Com puter graphics III – Light reflection, BRDF Jaroslav Křivánek, MFF UK Jaroslav.Krivanek@mff.cuni.cz
Basic radiom etric quantities Image: Wojciech Jarosz CG III (NPGR010) - J. Křivánek 2015
Interaction of light with a surface Absorption Reflection Transmission / refraction Reflective properties of materials determine to incom ing the relation of reflected radiance L r radiance L i , and therefore the appearance of the object: color, glossiness, etc. CG III (NPGR010) - J. Křivánek 2015
Interaction of light with a surface Same illumination Different materials Source: MERL BRDF database CG III (NPGR010) - J. Křivánek 2015
BRDF Bidirectional reflectance distribution function (cz: Dvousměrová distribuční funkce odrazu ) L i ( ω i ) n “ o utgoing” L r ( ω o ) d ω i θ i θ o “ i ncoming” “ r eflected” ω ω d L ( ) d L ( ) − ω → ω = = 1 r o r o f r ( ) [ sr ] ω ω ⋅ θ ⋅ ω i o d E ( ) L ( ) cos d i i i i i CG III (NPGR010) - J. Křivánek 2015
BRDF Mathematical model of the reflection properties of a surface Intuition Value of a BRDF = probability density , describing the event that a light energy “packet”, or “photon”, coming from direction ω i gets reflected to the direction ω o . [ ) ω → ω ∈ ∞ f ( ) 0 , Range: r i o CG III (NPGR010) - J. Křivánek 2015
BRDF Westin et al. Predicting Reflectance Functions from Complex Surfaces, SIGGRAPH 1992. The BRDF is a m odel of the bulk behavior of light on the microstructure when viewed from distance CG III (NPGR010) - J. Křivánek 2015
BRDF properties Helm holz reciprocity (always holds in nature, a physically-plausible BRDF model must follow it) ω → ω = ω → ω f ( ) f ( ) r i o r o i CG III (NPGR010) - J. Křivánek 2015
BRDF properties Energy conservation Reflected flux per unit area (i.e. radiosity B ) cannot be larger than the incoming flux per unit surface area (i.e. irradiance E ). ∫ ω θ ω L ( ) cos d B = r o o r = ∫ ω θ ω E L ( ) cos d i i i i [ ] ∫ ∫ ω → ω ω θ ω θ ω f ( ) L ( ) cos d cos d = r i o i i i i o o = ∫ ω θ ω L ( ) cos d i i i i ≤ 1 CG III (NPGR010) - J. Křivánek 2015
BRDF (an)isotropy Isotropic BRDF = invariant to a rotation around surface normal ( ) ( ) θ φ θ φ = θ φ + φ θ φ + φ f , ; , f , ; , r i i o o r i i o o ( ) = θ θ φ − φ f , , r i o o i CG III (NPGR010) - J. Křivánek 2015
Surfaces with anisotropic BRDF CG III (NPGR010) - J. Křivánek 2015
Anisotropic BRDF Different microscopic roughness in different directions (brushed metals, fabrics, … ) CG III (NPGR010) - J. Křivánek 2015
Isotropic vs. anisotropic BRDF Isotropic BRDFs have only 3 degrees of freedom Instead of φ i and φ o it is enough to consider only ∆φ = φ i – φ o But this is not enough to describe an anisotropic BRDF Description of an anisotropic BRDF φ i and φ o are expressed in a local coordinate fram e ( U , V , N ) U … tangent – e.g. the direction of brushing V … binormal N … surface normal … the Z axis of the local coordinate frame CG III (NPGR010) - J. Křivánek 2015
Reflection equation A.k.a. reflectance equation, illumination integral, OVTIGRE (“outgoing, vacuum, time-invariant, gray radiance equation”) “How much total light gets reflected in the direction ω o ?“ From the definition of the BRDF, we have ω = ω → ω ⋅ ω ⋅ θ ω d L ( ) f ( ) L ( ) cos d r o r i o i i i i CG III (NPGR010) - J. Křivánek 2015
Reflection equation Integrating the contributions d L r over the entire hemisphere: ∫ ω = ω ⋅ ω → ω ⋅ θ ω L ( x , ) L ( x , ) f ( x , ) cos d r o i i r i o i i H ( x ) upper hemisphere n L i ( x , ω i ) over x L o ( x , ω o ) d ω i θ i θ o L r ( x , ω o ) CG III (NPGR010) - J. Křivánek 2015
Reflection equation Evaluating the reflectance equation renders images!!! Direct illumination Environment maps Area light sources etc. CG III (NPGR010) - J. Křivánek 2015
Reflectance Ratio of the incom ing and outgoing flux A.k.a. „albedo“ (used mostly for diffuse reflection) Hem ispherical-hem ispherical reflectance See the “Energy conservation” slide Hem ispherical-directional reflectance The amount of light that gets reflected in direction ω o when illuminated by the unit, uniform incoming radiance. ∫ ρ ω = ω = ω → ω θ ω x ( ) a ( ) f ( , ) cos d o o r i o i i H ( x ) CG III (NPGR010) - J. Křivánek 2015
Hem ispherical-directional reflectance Nonnegative [ ] ρ ω o ∈ ( ) 0 , 1 Less than or equal to 1 (energy conservation) Equal to directional-hem ispherical reflectance What is the percentage of the energy coming from the incoming direction ω i that gets reflected (to any direction)?“ Equality follows from the Helmholz reciprocity CG III (NPGR010) - J. Křivánek 2015
CG III (NPGR010) - J. Křivánek 2015
CG III (NPGR010) - J. Křivánek 2015
BRDF com ponents General BRDF Ideal diffuse Ideal specular Glossy, (Lambertian) directional diffuse CG III (NPGR010) - J. Křivánek 2015
Ideal diffuse reflection
Ideal diffuse reflection CG III (NPGR010) - J. Křivánek 2015
Ideal diffuse reflection A.k.a. Lambertian reflection Johann Heinrich Lambert, „Photometria“, 1760. Postulate: Light gets reflected to all directions with the same probability, irrespective of the direction it came from The corresponding BRDF is a constant function (independent of ω i , ω o ) ω → ω = f ( ) f r , d i o r , d CG III (NPGR010) - J. Křivánek 2015
Ideal diffuse reflection Reflection on a Lambertian surface: ∫ ω = ω θ ω L ( ) f L ( ) cos d o o r , d i i i i x H ( ) = f E r , d irradiance View independent appearance Outgoing radiance L o is independent of ω o Reflectance (derive) ρ = π ⋅ f , d r d CG III (NPGR010) - J. Křivánek 2015
Ideal diffuse reflection Mathematical idealization that does not exist in nature The actual behavior of natural materials deviates from the Lambertian assumption especially for grazing incidence angles CG III (NPGR010) - J. Křivánek 2015
White-out conditions Under a covered sky we cannot tell the shape of a terrain covered by snow We do not have this problem close to a localized light source. Why? CG III (NPGR010) - J. Křivánek 2015
White-out conditions We assume sky radiance independent of direction (covered sky) ω = sky L ( x , ) L i i We also assume Lambertian reflection on snow Reflected radiance given by: = ρ ⋅ snow snow sky L L o d i White-out!!! CG III (NPGR010) - J. Křivánek 2015
Ideal m irror reflection
Ideal m irror reflection CG III (NPGR010) - J. Křivánek 2015
CG III (NPGR010) - J. Křivánek 2015
CG III (NPGR010) - J. Křivánek 2015 Nishino, Nayar: Eyes for Relighting, SIGGRAPH 2004
The law of reflection n φ o θ i θ o φ i θ o = θ i φ o = (φ i + π) mod 2 π Direction of the reflected ray (derive the formula) ω = ω ⋅ − ω n ) n 2 ( o i i CG III (NPGR010) - J. Křivánek 2015
Digression: Dirac delta distribution Definition (informal): Image : Wikipedia The following holds for any f : Delta distribution is not a function (otherwise the integrals would = 0) CG III (NPGR010) - J. Křivánek 2015
BRDF of the ideal m irror BRDF of the ideal mirror is a Dirac delta distribution We want: n θ o = θ i θ ϕ = θ θ ϕ ± π θ i L ( , ) R ( ) L ( , ) θ o r o o i i o o Fresnel reflectance (see below) δ θ − θ δ ϕ − ϕ ± π (cos cos ) ( ) θ ϕ θ ϕ = R θ i o i o f ( , ; , ) ( ) θ r , m i i o o i cos i CG III (NPGR010) - J. Křivánek 2015
BRDF of the ideal m irror BRDF of the ideal mirror is a Dirac delta distribution Varification: ∫ θ ϕ = θ ω L ( , ) f (.) L (.) cos d r o o r , m i i i δ θ − θ δ ϕ − ϕ ± π (cos cos ) ( ) ∫ = θ θ ϕ θ ω i o i o R ( ) L ( , ) cos d θ i i i i i i cos i = θ θ ϕ ± π R ( ) L ( , ) i i r r CG III (NPGR010) - J. Křivánek 2015
CG III (NPGR010) - J. Křivánek 2015
Ideal refraction
Ideal refraction CG III (NPGR010) - J. Křivánek 2015
Ideal refraction Index of refraction η Water 1.33, glass 1.6, diamond 2.4 Often depends on the wavelength ω i Snell’s law η θ = η θ sin sin η i i i o o η o ω o CG III (NPGR010) - J. Křivánek 2015
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