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