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Fabricating BRDFs at High Spatial
Resolution Using Wave Optics
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
Applications:
- Architecture
- Product design
- Security markers visible under
certain illumination conditions
- Camouflage
- Photometric stereo
(Johnson&Adelson 09)
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BRDF (Bidirectional Reflectance Distribution Function)
3
𝑹 ℓ, 𝒗 =?
z
ℓ
𝒗
Dot (pixel) unit
on surface
x
4
Fabricating spatially varying BRDF
Reflectance
𝒗𝒙
𝒗𝒙
(ℓ𝒙 = ℓ𝒚 = 𝟎)
Shiny
𝒗𝒚
Diffuse
𝒗𝒚
Controlling reflectance via surface micro-structure5
Reflectance
𝒗𝒚
𝒗𝒙
𝒗𝒙
(ℓ𝒙 = ℓ𝒚 = 𝟎)
Shiny
𝒗𝒚
Diffuse
What surface microstructure produces
certain reflectances?
Surface micro
structure
Previous work: BRDF fabrication using microfacets theory (Weyrich et al. 09)
Reflectance
Surface
3cm
Surface:
Limited oriented
spatial resolution
planner
facets
Dot size
~ 3cm x 3cm
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Micro-facet model: limitations
0.3cm
0.03cm
Reflectance
Surface
scale
3cm
Wave effects at small scales
=> Substantial deviation from
geometric optics prediction
0.003cm
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Previous work: BRDF design
• Weyrich et al. (2009);
Fabricating microgeometry for custom surface reflectance.
• Matusik et al. (2009); Printing spatially-varying reflectance
• 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
• Wave models for BRDF:
He et al. 91; Nayar et al. 91; Stam 99; Cuypers et al. 12
No practical surface construction
• Holography
e.g. Yaroslavsky 2004; Benton and Bove 2008
Specific illumination conditions
(often coherent), not general BRDF
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Contributions:
• Extra high resolution fabrication
• Analyze wave effects under natural
illumination
• Analyze spatial-angular resolution tradeoffs
• Practical surface design algorithm compatible
with existing micro-fabrication technology
0.1mm
3cm
Photolithography and its limitations
Surface should be stepwise constant with a small
number of different depth values
z
x
Geometric
opticsdepth
predicts:
surface is a mirror
Prototype: Binary
values
Restricts
achievable
BRDFseffects
Wave optics:
variety
of reflectance
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12
Preview: reflectance = Fourier transform
𝟐
𝒗𝒙
𝒗𝒚 Narrow
𝒗𝒙
𝒗𝒙
Anisotropic
𝒗𝒚
Shiny
Wide
𝓕
Reflectance
𝒗𝒚 Wide
Diffuse
Surface
micro-structure
Narrow
Background: understanding light scattering
1. Coherent illumination: laser in physics lab
2. Incoherent illumination: natural world
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Wave effects on light scattering
z
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ℓ
x
Surface scattering – Fourier transform
𝑹 ℓ, 𝒗 =
Fourier
transform
z
𝒊𝒛(𝒙)
𝒆
ℱ(ℓ𝒙+𝒗𝒙) 𝒆
𝒗
ℓ
ℓ𝒙
𝒗𝒙
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2
See also:
He et al. 91
Stam 99
x
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Inverse width relationship
𝑹 ℓ, 𝒗 =
ℱℓ𝒙 +𝒗𝒙
𝒊𝒛(𝒙)
𝒆
2
Narrow (shiny)
reflectance
Wide surface
features
x
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Inverse width relationship
𝑹 ℓ, 𝒗 =
ℱℓ𝒙 +𝒗𝒙
𝒊𝒛(𝒙)
𝒆
2
Wide (diffuse)
reflectance
Narrow surface
features
x
Inverse width relationship
2
𝑹 ℓ, 𝒗 =
ℱℓ𝒙 +𝒗𝒙
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𝒊𝒛(𝒙)
𝒆
impulse (mirror)
reflectance
Flat surface
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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Speckles
20
Noisy reflectance from an
inexact surface
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
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
*
Illumination
angle
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Coherent
reflectance
Angular
Convolution
x
Incoherent reflectance: blurring coherent
reflectance by source angle
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Reflectance averaged
over illumination angle
is smooth
x
Challenge: avoiding speckles
Our analysis:
• Angular v.s. spatial resolution tradeoffs.
• Partial coherence.
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Angular resolution => Spatial coherence resolution
𝟏
∆𝒄 ∝
∆𝒂
Size of spatial unit over which illumination is coherent
𝚫𝒂
𝚫𝒄
x
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Angular resolution => Spatial coherence resolution
Each coherent region
emits a coherent field with
speckles
𝚫𝒂
𝚫𝒄
x
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Angular resolution => Spatial coherence resolution
Each coherent region
emits a coherent field with
speckles
𝚫𝒂
𝚫𝒄
x
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Angular resolution => Spatial coherence resolution
Each coherent region
emits a coherent field with
speckles
𝚫𝒂
𝚫𝒄
x
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Angular resolution => Spatial coherence resolution
Averaging different noisy
reflectances from multiple
coherent regions
=> smooth reflectance.
𝚫𝒂
𝚫𝒄
x
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Angular resolution => Spatial coherence resolution
Reflectance is smooth only if
∆𝒄 ≪ desired dot size
Coherent size
𝚫𝒄
𝚫𝒂
Dot size
𝚫𝒄
x
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Angular resolution => Spatial coherence resolution
Reflectance is smooth only if
∆𝒄 ≪ desired dot size
Coherent size
𝚫𝒄
𝚫𝒂
𝚫𝒄
Dot size
𝚫𝒄
x
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Angular resolution => Spatial coherence resolution
Reflectance is smooth only if
∆𝒄 ≪ desired dot size
Coherent size
𝚫𝒄
Human eye resolution
+ typical angle of 𝚫
natural sources.
𝒂
=> Smooth reflectance
(see paper)
𝚫𝒄
Dot size
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 Fourier power spectrum to
produce BRDF
- Pseudo randomness is sufficient
Surface tiling algorithm
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Randomly sample steps:
• Step width ~ 𝒑𝒂
• Step height ~ 𝒑𝒛 (uniform)
z
𝒛𝟏
𝒂𝟏
𝒛𝟐
𝒂𝟐
𝒛𝟑
𝒂𝟑
𝒛𝟒
𝒂𝟒
𝒛𝟓
𝒂𝟓
x
z
x
Surface tiling algorithm
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Randomly sample steps:
• Step width ~ 𝒑𝒂
• Step height ~ 𝒑𝒛 (uniform)
Coherent illumination
=> noisy reflectance
x
Surface tiling algorithm
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Randomly sample steps:
• Step width ~ 𝒑𝒂
• Step height ~ 𝒑𝒛 (uniform)
Width distribution 𝒑𝒂
defines reflectance:
𝑹 ℓ𝒙 , 𝒗𝒙 =
𝑬𝒑𝒂
𝒔𝒊𝒏𝒄𝟐
ℓ 𝒙 + 𝒗𝒙
𝒂−𝟏
Incoherent illumination
+ resolution conditions:
coherent size ≪ dot size
=> smooth reflectance
x
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Surface sampling
Step size distribution
𝒂
𝟐𝝁𝒎
𝓕
𝒂
𝟒𝝁𝒎
𝒂
Shiny
𝟖𝝁𝒎
𝟐
Glossy
𝒑𝒂
𝒑𝒂
Reflectance
Diffuse
𝒑𝒂
Sampled surface
micro-structure
BRDFs produced by our approach
Isotropic
Anisotropic
Anti-mirror
Anisotropic anti-mirrors
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Fabrication results
20mm
Electron microscope
scanning of
fabricated surface
Imaging reflectance from fabricated surface
Specular spike, artifact of
binary depth prototype,
can be removed with more
etching passes (see paper)
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Imaging under white illumination at
varying directions
wafer
Moving
light
Vertical illumination
Anisotropic BRDFs at
opposite orientations
Horizontal illumination
Negative image
Vertical
Horizontal
Negative image
Narrow AntiIsotropic mirror
large incident angle:
Anti-mirror kids: bright
Background: dark
Small incident angle:
Anti-mirror kids: dark
Background: bright
Limitations
• Color and albedo cannot be controlled
• Binary height restrictions:
Specular spike
BRDF must be symmetric
Simulation: eliminated with ≥ 4 different depths
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Summary
• Spatially varying BRDF at high spatial resolution (220 dpi).
• Analyze wave effects under natural illumination.
• Account for photolithography limitations.
• Pseudo randomness replaces sophisticated surface design.
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20mm
Thank you!
Wafer available after session