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

CG III (NPGR010) - J. Křivánek 2015

Image: Wojciech Jarosz

Interaction of light with a surface

 Absorption  Reflection  Transmission / refraction  Reflective properties of materials determine

 the relation of reflected radiance Lr

to incom ing

radiance Li , 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

 Bidirectional reflectance distribution function  (cz: Dvousměrová distribuční funkce odrazu)

dωi Lr(ωo) θo n Li(ωi) θi

] sr [ d cos ) ( ) ( d ) ( d ) ( d ) (

1 i i i i

• r

i

• r
• i

−

⋅ ⋅ = = → ω θ ω ω ω ω ω ω L L E L fr

BRDF

“incoming” “outgoing” “reflected”

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 ωi gets reflected to the direction ωo.

 Range:

[

)

∞ ∈ → , ) (

• i

ω ω

r

f

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

• n 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)

CG III (NPGR010) - J. Křivánek 2015

) ( ) (

i

• i

ω ω ω ω → = →

r r

f f

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).

CG III (NPGR010) - J. Křivánek 2015

[ ]

1 cos ) ( cos cos ) ( ) ( cos ) ( cos ) (

• i

≤ = → = = =

∫ ∫ ∫ ∫ ∫

i i i i

• i

i i i r i i i i r

• r

d L d d L f d L d L E B ω θ ω ω θ ω θ ω ω ω ω θ ω ω θ ω

BRDF (an)isotropy

 Isotropic BRDF = invariant to a rotation around

surface normal

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( ) ( ) ( )

i

• i
• i

i

• i

i

, , , ; , , ; , φ φ θ θ φ φ θ φ φ θ φ θ φ θ − = + + =

r r r

f f f

Surfaces with anisotropic BRDF

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Anisotropic BRDF

 Different microscopic roughness in different directions

(brushed metals, fabrics, … )

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Isotropic vs. anisotropic BRDF

 Isotropic BRDFs have only 3 degrees of freedom

 Instead of φi and φo it is enough to consider only ∆φ = φi – φo  But this is not enough to describe an anisotropic BRDF

 Description of an anisotropic BRDF

 φi and φo 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 ωo?“  From the definition of the BRDF, we have

i r

L f L ω θ ω ω ω ω d cos ) ( ) ( ) ( d

i i i

• i
• r

⋅ ⋅ → =

CG III (NPGR010) - J. Křivánek 2015

Reflection equation

 Integrating the contributions dLr over the entire

hemisphere:

∫

⋅ → ⋅ =

) ( i i

• i

i i

• r

d cos ) , ( ) , ( ) , (

x

x x x

H r

f L L ω θ ω ω ω ω dωi Lo(x, ωo) θo n Li(x, ωi) θi Lr(x, ωo)

upper hemisphere

• ver x

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Reflection equation

 Evaluating the reflectance equation renders images!!!

 Direct illumination 

Environment maps



Area light sources



etc.

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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 ωo when

illuminated by the unit, uniform incoming radiance.

∫

→ = =

) ( i i

• i
• d

cos ) , ( ) ( ) (

x

x

H r

f a ω θ ω ω ω ω ρ

CG III (NPGR010) - J. Křivánek 2015

Hem ispherical-directional reflectance

 Nonnegative  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 ωi that gets reflected (to any direction)?“

 Equality follows from the Helmholz reciprocity

[ ]

1 , ) (

• ∈

ω ρ

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BRDF com ponents

General BRDF Ideal diffuse (Lambertian) Ideal specular Glossy, directional diffuse

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Ideal diffuse reflection

Ideal diffuse reflection

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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 ωi , ωo)

d r d r

f f

,

• i

,

) ( = → ω ω

CG III (NPGR010) - J. Křivánek 2015

Ideal diffuse reflection

 Reflection on a Lambertian surface:  View independent appearance

 Outgoing radiance Lo is independent of ωo

 Reflectance (derive)

E f L f L

d r H d r , ) ( i i i i ,

• d

cos ) ( ) ( = =

∫

x

ω θ ω ω

d r d

f , ⋅ = π ρ

irradiance

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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

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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?

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White-out conditions

 We assume sky radiance independent of direction

(covered sky)

 We also assume Lambertian reflection on snow  Reflected radiance given by:

CG III (NPGR010) - J. Křivánek 2015

sky i i

) , ( L L = ω x

sky i snow snow

• L

L

d

⋅ = ρ

White-out!!!

Ideal m irror reflection

Ideal m irror reflection

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Nishino, Nayar: Eyes for Relighting, SIGGRAPH 2004

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The law of reflection

 Direction of the reflected ray (derive the formula)

CG III (NPGR010) - J. Křivánek 2015

i i

• ω

ω ω − ⋅ = n n) ( 2 θo n θi θo = θi φo = (φi + π) mod 2π

φo φi

Digression: Dirac delta distribution

 Definition (informal):  The following holds for any f:  Delta distribution is not a function (otherwise the

integrals would = 0)

CG III (NPGR010) - J. Křivánek 2015 Image: Wikipedia

BRDF of the ideal m irror

 BRDF of the ideal mirror is a Dirac delta distribution

CG III (NPGR010) - J. Křivánek 2015

) , ( ) ( ) , (

• i

i

• r

π ϕ θ θ ϕ θ ± = L R L

i

• i
• i

i

• i

i ,

cos ) ( ) cos (cos ) ( ) , ; , ( θ π ϕ ϕ δ θ θ δ θ ϕ θ ϕ θ ± − − = R f

m r

θo n θi θo = θi

Fresnel reflectance (see below) We want:

BRDF of the ideal m irror

 BRDF of the ideal mirror is a Dirac delta distribution  Varification:

CG III (NPGR010) - J. Křivánek 2015

) , ( ) ( cos ) , ( cos ) ( ) cos (cos ) ( cos (.) (.) ) , (

r r i i i i i i i i

• i
• i

i i i i ,

• r

π ϕ θ θ ω θ ϕ θ θ π ϕ ϕ δ θ θ δ θ ω θ ϕ θ ± = ± − − = =

∫ ∫

L R d L R d L f L

m r

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Ideal refraction

Ideal refraction

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ηi ηo ωi ωo

Ideal refraction

 Index of refraction η

 Water 1.33, glass 1.6, diamond 2.4  Often depends on the wavelength

 Snell’s law

• i

i

θ η θ η sin sin =

CG III (NPGR010) - J. Křivánek 2015

Ideal refraction

 Direction of the refracted ray:

( )n

) cos 1 ( 1 cos

i 2 2 io i io i io

• θ

η θ η ω η ω − − + − − =

• i

io

η η η =

if < 0, total internal reflection Critical angle:

        =

i

• c

i,

arcsin η η θ

Image: wikipedia

CG III (NPGR010) - J. Křivánek 2015

Ideal refraction

 Change of radiance

 Follows from the conservation of energy (flux)  When going from an optically rarer to a more dense

medium, light energy gets “compressed” in directions => higher energy density => higher radiance

2 2 i

• i
• L

L η η =

CG III (NPGR010) - J. Křivánek 2015

 BRDF of the ideal refraction is a delta distribution:

BRDF of ideal refraction

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i

• i
• i

i i 2 i 2

• i

i

cos ) ( ) sin sin ( )) ( 1 ( ) , ; , ( θ π ϕ ϕ δ θ η θ η δ θ η η ϕ θ ϕ θ ± − − − = R ft

Fresnel transmittance Change of radiance Snell’s law Refracted ray stays in the incidence plane

Fresnel equations

Fresnel equations

 Read [frenel]  Ratio of the transmitted and reflected light depends on

the incident direction

 From above – more transmission  From the side – more reflection

 Extremely important for realistic rendering of glass,

water and other smooth dielectrics

 Not to be confused with

Fresnel lenses (used in lighthouses)

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Fresnel equations

 Dielectrics

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Image: Wikipedia

Fresnel equations

 Dielectrics

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Fresnel equations

From above

• little reflection
• more transmission

From the side

• little transmission
• more reflection

Try for yourself!!!

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Fresnel equations

 Metals

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Glossy reflection

Glossy reflection

 Neither ideal diffuse nor ideal mirror  All real materials in fact fall in this

category

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Surface roughness and blurred reflections

 The rougher the blurrier

Microscopic surface roughness

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BRDF m odels

BRDF m odeling



BRDF is a model

• f the bulk

behavior of light when viewing a surface from distance



BRDF m odels



Empirical



Physically based



Approximation

• f measured

data

(a.k.a meso-scale)

CG III (NPGR010) - J. Křivánek 2015

Em pirical BRDF m odels

 An arbitrary formula that takes ωi and ωo as arguments  ωi and ωo are sometimes denoted L (Light direction) a V

(Viewing direction)

 Example: Phong model  Arbitrary shading calculations (shaders)

CG III (NPGR010) - J. Křivánek 2015

Phong shading m odel

( )

n s d

R V k L N k I C ) ( ) ( ⋅ + ⋅ = L R V N L N L N R − ⋅ = ) ( 2

CG III (NPGR010) - J. Křivánek 2015

Phong shading m odel in the radiom etric notation

( )

r n s d

k k L L θ θ ω ω cos cos ) ( ) (

i i i

• +

=

i i

• r

ω ω ω θ − ⋅ = ⋅ = n n r r ) ( 2 cos BRDF

i i

• r

L L f θ cos =

i r n s d Orig Phong r

k k f θ θ cos cos + = Original shading model

ωi r ωo n

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Physically-plausible Phong BRDF

 Modification to ensure reciprocity (symmetry) and

energy conservation

 Energy conserved when  It is still an empirical formula (i.e. it does not follow from

physical considerations), but at least it fulfills the basic properties of a BRDF

r n s d r

n f θ ρ π π ρ cos 2 2

modif Phong

+ + = 1 ≤ +

s d

ρ ρ

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Physically-plausible BRDF m odels

 E.g. Torrance-Sparrow / Cook-Torrance model  Based on the microfacet theory

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Torrance-Sparrow BRDF

 Analytically derived  Used for modeling glossy surfaces (as the Phong model)

 Corresponds more closely to reality than Phong  Derived from a physical model of the surface

microgeometry (as opposed to “because it looks good”- approach used for the Phong model)

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Torrance-Sparrow BRDF

 Assumes that the macrosurface consists of randomly

• riented microfacets

 We assume that each microfacet behaves as an ideal

mirror.

 We consider 3 phenomena:

Shadowing Masking Reflection

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Torrance-Sparrow BRDF

( ) ( , ) ( ) 4cos( )cos( )

i i r h i r

F G D f θ ω ω θ θ θ =

Fresnel term Geom etry term Models shadowing and masking Microfacet distribution Part of the macroscopic surface visible by the light source Part of the macroscopic surface visible by the viewer

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Approxim ation of m easured data

 We can fit any BRDF model to the data  Some BRDF models have been specifically designed for

the purpose of fitting measured data, e.g. Ward BRDF, Lafortune BRDF

 Nonlinear optim ization required to find the BRDF

parameters

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BRDF m easurem ents – Gonio-reflectom eter

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BRDF m odels vs. m easured data

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BRDF m odels vs. m easured data

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BRDF m odels vs. m easured data

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BRDF m odels vs. m easured data

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BRDF m odels vs. m easured data

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BRDF, BTDF, BSDF: What’s up with all these abbreviations?

 BTDF

 Bidirectional transm ittance

distribution function

 Described light transmission

 BSDF = BRDF+BTDF

 Bidirectional scattering

distribution function

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SBRDF, BTF

 SV-BRDF …

Spatially Varying BRDF

 BRDF parameters are spatially varying (can be given by a

surface texture)

 BTF …

Bidirectional Texture Function

 Used for materials with complex structure  As opposed to the BRDF, models even the meso-scale

CG III (NPGR010) - J. Křivánek 2015

BSSRDF

 BRDF

 Light arriving at a point is reflected/ transmitted at the

same point

 No subsurface scattering considered

 BSSRDF

 Bi-directional surface scattering reflectance distribution

function

 Takes into account

scattering of light under the surface

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BSSRDF

 Sub-surface scattering makes surfaces looks “softer”

BRDF BSSRDF

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BSSRDF

BRDF BSSRDF

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