Computer Graphics - BRDFs & Texturing - Hendrik Lensch Computer Graphics WS07/08 – BRDFs and Texturing
Overview • Last time – Radiance – Light sources – Rendering Equation & Formal Solutions • Today – Bidirectional Reflectance Distribution Function (BRDF) – Reflection models – Projection onto spherical basis functions – Shading • Next lecture – Varying (reflection) properties over object surface: texturing Computer Graphics WS07/08 – BRDFs and Texturing
Reflection Equation - Reflectance • Reflection equation ∫ ω = ω ω ω L x f x L x θ d ω ( , ) ( , , ) ( , ) cos o o r i o i i i i Ω + • BRDF – Ratio of reflected radiance to incident irradiance ω dL x ( , ) ω ω = f x o o ( , , ) r o i ω dE x ( , ) i i Computer Graphics WS07/08 – BRDFs and Texturing
Bidirectional Reflectance Distribution Function • BRDF describes surface reflection for light incident from direction ( θ i , φ i ) observed from direction ( θ ο , φ ο ) • Bidirectional – Depends on two directions and position (6-D function) • Distribution function – Can be infinite • Unit [1/sr] ω dL x ( , ) ω ω = f x o o ( , , ) r o i ω dE x ( , ) i i ω dL x ( , ) = o o ω θ ω dL x d ( , ) cos i i i i Computer Graphics WS07/08 – BRDFs and Texturing
BRDF Properties • Helmholtz reciprocity principle – BRDF remains unchanged if incident and reflected directions are interchanged ω ω = ω ω f f ( , ) ( , ) r o i r i o • Smooth surface: isotropic BRDF – reflectivity independent of rotation around surface normal – BRDF has only 3 instead of 4 directional degrees of freedom θ θ ϕ − ϕ f r x ( , , , ) i o o i Computer Graphics WS07/08 – BRDFs and Texturing
BRDF Properties • Characteristics – BRDF units [sr --1 ] • Not intuitive – Range of values: • From 0 (absorption) to ∞ (reflection, δ -function) – Energy conservation law • No self-emission • Possible absorption ∫ ω ω θ ω ≤ ∀ θ ϕ f x d ( , , ) cos 1 , r o i o o Ω – Reflection only at the point of entry ( x i = x o ) • No subsurface scattering Computer Graphics WS07/08 – BRDFs and Texturing
BRDF Measurement • Gonio-Reflectometer • BRDF measurement – point light source position ( θ , ϕ ) – light detector position ( θ o , ϕ o ) • 4 directional degrees of freedom • BRDF representation – m incident direction samples ( θ , ϕ ) – n outgoing direction samples ( θ o , ϕ o ) – m*n reflectance values (large!!!) Stanford light gantry Computer Graphics WS07/08 – BRDFs and Texturing
Reflectance • Reflectance may vary with – Illumination angle – Viewing angle – Wavelength – (Polarization, ...) • Variations due to Aluminum; λ =2.0 μ m – Absorption – Surface micro-geometry – Index of refraction / dielectric constant – Scattering Aluminum; λ =0.5 μ m Magnesium; λ =0.5 μ m Computer Graphics WS07/08 – BRDFs and Texturing
BRDF Modeling • Phenomenological approach – Description of visual surface appearance • Ideal specular reflection – Reflection law – Mirror Glossy reflection • – Directional diffuse – Shiny surfaces • Ideal diffuse reflection – Lambert’s law – Matte surfaces Computer Graphics WS07/08 – BRDFs and Texturing
Reflection Geometry • Direction vectors (normalize): N – N: surface normal -( - (I I• •N N) )N N R(I) – I: vector to the light source – V: viewpoint direction vector I -( (I I• •N N) )N N - – H: halfway vector H= (I + V) / |I + V| – R(I): reflection vector I R(I)= I - 2(I•N)N – Tangential surface: local plane Top view R(V) H N R(I) N I R(I) I R(V) V V H Computer Graphics WS07/08 – BRDFs and Texturing
Ideal Specular Reflection • Angle of reflectance equal to angle of incidence • Reflected vector in a plane with incident ray and surface normal vector R +(- I ) = 2 cos θ N = -2( I • N ) N R ( I ) = I - 2( I • N ) N I R N ϕ o I θ ϕ θ o θ = θ o ϕ = ϕ o + 180 ° Computer Graphics WS07/08 – BRDFs and Texturing
Mirror BRDF • Dirac Delta function δ (x) – δ (x) : zero everywhere except at x=0 – Unit integral iff integration domain contains zero (zero otherwise) δ θ − θ (cos cos ) ω ω = ρ θ ⋅ ⋅ δ ϕ − ϕ ± π i o f x ( , , ) ( ) ( ) r m o i s i i o θ , cos i ∫ ω = ω ω θ ϕ θ ω = ρ θ θ ϕ ± π L x f x L d L ( , ) ( , , ) ( , ) cos ( ) ( , ) o o r m o i i i i i i s i i o o , Ω + • Specular reflectance ρ s – Ratio of reflected radiance in specular N direction and incoming radiance L R – Dimensionless quantity between 0 and 1 ( ) Φ θ ( ) θ i θ o ρ θ = o o ( ) s i Φ θ i i Computer Graphics WS07/08 – BRDFs and Texturing
Diffuse Reflection • Light equally likely to be reflected in any output direction (independent of input direction) • Constant BRDF ω ω = = f x k ( , , ) const r d o i d , ∫ ∫ ω = ω θ ω = ω θ ω = L x k L x d k L x d k E ( , ) ( , ) cos ( , ) cos o o d i i i i d i i i i d Ω Ω – k d : diffuse coefficient, material property [1/sr] I L o = const N Computer Graphics WS07/08 – BRDFs and Texturing
Lambertian Diffuse Reflection ∫ ∫ = ω θ ω = θ ω = π B L x d L d L • Radiosity ( , ) cos cos o o o o o o o o Ω Ω B ρ = = π k • Diffuse Reflectance d d E • Lambert’s Cosine Law = ρ = ρ θ B E E cos d d i i • For each light source I – L r,d = k d L i cos θ i = k d L i (I•N) N L r,d L r,d θ i Computer Graphics WS07/08 – BRDFs and Texturing
Lambertian Objects Self-Luminous Eye-light illuminated spherical Lambertian Light Source Spherical Lambertian Reflector Φ ∝ ⋅ Ω Φ ∝ ⋅ ϕ ⋅ Ω L 0 d L d cos 0 1 0 ϕ d Ω d Ω Computer Graphics WS07/08 – BRDFs and Texturing
Lambertian Objects II The Sun The Moon • Absorption in photosphere • Surface covered with fine dust • Path length through photosphere • Dust on TV visible best from longer from the Sun’s rim slanted viewing angle ⇒ Neither the Sun nor the Moon are Lambertian Computer Graphics WS07/08 – BRDFs and Texturing
“Diffuse” Reflection • Theoretical explanation – Multiple scattering • Experimental realization – Pressed magnesium oxide powder – Almost never valid at high angles of incidence Paint manufacturers attempt to create ideal diffuse paints Computer Graphics WS07/08 – BRDFs and Texturing
Glossy Reflection Computer Graphics WS07/08 – BRDFs and Texturing
Glossy Reflection • Due to surface roughness • Empirical models – Phong – Blinn-Phong • Physical models – Blinn – Cook & Torrance Computer Graphics WS07/08 – BRDFs and Texturing
Phong Reflection Model • Cosine power lobe ( ) k ω ω = ⋅ f x k R I V e ( , , ) ( ) r o i s – L r,s = L i k s cos ke θ RV • Dot product & power • Not energy conserving/reciprocal • Plastic-like appearance R(V) θ HN θ H HN R(I) N N I θ RV θ I R(I) RV V V H Computer Graphics WS07/08 – BRDFs and Texturing
Phong Exponent k e ( ) k ω ω = ⋅ f x k R I V e ( , , ) ( ) r o i s • Determines size of highlight Computer Graphics WS07/08 – BRDFs and Texturing
Blinn-Phong Reflection Model • Blinn-Phong reflection model ( ) k ω ω = ⋅ f x k H N e ( , , ) r o i s – L r,s = L i k s cos ke θ HN – θ RV ⇒ θ HN – Light source, viewer far away – I, R constant: H constant θ HN less expensive to compute R(V) θ HN θ H HN R(I) N N I θ RV θ I R(I) RV V V H Computer Graphics WS07/08 – BRDFs and Texturing
Phong Illumination Model • Extended light sources: l point light sources ∑ ∑ = + ⋅ + ⋅ k k L k L I N k L R I V L ( ) ( ( ) ) e ( Phong) a i a d l l s l l r , l l ∑ ∑ = + ⋅ + ⋅ k k L k L I N k L H N L ( ) ( ) e (Blinn) a i a d l l s l l r , l l • Color of specular reflection equal to light source • Heuristic model – Contradicts physics – Purely local illumination • Only direct light from the light sources • No further reflection on other surfaces • Constant ambient term • Often: light sources & viewer assumed to be far away Computer Graphics WS07/08 – BRDFs and Texturing
Microfacet Model • Isotropic microfacet collection • Microfacets assumed as perfectly smooth reflectors • BRDF – Distribution of microfacets • Often probabilistic distribution of orientation or V-groove assumption – Planar reflection properties – Self-masking, shadowing Computer Graphics WS07/08 – BRDFs and Texturing
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