Light Fields in Ray and Wave Optics Introduction to Light Fields: - - PowerPoint PPT Presentation

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Light Fields in Ray and Wave Optics Introduction to Light Fields: - - PowerPoint PPT Presentation

Nov 24, 2023 124 likes •1.28k views

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

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SLIDE 1

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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SLIDE 2

Using Wigner Distributions to Explain Light Fields

Zhengyun Zhang Stanford University

IEEE Computer Society Conference on Computer Vision and Pattern Recognition 2009

CVPR 2009 Short Course Light Fields: Present and Future (Computational Photography)

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SLIDE 3

Light Fields and Wave Optics

Zhengyun Zhang Stanford University

IEEE Computer Society Conference on Computer Vision and Pattern Recognition 2009

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SLIDE 4

Why Study Light Fields Using Wave Optics?

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SLIDE 5

Why Study Light Fields Using Wave Optics?

macro micro

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SLIDE 6

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 7

Outline

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

the Wigner distribution

  • applications

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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)

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SLIDE 10

Light Fields

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

reference plane

position

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SLIDE 11

Light Fields

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

reference plane

position direction

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SLIDE 12

Light Fields

  • Radiance per ray
  • Ray parametrization:
  • Position

: s, x, r

  • Direction : u, θ, s

Reference plane

position direction

Goal: Representing propagation, interaction and image formation of light using purely position and angle parameters

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SLIDE 13

Light Fields

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

reference plane

position direction

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SLIDE 14

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 15

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 16

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 17

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 18

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 19

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 20

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 21

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 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
  • direction
  • axial
  • oblique
  • more oblique

Position and Direction in Wave Optics

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SLIDE 26
  • direction
  • axial
  • oblique
  • more oblique

Position and Direction in Wave Optics

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SLIDE 27
  • direction
  • axial
  • oblique
  • more oblique

Position and Direction in Wave Optics

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SLIDE 28

Position and Direction in Wave Optics

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SLIDE 29

Position and Direction in Wave Optics

axial

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SLIDE 30

Position and Direction in Wave Optics

zero spatial frequency axial

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SLIDE 31

Position and Direction in Wave Optics

zero spatial frequency axial

  • blique

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SLIDE 32

Position and Direction in Wave Optics

zero spatial frequency axial low spatial frequency

  • blique

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SLIDE 33

Position and Direction in Wave Optics

zero spatial frequency axial low spatial frequency

  • blique

more oblique

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SLIDE 34

Position and Direction in Wave Optics

zero spatial frequency axial low spatial frequency

  • blique

higher spatial frequency more oblique

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SLIDE 35

Position and Direction in Wave Optics

plane waves

?

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SLIDE 36

Position and Direction in Wave Optics

plane waves

?

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SLIDE 37

Position and Direction in Wave Optics

aperture = 128 wavelengths

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SLIDE 38

Position and Direction in Wave Optics

aperture = 64 wavelengths

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SLIDE 39

Position and Direction in Wave Optics

aperture = 32 wavelengths

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SLIDE 40

Position and Direction in Wave Optics

aperture = 16 wavelengths

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SLIDE 41

Position and Direction in Wave Optics

aperture = 8 wavelengths

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SLIDE 42

Position and Direction in Wave Optics

aperture = 4 wavelengths

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SLIDE 43

Position and Direction in Wave Optics

aperture = 2 wavelengths

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SLIDE 44

Position and Direction in Wave Optics

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SLIDE 45

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 46

2D Wigner Distribution

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SLIDE 47

2D Wigner Distribution

x h(x)

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SLIDE 48

2D Wigner Distribution

Fourier

  • h (x) e−j2πfxxdx

x h(x)

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SLIDE 49

2D Wigner Distribution

Fourier

  • h (x) e−j2πfxxdx

x h(x) fx H(fx)

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SLIDE 50

2D Wigner Distribution

Fourier

  • h (x) e−j2πfxxdx

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

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SLIDE 51

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 52

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 53

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 54

x fξ Wh (x, fξ)

2D Wigner Distribution

  • projection along frequency

yields power

  • projection along position

yields spectral power

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SLIDE 55

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 56

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 57

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 58

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 59

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 60

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 61

Observable Light Field

  • move aperture

across plane

  • look at

directional spread

  • continuous

form of plenoptic camera

scene

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SLIDE 62

Observable Light Field

  • move aperture

across plane

  • look at

directional spread

  • continuous

form of plenoptic camera

scene

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SLIDE 63

Observable Light Field

  • move aperture

across plane

  • look at

directional spread

  • continuous

form of plenoptic camera

scene

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SLIDE 64

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 65

WDF

Traditional light field

Augmented LF Observable LF Rihaczek Distribution Function

Space of LF representations Time-frequency representations Phase space representations Quasi light field

incoherent coherent

Other LF representations Other LF representations

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SLIDE 66

Property of the Representation

Constant along rays Non-negativity Coherence Wavelength Interference Cross term Traditional LF always constant always positive only incoherent zero no Observable LF nearly constant always positive any coherence state any yes Augmented LF only in the paraxial region positive and negative any any yes WDF

  • nly in the

paraxial region positive and negative any any yes Rihaczek DF no; linear drift complex any any reduced

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SLIDE 67

Benefits & Limitations of the Representation

Ability to propagate Modeling wave optics Simplicity of computation Adaptability to current pipe line Near Field Far Field Traditional LF x-shear no very simple high no yes Observable LF not x-shear yes modest low yes yes Augmented LF x-shear yes modest high no yes WDF x-shear yes modest low yes yes Rihaczek DF x-shear yes better than WDF, not as simple as LF low no yes

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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

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SLIDE 69

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 70

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 71

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 72

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 73

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

  • 37
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SLIDE 74

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 75

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 76

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 77

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 78

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 79

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 80

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 81

Application - Refocusing

s u

light field

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SLIDE 82

Application - Refocusing

s u Isaksen

  • et. al

2000

light field

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SLIDE 83

Application - Refocusing

s u image at z=0 Isaksen

  • et. al

2000

light field

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SLIDE 84

Application - Refocusing

s u image at z=z0 Isaksen

  • et. al

2000

light field

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SLIDE 85

Application - Refocusing

s u

Fourier

fs fu Isaksen

  • et. al

2000

light field light field spectrum

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SLIDE 86

Application - Refocusing

s u

Fourier

fs fu Isaksen

  • et. al

2000 Ng 2005

light field light field spectrum

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SLIDE 87

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 88

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 89

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 90

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 91

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 92

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 93

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 94

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 95

Application - Wavefront Coding

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

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SLIDE 96

Application - Wavefront Coding

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

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SLIDE 97

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 98

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 99

Application - Wavefront Coding

ambiguity function slices corresponding to various depths

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SLIDE 100

Application - Wavefront Coding

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SLIDE 101

Application - Wavefront Coding

s u point

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SLIDE 102

Application - Wavefront Coding

s u s u point before phase plate

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SLIDE 103

Application - Wavefront Coding

s u s u point after phase plate

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SLIDE 104

Application - Wavefront Coding

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

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SLIDE 105

Application - Wavefront Coding

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

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SLIDE 106

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 107

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 108

Application - Wavefront Coding

Fourier transform

  • f light field

slices corresponding to various depths

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SLIDE 109

Application - Wavefront Coding

Wigner distribution for cubic phase plate system

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SLIDE 110

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 111

Further Reading

  • http://scripts.mit.edu/~raskar/lightfields/

Wiki for this course

  • Z. Zhang, M. Levoy, “Wigner Distributions

and How They Relate to the Light Field”, ICCP 2009

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SLIDE 112

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 113

Acknowledgements

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

Instruments and NSF Grant CCF-0540872

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SLIDE 114

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

51