Computer Graphics - Material Models - Philipp Slusallek & Arsne - PowerPoint PPT Presentation

Computer Graphics - Material Models - Philipp Slusallek & Arsène Pérard-Gayot

Overview • Last time – Light: radiance & light sources – Light transport: rendering equation & formal solutions • Today – Reflectance properties: • Material models • Bidirectional Reflectance Distribution Function (BRDF) • Reflection models – Shading • Next lecture – Varying (reflection) properties over object surfaces: texturing 2

REFLECTANCE PROPERTIES 3

Appearance Samples • How do materials reflect light? Opaque Translucency - subsurface scattering 4

Material Samples • Anisotropic anisotropic 5

Material Samples • Complex surface meso-structure 6

Material Samples • Fibers 7

Material Samples • Photos of samples with light source at exactly the same position diffuse glossy mirror 8

How to describe materials? • Reflection properties • Mechanical, chemical, electrical properties • Surface roughness • Geometry/meso-structure • Goal: relightable representation of appearance 9

Reflection Equation - Reflectance • Reflection equation 𝑀 𝑝 𝑦, 𝜕 𝑝 = 𝑔 𝑠 𝜕 𝑗 , 𝑦, 𝜕 𝑝 𝑀 𝑗 𝑦, 𝜕 𝑗 𝑑𝑝𝑡𝜄 𝑗 𝑒𝜕 𝑗 Ω + • BRDF – Ratio of reflected radiance to incident irradiance 𝑠 𝜕 𝑗 , 𝑦, 𝜕 𝑝 = 𝑒𝑀 𝑝 𝑦, 𝜕 𝑝 𝑔 𝑒𝐹 𝑗 (𝑦, 𝜕 𝑗 ) 10

BRDF • BRDF describes surface reflection – for light incident from direction 𝝏 𝒋 = 𝜾 𝒋 , 𝝌 𝒋 – observed from direction 𝝏 𝒑 = 𝜾 𝒑 , 𝝌 𝒑 • Bidirectional – Depends on 2 directions 𝜕 𝑗 , 𝜕 𝑝 and position x (6-D function) 𝑠 𝜕 𝑗 , 𝑦, 𝜕 𝑝 = 𝑒𝑀 𝑝 𝑦, 𝜕 𝑝 𝑒𝑀 𝑝 (𝑦, 𝜕 𝑝 ) 𝑔 𝑒𝐹 𝑗 (𝑦, 𝜕 𝑗 ) = 𝑀 𝑗 𝑦, 𝜕 𝑗 𝑑𝑝𝑡𝜄 𝑗 𝑒𝜕 𝑗 11

BRDF Properties • Helmholtz reciprocity principle – BRDF remains unchanged if incident and reflected directions are interchanged – Due to physical law of time reversal 𝑔 𝑠 𝜕 𝑗 , 𝜕 𝑝 = 𝑔 𝑠 (𝜕 𝑝 , 𝜕 𝑗 ) • Smooth surface: isotropic BRDF – Reflectivity independent of rotation around surface normal – BRDF has only 3 instead of 4 directional degrees of freedom 𝑔 𝑠 (𝜄 𝑗 , 𝑦, 𝜄 𝑝 , 𝜒 𝑝 − 𝜒 𝑗 ) 12

BRDF Properties • Characteristics – BRDF units • Inverse steradian: 𝑡𝑠 −1 (not intuitive) – Range of values: distribution function is positive, can be infinite • From 0 (total absorption) to ∞ (reflection, 𝜀 -function) – Energy conservation law • No self-emission • Possible absorption 𝑔 𝑠 𝜕 𝑗 , 𝑦, 𝜕 𝑝 𝑑𝑝𝑡𝜄 𝑝 𝑒𝜕 𝑝 ≤ 1, ∀𝜕 𝑗 Ω + • Reflection only at the point of entry ( 𝒚 𝒋 = 𝒚 𝒑 ) – No subsurface scattering 13

Standardized Gloss Model • Industry often uses only a subset of BRDF values – Reflection only measured at discrete set of angles 14

Reflection of an Opaque Surface 15

Reflection of an Opaque Surface ω 𝑝 ω 𝑝 ω 𝑝 16

Isotropic BRDF – 3D • Invariant with respect to rotation about the normal – Only depends on azimuth difference to incoming angle 𝑔 𝜄 𝑗 , 𝜒 𝑗 → 𝜄 𝑝 , 𝜒 𝑝 ⟹ 𝑠 𝑔 𝑠 𝜄 𝑗 → 𝜄 𝑝 , (𝜒 𝑗 −𝜒 𝑝 ) = 𝑔 𝑠 𝜄 𝑗 → 𝜄 𝑝 , Δ𝜒 ω 𝑝 ω 𝑗 𝑦 Δϕ 17

Homogeneous BRDF – 4D • Homogeneous bidirectional reflectance distribution function – Ratio of reflected radiance to incident irradiance 𝑠 𝜕 𝑗 → 𝜕 𝑝 = 𝑒𝑀 𝑝 𝜕 𝑝 𝑔 𝑒𝐹 𝑗 (𝜕 𝑗 ) ω 𝑝 ω 𝑗 18

Spatially Varying BRDF – 6D • Heterogeneous materials (standard model for BRDF) – Dependent on position, and two directions – Reflection at the point of incidence 𝑔 𝑠 𝑦, 𝜕 𝑗 → 𝜕 𝑝 ω 𝑝 ω 𝑗 𝑦 19

Homogeneous BSSRDF – 6D • Homogeneous bidirectional scattering surface reflectance distribution function – Assumes a homogeneous and flat surface – Only depends on the difference vector to the outgoing point 𝑔 𝑠 Δ𝑦, 𝜕 𝑗 → 𝜕 𝑝 ω 𝑝 ω 𝑝 ω 𝑗 Δ𝑦 𝑦 𝑗 𝑦 𝑝 20

BSSRDF – 8D • Bidirectional scattering surface reflectance distribution function 𝑔 𝑠 (𝑦 𝑗 , 𝜕 𝑗 ) → (𝑦 𝑝 , 𝜕 𝑝 ) ω 𝑝 ω 𝑗 𝑦 𝑗 𝑦 𝑝 21

Generalization – 9D • Generalizations – Add wavelength dependence 𝑔 𝑠 𝜇, (𝑦 𝑗 , 𝜕 𝑗 ) → (𝑦 𝑝 , 𝜕 𝑝 ) ω 𝑝 ω 𝑗 𝑦 𝑗 𝑦 𝑝 22

Generalization – 10D • Generalizations – Add wavelength dependence – Add fluorescence • Change to longer wavelength during scattering 𝑔 𝑦 𝑗 , 𝜕 𝑗 , 𝜇 𝑗 → 𝑦 𝑝 , 𝜕 𝑝 , 𝜇 𝑝 𝑠 ω 𝑝 ω 𝑗 𝑦 𝑗 𝑦 𝑝 23

Generalization – 11D • Generalizations – Add wavelength dependence – Add fluorescence (change to longer wavelength for reflection) – Time varying surface characteristics 𝑔 𝑠 𝑢, 𝑦 𝑗 , 𝜕 𝑗 , 𝜇 𝑗 → 𝑦 𝑝 , 𝜕 𝑝 , 𝜇 𝑝 ω 𝑝 ω 𝑗 𝑦 𝑗 𝑦 𝑝 24

Generalization – 12D • Generalizations – Add wavelength dependence – Add fluorescence (change to longer wavelength for reflection) – Time varying surface characteristics – Phosphorescence • Temporal storage of light 𝑔 𝑦 𝑗 , 𝜕 𝑗 , 𝑢 𝑗 , 𝜇 𝑗 → 𝑦 𝑝 , 𝜕 𝑝 , 𝑢 𝑝 , 𝜇 𝑝 𝑠 ω 𝑝 ω 𝑗 𝑦 𝑗 𝑦 𝑝 25

Reflectance • Reflectance may vary with – Illumination angle – Viewing angle – Wavelength – (Polarization, ...) • Variations due to Aluminum; λ=2.0μm – Absorption – Surface micro-geometry – Index of refraction / dielectric constant – Scattering Aluminum; λ=0.5μm Magnesium; λ=0.5μm 26

BRDF Measurement • Gonio-Reflectometer • BRDF measurement – Point light source position (𝜄 𝑗 , 𝜒 𝑗 ) – Light detector position (𝜄 𝑝 , 𝜒 𝑝 ) • 4 directional degrees of freedom • BRDF representation – m incident direction samples – n outgoing direction samples – m*n reflectance values (large!!!) Stanford light gantry 27

Rendering from Measured BRDF • Linearity, superposition principle – Complex illumination: integrating light distribution against BRDF – Sampled computation: superimposing many point light sources • Interpolation – Look-up of BRDF values during rendering – Sampled BRDF must be filtered • BRDF Modeling – Fitting of parameterized BRDF models to measured data • Continuous, analytic function • No interpolation Spherical Harmonics • Typically fast evaluation Red is positive, green negative [Wikipedia] • Representation in a basis – Most appropriate: Spherical harmonics • Ortho-normal function basis on the sphere – Mathematically elegant filtering, illumination-BRDF integration 28

BRDF Modeling • Phenomenological approach – Description of visual surface appearance – Composition of different terms: • Ideal diffuse reflection – Lambert’s law, interactions within material – Matte surfaces • Ideal specular reflection – Reflection law, reflection on a planar surface – Mirror • Glossy reflection – Directional diffuse, reflection on surface that is somewhat rough – Shiny surface – Glossy highlights 29

Reflection Geometry • Direction vectors (normalize): 𝑶 – 𝑂 : Surface normal (𝑱 ⋅ 𝑶)𝑶 𝑺(𝑱) – 𝐽 : Light source direction vector – 𝑊 : Viewpoint direction vector −𝑱 – 𝑆(𝐽) : Reflection vector • 𝑆(𝐽) = −𝐽 + 2(𝐽 ⋅ 𝑂)𝑂 𝟑(𝑱 ⋅ 𝑶)𝑶 – 𝐼 : Halfway vector −𝑱 • 𝐼 = (𝐽 + 𝑊) / |𝐽 + 𝑊| Top view • Tangential surface: local plane 𝑺(𝑾) 𝑰 𝑺(𝑱) 𝑶 𝑱 𝑶 𝑺(𝑱) 𝑱 𝑺(𝑾) 𝑾 𝑾 𝑰 30

Ideal Specular Reflection • Angle of reflectance equal to angle of incidence • Reflected vector in a plane with incident ray and surface normal vector 𝑆 + 𝐽 = 2 cos 𝜾 𝑂 = 2 𝐽 ⋅ 𝑂 𝑂 ⟹ 𝑆(𝐽) = −𝐽 + 2(𝐽 ⋅ 𝑂) 𝑂 I 𝑺 𝑂 cos 𝜄 𝜒 𝑝 −𝑱 𝜄 𝑗 𝜄 𝑝 𝜒 𝑗 𝜄 𝑗 = 𝜄 𝑝 𝜒 𝑝 = 𝜒 𝑗 + 180° 31

Mirror BRDF • Dirac Delta function 𝜺 𝒚 – 𝜺 𝒚 : zero everywhere except at 𝑦 = 0 – Unit integral iff domain contains 𝑦 = 0 (else zero) 𝜀(𝑑𝑝𝑡𝜄 𝑗 − 𝑑𝑝𝑡𝜄 𝑝 ) 𝑔 𝑠,𝑛 𝜕 𝑗 , 𝑦, 𝜕 𝑝 = 𝜍 𝑡 𝜄 𝑗 𝜀 𝜒 𝑗 − 𝜒 𝑝 ± 𝜌 cos 𝜄 𝑗 𝑀 𝑝 𝑦, 𝜕 𝑝 = 𝑔 𝑠,𝑛 𝜕 𝑗 , 𝑦, 𝜕 𝑝 𝑀 𝑗 𝑦, 𝜕 𝑗 𝑑𝑝𝑡𝜄 𝑗 𝑒𝜕 𝑗 = Ω + 𝜍 𝑡 𝜄 𝑝 𝑀 𝑗 (𝑦, 𝜄 𝑝 , 𝜒 𝑝 ± 𝜌) • Specular reflectance 𝜍 𝑡 N – Ratio of reflected radiance in specular L direction and incoming radiance R – Dimensionless quantity between 0 and 1 𝜍 𝑡 𝑦, 𝜄 𝑗 = 𝑀 𝑝 (𝑦, 𝜄 𝑝 )  i  o 𝑀 𝑗 (𝑦, 𝜄 𝑝 ) 32