Fourier

Papoulis 1974

light field light field spectrum Wigner distribution ambiguity function

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SLIDE 89

Application - Wavefront Coding

Dowski and Cathey 1995 same aberrant blur regardless of depth of focus

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SLIDE 90

Application - Wavefront Coding

Dowski and Cathey 1995 same aberrant blur regardless of depth of focus point in scene

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SLIDE 91

Application - Wavefront Coding

Dowski and Cathey 1995 same aberrant blur regardless of depth of focus cubic phase plate point in scene

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SLIDE 92

Application - Wavefront Coding

Dowski and Cathey 1995 same aberrant blur regardless of depth of focus cubic phase plate point in scene small change in blur shape

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SLIDE 93

Application - Wavefront Coding

ambiguity function slices corresponding to various depths

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SLIDE 94

Application - Wavefront Coding

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SLIDE 95

Application - Wavefront Coding

s u point

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SLIDE 96

Application - Wavefront Coding

s u s u point before phase plate

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SLIDE 97

Application - Wavefront Coding

s u s u point after phase plate

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SLIDE 98

Application - Wavefront Coding

s u s u s u point after phase plate at image plane

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SLIDE 99

Application - Wavefront Coding

s u s u s u point after phase plate at image plane

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SLIDE 100

Application - Wavefront Coding

s u

  • refocusing in

ray space is shearing

  • shearing of a parabola

results in translation

  • blur shape invariant

to refocusing

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SLIDE 101

Application - Wavefront Coding

s u

  • refocusing in

ray space is shearing

  • shearing of a parabola

results in translation

  • blur shape invariant

to refocusing

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SLIDE 102

Application - Wavefront Coding

Fourier transform

  • f light field

slices corresponding to various depths

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SLIDE 103

Application - Wavefront Coding

Wigner distribution for cubic phase plate system

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SLIDE 104

Conclusions

  • light field’s position and direction =

wave optics’s position and frequency

  • observable light field =

blurred Wigner distribution (equal at zero wavelength limit)

  • analysis using light fields and

Wigner distribution interchangeable

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SLIDE 105

Future Work

  • analyze various light field capture and

generation systems using wave optics

  • rendering wave optics phenomena
  • adapt more ideas from
  • ptics community

and vice versa!

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SLIDE 106

Acknowledgements

  • Anat Levin, Fredo Durand and Bill Freeman
  • Stanford Graduate Fellowship from Texas

Instruments and NSF Grant CCF-0540872