Report

1 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) 2 BRDF (Bidirectional Reflectance Distribution Function) 3 ℓ, =? z ℓ Dot (pixel) unit on surface x 4 Fabricating spatially varying BRDF Reflectance (ℓ = ℓ = ) Shiny Diffuse Controlling reflectance via surface micro-structure5 Reflectance (ℓ = ℓ = ) Shiny 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 6 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 7 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 8 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 9 10 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 11 12 Preview: reflectance = Fourier transform Narrow Anisotropic Shiny Wide Reflectance Wide Diffuse Surface micro-structure Narrow Background: understanding light scattering 1. Coherent illumination: laser in physics lab 2. Incoherent illumination: natural world 13 Wave effects on light scattering z 14 ℓ x Surface scattering – Fourier transform ℓ, = Fourier transform z () ℱ(ℓ+) ℓ ℓ 15 2 See also: He et al. 91 Stam 99 x 16 Inverse width relationship ℓ, = ℱℓ + () 2 Narrow (shiny) reflectance Wide surface features x 17 Inverse width relationship ℓ, = ℱℓ + () 2 Wide (diffuse) reflectance Narrow surface features x Inverse width relationship 2 ℓ, = ℱℓ + 18 () 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 19 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 21 22 Incoherent illumination: Point source=> Area source Area source = collection of independent coherent point sources ℓ x Incoherent reflectance: blurring coherent reflectance by source angle * Illumination angle 23 Coherent reflectance Angular Convolution x Incoherent reflectance: blurring coherent reflectance by source angle 24 Reflectance averaged over illumination angle is smooth x Challenge: avoiding speckles Our analysis: • Angular v.s. spatial resolution tradeoffs. • Partial coherence. 25 26 Angular resolution => Spatial coherence resolution ∆ ∝ ∆ Size of spatial unit over which illumination is coherent x 29 Angular resolution => Spatial coherence resolution Each coherent region emits a coherent field with speckles x 30 Angular resolution => Spatial coherence resolution Each coherent region emits a coherent field with speckles x 31 Angular resolution => Spatial coherence resolution Each coherent region emits a coherent field with speckles x 32 Angular resolution => Spatial coherence resolution Averaging different noisy reflectances from multiple coherent regions => smooth reflectance. x 33 Angular resolution => Spatial coherence resolution Reflectance is smooth only if ∆ ≪ desired dot size Coherent size Dot size x 34 Angular resolution => Spatial coherence resolution Reflectance is smooth only if ∆ ≪ desired dot size Coherent size Dot size x 35 Angular resolution => Spatial coherence resolution Reflectance is smooth only if ∆ ≪ desired dot size Coherent size Human eye resolution + typical angle of natural sources. => Smooth reflectance (see paper) 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 38 Randomly sample steps: • Step width ~ • Step height ~ (uniform) z x z x Surface tiling algorithm 39 Randomly sample steps: • Step width ~ • Step height ~ (uniform) Coherent illumination => noisy reflectance x Surface tiling algorithm 40 Randomly sample steps: • Step width ~ • Step height ~ (uniform) Width distribution defines reflectance: ℓ , = ℓ + − Incoherent illumination + resolution conditions: coherent size ≪ dot size => smooth reflectance x 41 Surface sampling Step size distribution Shiny Glossy Reflectance Diffuse Sampled surface micro-structure BRDFs produced by our approach Isotropic Anisotropic Anti-mirror Anisotropic anti-mirrors 42 43 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) 44 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 49 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. 51 52 20mm Thank you! 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