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

Computer Graphics - Material Models - Philipp Slusallek

REFLECTANCE PROPERTIES 2

Appearance Samples • How do materials reflect light? – At the same point / in the neighborhood (subsurface scattering) 3

Material Samples • Anisotropic surfaces anisotropic 4

Material Samples • Complex surface meso-structure 5

Material Samples • Lots of details: Fibers 6

Material Samples • Photos of samples with light source at exactly the same position diffuse glossy mirror 7

How to describe materials? • Surface roughness – Cause of different reflection properties: • Perfectly smooth: Mirror reflection • Slightly rough: Glossy highlights • Very rough: Diffuse reflection, light reflected many times, looses directionality • Geometry – Macro structure: Described as explicit geometry (e.g. triangles) – Micro structure: Captured in scattering function (BRDF) – Meso structure: Difficult to handle: integrate into BRDF (offline simulation), use geometry and simulate (online) • Representation of reflection properties – Bidirectional reflection distribution function (BRDF) • For reflections at a single point (approx.) – More complex scattering functions (e.g. subsurface scattering) • Goal: Relightable representation of appearance 8

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 𝑒𝑀 𝑝 𝑦,𝜕 𝑝 1 𝑔 𝑠 𝜕 𝑗 , 𝑦, 𝜕 𝑝 = Units: 𝑒𝐹 𝑗 (𝑦,𝜕 𝑗 ) 𝑡𝑠 9

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 10

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 ) • No surface structure: Isotropic BRDF – Reflectivity independent of rotation around surface normal – BRDF has only 3 instead of 4 directional degrees of freedom 𝑔 𝑠 (𝑦, 𝜄 𝑗 , 𝜄 𝑝 , 𝜒 𝑝 − 𝜒 𝑗 ) 11

BRDF Properties • Characteristics – BRDF units • Inverse steradian: 𝑡𝑠 −1 (not really intuitive) – Range of values: distribution function is positive, can be infinite • From 0 (no reflection in that direction) • to ∞ (perfect reflection into exactly one direction, 𝜀 -function) – Energy conservation law • Absorption physically unavoidable, no self-emission • Integral of 𝑔 𝑠 over outgoing directions integrates to less than one – For any incoming direction 𝑔 𝑠 𝜕 𝑗 , 𝑦, 𝜕 𝑝 𝑑𝑝𝑡𝜄 𝑝 𝑒𝜕 𝑝 ≤ 1, ∀𝜕 𝑗 Ω + • Reflection only at the point of entry ( 𝒚 𝒋 = 𝒚 𝒑 ) – Ignoring subsurface scattering (SSS) 12

Standardized Gloss Model • Industry often uses only a subset of BRDF values – Reflection only measured at discrete set of angles, in plane 13

Reflection on an Opaque Surface • BRDF is often shown as a slice of the 6D function – Given point 𝑦 and given incident direction 𝜕 𝑗 • Show 2D polar lot (intensity as length of vector from origin) – Often consists of some mostly diffuse component (here small) • and a somewhat glossy component (here rather large) Glossy cone Diffuse hemisphere 14

Reflection on an Opaque Surface • 2D plot varies with incident direction – (and possibly location) ω 𝑝 ω 𝑝 ω 𝑝 𝜕 𝑗 15

Homog. & Isotropic BRDF – 3D • Invariant with respect to rotation about the normal – Homogeneous and isotropic across surface – 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 16

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 17

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 18

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 19

BSSRDF – 8D • Bidirectional scattering surface reflectance distribution function 𝑔 𝑠 (𝑦 𝑗 , 𝜕 𝑗 ) → (𝑦 𝑝 , 𝜕 𝑝 ) ω 𝑝 ω 𝑗 𝑦 𝑗 𝑦 𝑝 20

Generalization – 9D • Generalizations – Add wavelength dependence 𝑔 𝑠 𝜇, (𝑦 𝑗 , 𝜕 𝑗 ) → (𝑦 𝑝 , 𝜕 𝑝 ) ω 𝑝 ω 𝑗 𝑦 𝑗 𝑦 𝑝 21

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 22

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 23

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 24

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 oxide; λ=0.5μm 25

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!!!) – Additional position dependent (6D) Stanford light gantry 26

Rendering from Measured BRDF • Linearity, superposition principle – Continuous illumin.: integrating light distribution against BRDF – Sampled illumination: 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 27

BRDF Modeling • Phenomenological approach (not physically correct) – 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 28

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 29

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° 30