Com puter graphics III Light reflection, BRDF Jaroslav Kivnek, MFF - PowerPoint PPT Presentation

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