Computer graphics III – Light reflection, BRDF Jaroslav Křivánek, MFF UK Jaroslav.Krivanek@mff.cuni.cz
Recap – Basic radiometric quantities Image: Wojciech Jarosz CG III (NPGR010) - J. Křivánek
Interaction of light with a surface ◼ Absorption ◼ Reflection ◼ Transmission / refraction ◼ Reflective properties of materials determine ❑ the relation of reflected radiance L r to incoming radiance L i , and therefore ❑ the appearance of the object: color, glossiness, etc. CG III (NPGR010) - J. Křivánek
Interaction of light with a surface ◼ Same illumination ◼ Different materials Source: MERL BRDF database CG III (NPGR010) - J. Křivánek
Recall the Phong shading model N R L V ( ) = + n C I k ( N L ) k ( V R ) d s = − R 2 ( N L ) N L CG III (NPGR010) - J. Křivánek
I) Adopt radiometric notation n r i i r o ( ) = + n L ( ) L ( ) k cos k cos o o i i d i s r = = − cos 2 ( ) r r n n r o i i Exact same thing as on the previous slide – just using physically-based notation. CG III (NPGR010) - J. Křivánek
BRDF corresponding to the original Phong shading model n r i i r o n L cos = = + o Phong Orig f r f k k BRDF: r r d s L cos cos i i i General definition of a BRDF Application of this definition to the Phong shading formula. CG III (NPGR010) - J. Křivánek
BRDF – Formal definition ◼ B idirectional R eflectance D istribution F unction n L i ( i ) „ o utgoing “ L r ( o ) d i i o „ i ncoming“ „ r eflected “ d L ( ) − → = 1 r o f r ( ) [ sr ] i o L ( ) cos d i i i i CG III (NPGR010) - J. Křivánek
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 i gets reflected to the direction o . ) → ◼ Range: f ( ) 0 , r i o CG III (NPGR010) - J. Křivánek
BRDF Westin et al. Predicting Reflectance Functions from Complex Surfaces, SIGGRAPH 1992. ◼ The BRDF is a model of the bulk behavior of light on the microstructure when viewed from distance CG III (NPGR010) - J. Křivánek
Surface roughness and blurred reflections ◼ The rougher the blurrier Microscopic surface roughness CG III (NPGR010) - J. Křivánek
Surface appearance and the BRDF Appearance BRDF lobe (for four different viewing directions) CG III (NPGR010) - J. Křivánek Souce: Ngan et al. Experimental analysis of BRDF models, http://people.csail.mit.edu/addy/research/brdf/
Surface appearance and the BRDF Appearance BRDF lobe (for four different viewing directions) CG III (NPGR010) - J. Křivánek Souce: Ngan et al. Experimental analysis of BRDF models, http://people.csail.mit.edu/addy/research/brdf/
Surface appearance and the BRDF Appearance BRDF lobe (for four different viewing directions) CG III (NPGR010) - J. Křivánek Souce: Ngan et al. Experimental analysis of BRDF models, http://people.csail.mit.edu/addy/research/brdf/
Surface appearance and the BRDF Appearance BRDF lobe (for four different viewing directions) CG III (NPGR010) - J. Křivánek Souce: Ngan et al. Experimental analysis of BRDF models, http://people.csail.mit.edu/addy/research/brdf/
BRDF properties ◼ Helmholz reciprocity (always holds in nature, a physically-plausible BRDF model must follow it) → = → f ( ) f ( ) r i o r o i CG III (NPGR010) - J. Křivánek
BRDF properties ◼ Energy conservation ❑ A patch of surface cannot reflect more light energy than it receives CG III (NPGR010) - J. Křivánek
BRDF (an)isotropy ◼ Isotropic BRDF = invariant to a rotation around surface normal ( ) ( ) = + + f , ; , f , ; , r i i o o r i i o o ( ) = − f , , r i o o i CG III (NPGR010) - J. Křivánek
Surfaces with anisotropic BRDF CG III (NPGR010) - J. Křivánek
Anisotropic BRDF ◼ Different microscopic roughness in different directions (brushed metals, fabrics , …) CG III (NPGR010) - J. Křivánek
Isotropic vs. anisotropic BRDF ◼ Isotropic BRDFs have only 3 degrees of freedom ❑ Instead of i and o it is enough to consider only D = i – o ❑ But this is not enough to describe an anisotropic BRDF ◼ Description of an anisotropic BRDF ❑ i and o are expressed in a local coordinate frame ( 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
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 o ?“ ◼ From the definition of the BRDF, we have = → d L ( ) f ( ) L ( ) cos d r o r i o i i i i CG III (NPGR010) - J. Křivánek
Reflection equation ◼ Total reflected radiance: integrate contributions of incident radiance, weighted by the BRDF, over the hemisphere = → L ( ) L ( ) f ( ) cos d r o i i r i o i i H ( ) x upper hemisphere over x = න 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
Energy conservation – More rigorous ◼ Reflected flux per unit area (i.e. radiosity B ) cannot be larger than the incoming flux per unit surface area (i.e. irradiance E ). L ( ) cos d B = r o o o = E L ( ) cos d i i i i → f ( ) L ( ) cos d cos d = r i o i i i i o o = L ( ) cos d i i i i 1 CG III (NPGR010) - J. Křivánek
Reflectance ◼ Ratio of the incoming and outgoing flux ❑ A.k.a . „albedo“ ( used mostly for diffuse reflection) ◼ Hemispherical-hemispherical reflectance ❑ See the “Energy conservation” slide ◼ Hemispherical-directional reflectance ❑ The amount of light that gets reflected in direction o when illuminated by the unit, uniform incoming radiance. = = → ( ) a ( ) f ( ) cos d o o r i o i i H ( x ) CG III (NPGR010) - J. Křivánek
Hemispherical-directional reflectance ◼ Nonnegative o ( ) 0 , 1 ◼ Less than or equal to 1 (energy conservation) ◼ Equal to directional-hemispherical reflectance ❑ What is the percentage of the energy coming from the incoming direction i that gets reflected (to any direction )?“ ❑ Equality follows from the Helmholz reciprocity CG III (NPGR010) - J. Křivánek
CG III (NPGR010) - J. Křivánek
CG III (NPGR010) - J. Křivánek
BRDF components General BRDF Ideal diffuse Ideal specular Glossy, (Lambertian) directional diffuse CG III (NPGR010) - J. Křivánek
Ideal diffuse reflection
Ideal diffuse reflection CG III (NPGR010) - J. Křivánek
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 i , o ) → = f ( ) f r , d i o r , d CG III (NPGR010) - J. Křivánek
Ideal diffuse reflection ◼ Reflection on a Lambertian surface: = L ( ) f L ( ) cos d o o r , d i i i i H ( x ) = f E r , d irradiance ◼ View independent appearance ❑ Outgoing radiance L o is independent of o ◼ Reflectance (derive) = f , d r d CG III (NPGR010) - J. Křivánek
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
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
White-out conditions ◼ We assume sky radiance independent of direction (covered sky) = sky L ( , ) L x i i ◼ We also assume Lambertian reflection on snow ◼ Reflected radiance given by: = snow snow sky L L o d i White-out!!! CG III (NPGR010) - J. Křivánek
Ideal mirror reflection
Ideal mirror reflection CG III (NPGR010) - J. Křivánek
CG III (NPGR010) - J. Křivánek
CG III (NPGR010) - J. Křivánek Nishino, Nayar: Eyes for Relighting, SIGGRAPH 2004
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