Transcript Slides
Fabricating BRDFs at High Spatial Resolution Using Wave Optics
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Anat Levin, Daniel Glasner, Ying Xiong, Fredo Durand, Bill Freeman, Wojciech Matusik, Todd Zickler.
Weizmann Institute, Harvard University, MIT
Appearance fabrication Goal: Fabricating surfaces with user defined appearance
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Applications: - Architecture Product design Security markers visible under
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certain illumination conditions Camouflage - Photometric stereo (Johnson&Adelson 09)
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z
BRDF (Bidirectional Reflectance Distribution Function)
𝑹 ℓ , 𝒗 =
?
3 ℓ
Dot (pixel) unit on surface
𝒗
x
Fabricating spatially varying BRDF Reflectance
𝒗 𝒚 𝒗 𝒙 𝒗 𝒚 𝒗 𝒙
(
ℓ 𝒙 = ℓ 𝒚 = 𝟎
)
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Controlling reflectance via surface micro-structure
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What surface micro structure produces certain reflectances?
Surface micro structure Reflectance
𝒗 𝒚 𝒗 𝒙 𝒗 𝒚 𝒗 𝒙
(
ℓ 𝒙 = ℓ 𝒚 = 𝟎
)
Previous work: BRDF fabrication using micro facets theory (Weyrich et al. 09) 3cm
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Micro-facet model: limitations
3cm 0.3cm
0.03cm
0.003cm
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Wave effects at small scales => Substantial deviation from geometric optics prediction
Previous work: BRDF design
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Weyrich et al. (2009); Fabricating microgeometry for custom surface reflectance.
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Matusik et al. (2009); Printing spatially-varying reflectance
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Finckh et al. (2010); Geometry construction from caustic images
• • • •
Dong et al. (2010); Fabricating spatially-varying subsurface scattering.
Papas et al (2011); Goal-based caustics.
Malzbender et al. (2012); Printing reflectance functions Lan et al. (2013); Bi-Scale Appearance Fabrication
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Previous work: Wave scattering
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Wave models for BRDF: He et al. 91; Nayar et al. 91; Stam 99; Cuypers et al. 12 No practical surface construction
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Holography e.g. Yaroslavsky 2004; Benton and Bove 2008 Specific illumination conditions (often coherent), not general BRDF
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Contributions:
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Extra high resolution fabrication
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Analyze wave effects under natural illumination
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Analyze spatial-angular resolution tradeoffs
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Practical surface design algorithm compatible with existing micro-fabrication technology
Photolithography and its limitations Surface should be stepwise constant with a small number of different depth values
z x
Preview: reflectance = Fourier transform Surface micro-structure Narrow Reflectance
𝒗 𝒚
Wide Wide
𝓕 𝟐 𝒗 𝒙 𝒗 𝒚
Narrow
𝒗 𝒙 𝒗 𝒚 𝒗 𝒙 12
Background: understanding light scattering
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1. Coherent illumination: laser in physics lab 2. Incoherent illumination: natural world
Wave effects on light scattering
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z
ℓ
x
z
𝑹
Surface scattering – Fourier transform 2
ℓ , 𝒗 =
Fourier transform
ℱ ( ℓ 𝒙 + 𝒗 𝒙 ) 𝒆 𝒊
𝒛(𝒙)
15 ℓ ℓ 𝒙 𝒗 𝒙 𝒗
See also: He et al. 91 Stam 99
x
Inverse width relationship
𝑹 ℓ , 𝒗 =
Narrow (shiny) reflectance
ℱ ℓ 𝒙 + 𝒗 𝒙 𝒆 𝒊
𝒛(𝒙)
2 Wide surface features
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x
Inverse width relationship
𝑹 ℓ , 𝒗 =
Wide (diffuse) reflectance
ℱ ℓ 𝒙 + 𝒗 𝒙 𝒆 𝒊
𝒛(𝒙)
2 Narrow surface features
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x
𝑹 ℓ , 𝒗
Inverse width relationship
2
= ℱ ℓ 𝒙 + 𝒗 𝒙 𝒆 𝒊
𝒛(𝒙)
impulse (mirror) reflectance Flat surface
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x
Reflectance design with coherent illumination:
Fourier power spectrum of surface height to produce reflectance
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Challenges: Complex non-linear optimization May not have a solution with stepwise constant heights Inexact solutions:
speckles
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Speckles Noisy reflectance from an inexact surface
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x
Reflectance design with coherent illumination:
Fourier power spectrum of surface height to produce reflectance
• • •
Challenges: Complex non-linear optimization May not have a solution with stepwise constant heights Inexact solutions:
speckles
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Our approach: Bypass problems utilizing
natural illumination
Pseudo random surface replaces optimization Need to model
partial coherence
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Incoherent illumination: Point source=> Area source Area source = collection of independent coherent point sources
ℓ 𝒗
x
Incoherent reflectance: blurring coherent reflectance by source angle
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Illumination angle * Angular Convolution Coherent reflectance
x
Incoherent reflectance: blurring coherent reflectance by source angle Reflectance averaged over illumination angle is smooth
x
Challenge: avoiding speckles Our analysis:
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Angular v.s. spatial resolution tradeoffs.
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Partial coherence.
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Angular resolution => Spatial coherence resolution
𝟏 ∆ 𝒄 ∝ ∆ 𝒂
Size of spatial unit over which illumination is coherent
𝚫 𝒂 𝚫 𝒄
x
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Angular resolution => Spatial coherence resolution Each coherent region emits a coherent field with speckles
𝚫 𝒂 𝚫 𝒄
x
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Angular resolution => Spatial coherence resolution Each coherent region emits a coherent field with speckles
𝚫 𝒂 𝚫 𝒄
x
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Angular resolution => Spatial coherence resolution Each coherent region emits a coherent field with speckles
𝚫 𝒂 𝚫 𝒄
x
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Angular resolution => Spatial coherence resolution Averaging different noisy reflectances from multiple coherent regions => smooth reflectance.
𝚫 𝒂 𝚫 𝒄
x
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Angular resolution => Spatial coherence resolution Reflectance is smooth only if
∆ 𝒄 ≪
desired dot size Coherent size
𝚫 𝒄 𝚫 𝒂
Dot size
𝚫 𝒄
x
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Angular resolution => Spatial coherence resolution Reflectance is smooth only if
∆ 𝒄 ≪
desired dot size Coherent size
𝚫 𝒄 𝚫 𝒂
Dot size
𝚫 𝒄 𝚫 𝒄
x
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Angular resolution => Spatial coherence resolution Reflectance is smooth only if
∆ 𝒄 ≪
desired dot size Human eye resolution Coherent size
𝚫 𝒄 𝚫 𝒂
(see paper) Dot size
𝚫 𝒄
x
Recap: Coherent BRDF = Fourier power spectrum of surface height.
Incoherent BRDF = Fourier power spectrum of surface height,
blurred
by illumination angle.
Next: Design surface height to produce desired BRDF.
Coherent design: Fourier power spectrum to produce BRDF - Complex non linear optimization Incoherent design:
Blurred
produce BRDF Fourier power spectrum to -
Pseudo randomness
is sufficient
Surface tiling algorithm
• •
Randomly sample steps: Step width ~
𝒑 𝒂
Step height ~
𝒑 𝒛
(uniform)
z
𝒛 𝟏 𝒂 𝟏 𝒂 𝟐 𝒛 𝟐 𝒛 𝟑 𝒂 𝟑 𝒛 𝟒 𝒂 𝟒 𝒛 𝟓 𝒂 𝟓
x z
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x
Surface tiling algorithm
• •
Randomly sample steps: Step width ~
𝒑 𝒂
Step height ~
𝒑 𝒛
(uniform) Coherent illumination => noisy reflectance
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x
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Surface tiling algorithm
• •
Randomly sample steps: Step width ~
𝒑 𝒂
Step height ~
𝒑 𝒛
(uniform) Width distribution
𝒑 𝒂
defines reflectance:
𝑹 ℓ 𝒙 , 𝒗 𝒙 = 𝑬 𝒑 𝒂 𝒔𝒊𝒏𝒄 𝟐 ℓ 𝒙 + 𝒗 𝒙 𝒂 −𝟏
Incoherent illumination + resolution conditions: coherent size
≪
dot size => smooth reflectance
x
Surface sampling
𝒑 𝒂
Step size distribution Sampled surface micro-structure
𝒑 𝒂 𝟐𝝁𝒎 𝒂 𝓕 𝟐 𝒂 𝟒𝝁𝒎 𝒑 𝒂
Reflectance
41 𝟖𝝁𝒎 𝒂
BRDFs produced by our approach Isotropic Anisotropic
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Anti-mirror Anisotropic anti-mirrors
Fabrication results
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20
m
m Electron microscope scanning of fabricated surface
Imaging reflectance from fabricated surface
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Specular spike, artifact of binary depth prototype, can be removed with more etching passes (see paper)
Imaging under white illumination at varying directions wafer Moving light
Vertical illumination Anisotropic BRDFs at Horizontal illumination
Negative image
opposite orientations
Vertical
Negative image
Horizontal
Narrow Isotropic Anti mirror large incident angle: Anti-mirror kids:
bright
Background:
dark
Small incident angle: Anti-mirror kids:
dark
Background:
bright
Limitations
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Color and albedo cannot be controlled
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Binary height restrictions: Specular spike BRDF must be symmetric Simulation: eliminated with
≥
4 different depths
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Summary
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Spatially varying BRDF at high spatial resolution (220 dpi).
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Analyze wave effects under natural illumination.
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Account for photolithography limitations.
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Pseudo randomness replaces sophisticated surface design.
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20
m
m
Thank you!
Wafer available after session
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