[PPT] - Computer graphics III Light reflection, BRDF Jaroslav Kivnek, MFF PowerPoint Presentation

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Computer graphics III – Light reflection, BRDF

Jaroslav Křivánek, MFF UK Jaroslav.Krivanek@mff.cuni.cz

SLIDE 2

Basic radiometric quantities

CG III (NPGR010) - J. Křivánek 2015

Image: Wojciech Jarosz

SLIDE 3

Interaction of light with a surface

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

 the relation of reflected radiance Lr

to incoming

radiance Li , and therefore

 the appearance of the object: color, glossiness, etc.

CG III (NPGR010) - J. Křivánek 2015

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Interaction of light with a surface

 Same illumination  Different materials Source: MERL BRDF database

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 Bidirectional Reflectance Distribution Function

dwi Lr(wo) qo n Li(wi) qi

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

1 i i i i

r
i



    w q w w w w L L fr

BRDF – Formal definition

„incoming“ „outgoing“ „reflected“

5

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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 wi gets reflected to the direction wo.

 Range:





   , ) (

i

w w

r

f

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BRDF

Westin et al. Predicting Reflectance Functions from Complex Surfaces, SIGGRAPH 1992.

 The BRDF is a model of the bulk behavior of light

n the microstructure when viewed from distance

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BRDF properties

 Helmholz reciprocity (always holds in nature, a

physically-plausible BRDF model must follow it)

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) ( ) (

i

i

w w w w   

r r

f f

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BRDF properties

 Energy conservation

 A patch of surface cannot reflect more light energy than it

receives

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BRDF (an)isotropy

 Isotropic BRDF = invariant to a rotation around

surface normal

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     

i

i
i

i

i

i

, , , ; , , ; ,   q q   q   q  q  q     

r r r

f f f

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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 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

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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 wo?“  From the definition of the BRDF, we have

i r

L f L w q w w w w d cos ) ( ) ( ) ( d

i i i

i
r

   

CG III (NPGR010) - J. Křivánek 2015

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Reflection equation

 Total reflected radiance: integrate contributions of incident

radiance, weighted by the BRDF, over the hemisphere



   

) ( i i

i

i i

r

d cos ) ( ) ( ) (

x H r

f L L w q w w w w

upper hemisphere over x

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=

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Reflection equation

 Evaluating the reflectance equation renders images!!!

 Direct illumination 

Environment maps



Area light sources



etc.

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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).

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

d L d d L f d L d L E B w q w w q w q w w w w q w w q w

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Surface appearance and the BRDF

18

Appearance BRDF lobe (for four different viewing directions)

Souce: Ngan et al. Experimental analysis of BRDF models, http://people.csail.mit.edu/addy/research/brdf/

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Surface appearance and the BRDF

19

Appearance BRDF lobe (for four different viewing directions)

Souce: Ngan et al. Experimental analysis of BRDF models, http://people.csail.mit.edu/addy/research/brdf/

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Surface appearance and the BRDF

20

Appearance BRDF lobe (for four different viewing directions)

Souce: Ngan et al. Experimental analysis of BRDF models, http://people.csail.mit.edu/addy/research/brdf/

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Surface appearance and the BRDF

21

Appearance BRDF lobe (for four different viewing directions)

Souce: Ngan et al. Experimental analysis of BRDF models, http://people.csail.mit.edu/addy/research/brdf/

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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 wo when

illuminated by the unit, uniform incoming radiance.



  

) ( i i

i
d

cos ) ( ) ( ) (

x H r

f a w q w w w w 

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Hemispherical-directional reflectance

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

 Equality follows from the Helmholz reciprocity

 

1 , ) (



w 

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BRDF components

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

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Ideal diffuse reflection

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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 wi , wo)

d r d r

f f

,

i

,

) (  w w

CG III (NPGR010) - J. Křivánek 2015

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Ideal diffuse reflection

 Reflection on a Lambertian surface:  View independent appearance

 Outgoing radiance Lo is independent of wo

 Reflectance (derive)

E f L f L

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

d

cos ) ( ) (  



x

w q w w

d r d

f ,   

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  w x

sky i snow snow

L

L

d

  

White-out!!!

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Ideal mirror reflection

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Ideal mirror 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

w

w w    n n) ( 2 qo n qi qo  qi o  i +  mod 2

o i

SLIDE 39

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

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BRDF of the ideal mirror

 BRDF of the ideal mirror is a Dirac delta distribution

CG III (NPGR010) - J. Křivánek 2015

) , ( ) ( ) , (

i

i

r

  q q  q   L R L

i

i
i

i

i

i ,

cos ) ( ) cos (cos ) ( ) , ; , ( q     q q  q  q  q     R f

m r

qo n qi qo  qi

Fresnel reflectance (see below) We want:

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BRDF of the ideal mirror

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

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

  q q w q  q q     q q  q w q  q       

 

L R d L R d L f L

m r

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Ideal refraction

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Ideal refraction

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hi ho wi wo

Ideal refraction

 Index of refraction h

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

 Snell’s law

i

i

q h q h sin sin 

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Ideal refraction

 Direction of the refracted ray:

 n

) cos 1 ( 1 cos

i 2 2 io i io i io

q

h q h w h w      

i

io

h h h 

if < 0, total internal reflection Critical angle:

        

i

c

i,

arcsin h h q

Image: wikipedia

CG III (NPGR010) - J. Křivánek 2015

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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 h h 

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 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 ( ) , ; , ( q     q h q h  q h h  q  q      R ft

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

SLIDE 49

Fresnel equations

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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

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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

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Glossy reflection

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

Surface roughness and blurred reflections

 The rougher the blurrier

Microscopic surface roughness

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BRDF models

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BRDF modeling



BRDF is a model

f the bulk

behavior of light when viewing a surface from distance



BRDF models



Empirical



Physically based



Approximation

f measured

data

(a.k.a meso-scale)

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Empirical BRDF models

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

(Viewing direction)

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

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Phong shading model

 

n s d

R V k L N k I C ) ( ) (     L R V N L N L N R    ) ( 2

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Phong shading model in the radiometric notation

 

r n s d

k k L L q q w w cos cos ) ( ) (

i i i





i i

r

w w w q      n n r r ) ( 2 cos

BRDF

i i

r

L L f q cos 

i r n s d Orig Phong r

k k f q q cos cos   Original shading model

wi r wo 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 q     cos 2 2

modif Phong

   1  

s d

 

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Physically-plausible BRDF models

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

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Microfacet 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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Microfacet 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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Microfacet BRDF

Microfacet theory [Cook et Torrance 1982]

A perfect mirror



Reflection in a single direction



Outgoing light visible surface normal aligned with the half vector



Half Vector: 𝐼 =

𝑀+𝑊 𝑀+𝑊

Aggregation of micro-mirrors (micro-facets)



Each micro-mirror have a micro-normal



How many micro-mirror have their micro-normal aligned so that 𝐼 = 𝑂 ?



Statistical distribution: Normal Distribution Function (NDF)

71

SLIDE 72

Microfacet BRDF

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

i i r h i r

F G D f q w w q q q 

Fresnel term Geometry 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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Approximation of measured 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 optimization required to find the BRDF

parameters

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BRDF measurements – Gonio-reflectometer

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UTIA University of Bonn Stanford

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Measured Material

 Techniques for speeding measurements



Mirrors



Objects coated by the material:



Sphere [Matusik et al 2003]



Cylinders [Ngan et al 2005]

10/24/2017 Efficient Representation for Measured Reflectance 75

[Matusik et al 2003]

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Surface appearance and the BRDF

76

Appearance BRDF lobe (for four different viewing directions)

Souce: Ngan et al. Experimental analysis of BRDF models, http://people.csail.mit.edu/addy/research/brdf/

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Surface appearance and the BRDF

77

Appearance BRDF lobe (for four different viewing directions)

Souce: Ngan et al. Experimental analysis of BRDF models, http://people.csail.mit.edu/addy/research/brdf/

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Surface appearance and the BRDF

78

Appearance BRDF lobe (for four different viewing directions)

Souce: Ngan et al. Experimental analysis of BRDF models, http://people.csail.mit.edu/addy/research/brdf/

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Surface appearance and the BRDF

79

Appearance BRDF lobe (for four different viewing directions)

Souce: Ngan et al. Experimental analysis of BRDF models, http://people.csail.mit.edu/addy/research/brdf/

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

 BTDF

 Bidirectional transmittance

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

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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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