lecture 3: rendering equation - cornell university · 1 lecture 3: rendering equation cs 6620,...

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Lecture 3: Rendering Equation

CS 6620, Spring 2009Kavita Bala

Computer ScienceCornell University

© Kavita Bala, Computer Science, Cornell University

Radiometry

• Radiometry: measurement of light energy

• Defines relation between– Power– Energy– Radiance– Radiosity

2


Hemispherical coordinates

• Defined a measure over hemisphere• dω = direction vector• Differential solid angle

ϕθθω ddrdAd sin2 ==


Radiance

• Radiance is radiant energy at x in direction θ: 5D function– : Power

per unit projected surface areaper unit solid angle

– units: Watt / m2.sr

Θ⊥=Θ→

ωddAPdxL

2

)(

)( Θ→xL

dA⊥

Θ

x

Θωd

3


Why is radiance important?

• Invariant along a straight line (in vacuum)

x1

x2

Θ

x


Why is radiance important?

• Response of a sensor (camera, human eye) is proportional to radiance

• Pixel values in image proportional to radiance received from that direction

eye

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Relationships• Radiance is the fundamental quantity

• Power:

• Radiosity:

dAdxLPArea

AngleSolid

⋅⋅⋅Θ→= ∫ ∫ Θωθcos)(

∫ Θ⋅⋅Θ→=

AngleSolid

dxLB ωθcos)(

Θ⊥=Θ→

ωddAPdxL

2

)(


Outline

• Light Model

• Radiance

• Materials: Interaction with light

• Rendering equation

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Materials - Three Forms

Ideal diffuse (Lambertian)

Idealspecular

Directionaldiffuse


Reflectance—Three Forms

Ideal diffuse (Lambertian)

Directionaldiffuse

Idealspecular

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• Bidirectional Reflectance Distribution Function

BRDF

DetectorDetectorLight Light SourceSource

NN

Θ

x

Ψ

2mW srm

W2

)()(),(

Ψ←Θ→

=Θ→ΨxdExdLxfr


Definition of BRDF

)()(),(

Ψ←Θ→

=Θ→ΨxdExdLxfr

ΨΨΨ←Θ→

=ωdNxL

xdL

x ),cos()()(

• 6D function? 4D function?– Why?

• Wavelength-dependent

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BRDF special case: ideal diffuse

Pure Lambertian

πρd

r xf =Θ→Ψ ),(

10 ≤≤ dρin

outd Energy

Energy=ρ


Properties of the BRDF

• Reciprocity:

• Therefore, notation:

• Important for bidirectional tracing

),(),( Ψ→Θ=Θ→Ψ xfxf rr

),( Θ↔Ψxfr

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Properties of the BRDF• Bounds:

∞≤Θ↔Ψ≤ ),(0 xfr

1),cos(),( ≤ΘΘ↔ΨΨ∀ ∫Θ

ΘωdNxf xr

• Energy conservation:


Outline

• Light Model

• Radiance

• Materials: Interaction with light

• Rendering equation

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Light Transport• Goal

– Describe steady-state radiance distribution in scene

• Assumptions:– Geometric Optics– Achieves steady state instantaneously

• Related:– Neutron Transport (neutrons)– Gas Dynamics (molecules)


Radiance represents equilibrium• Radiance values at all points in the scene

and in all directions expresses the equilibrium

• 4D function: only on surfaces

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Rendering Equation (RE)• RE describes energy transport in scene

• Input– Light sources– Surface geometry– Reflectance characteristics of surfaces

• Output: value of radiance at all surface points in all directions


Rendering Equation

)( Θ→xL

L

x

Θ

)( Θ→xLe

Le

x=

=

+

+ )( Θ→xLr

Lr

x

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Rendering Equation

)( Θ→xLe

Le

xx =

=

+

+

Lr

x

x

x

)( Θ→xL

LΘ


Rendering Equation

)( Θ→xLe

Le

x=

=

+

+

Lr

x

)( Ψ←xL ...∫hemisphere

x

)( Θ→xL

LΘ

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Rendering Equation

ΨΨΨ←Θ↔Ψ=Θ→ ωdNxLxfxdL xr ),cos()(),()(

ΨΨΨ←Θ↔Ψ=Θ→ ∫ ωdNxLxfxL xhemisphere

rr ),cos()(),()(

)()(),(

Ψ←Θ→

=Θ↔ΨxdExdLxfr

)(),()( Ψ←Θ↔Ψ=Θ→ xdExfxdL r


Rendering Equation

=

= )( Θ→xLe

Le

x +

+

Lr

x

)( Ψ←xL ΨΨΘ↔Ψ ωdxfr ),cos(),( xN∫hemisphere

• Applicable for each wavelength

x

)( Θ→xL

LΘ

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Rendering Equation

+Θ→=Θ→ )()( xLxL e

x ΨΨΘ↔ΨΨ←∫ ωdxfxLhemisphere

r ),cos(),()( xN

incoming radiance


Summary

• Geometric Optics

• Goal: – to compute steady-state radiance values in

scene

• Rendering equation: – mathematical formulation of problem that global

illumination algorithms must solve

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RE: Area Formulation

)( Ψ←xL

)( Ψ−→yL

Ray-casting function: whatis the nearest visible surfacepoint seen from x in direction Ψ?

)),(()(),(

Ψ−→Ψ=Ψ←Ψ=

xvpLxLxvpy

x

y

∫Ω

Ψ⋅⋅Ψ←⋅Θ↔Ψ+Θ→=Θ→x

dxLfxLxL xre ωθcos)()()()(


∫Ω



Rendering Equation

dAy

Ψωd

Ψ

x

2

cos

xy

yy

rdA

dθ

ω =Ψ

),( Ψ= xvpy

)),(()( Ψ−→Ψ=Ψ← xvpLxL

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Rendering Equation: visible surfaces

∫Ω



• Integration domain extended to ALL surface points by including visibility function

+Θ→=Θ→ )()( xLxL e

),( Ψ= xvpy

Integration domain = visible surface points y

Coordinate transform

∫surfaces all

on y)( Ψ−→⋅ yL)( Θ↔Ψrf y

xy

yx dA

r⋅⋅ 2

coscos

θθ


∫ ⋅⋅⋅

⋅Ψ−→⋅+=Θ→A

yxy

yxre dAyxV

ryLfLxL ),(

coscos)((...)(...))( 2

θθ

Rendering Equation: all surfaces

x

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Two forms of the RE• Hemisphere integration

• Area integration (over polygons from set A)

∫ ⋅⋅⋅

⋅Ψ−→⋅Θ↔Ψ+Θ→=Θ→A

yxy

yxre dAyxV

ryLfxLxL ),(

coscos)()()()( 2

θθ

∫Ω




Lighting Cues

glossy reflection

refraction

soft shadow

color bleeding

Global Illumination is important for realism

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Light Sources and Reflection Models


Outline• Light sources

– Light source characteristics– Types of sources

• Light reflection– Physics-based models– Empirical models

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Sources of light radiation• Thermal radiation (“blackbody”)

– Sun, tungsten & tungsten-halogen lamps; arc lamps

• Electric discharge– gas discharge lamps (neon, sodium, mercury vapor)– arc lamps, fluorescent lamps

• Other phenomena– fluorescence (fluorescent lamps, fluorescent dyes)– phosphorescence (CRTs); LEDs; lasers

© Kavita Bala, Computer Science, Cornell University400 500 600 700

1

2

3

4

Wavelength

Brig

htne

ss

Mercurylines

Phosphor emission

400 500 600 700

1

2

3

4

Wavelength

Brig

htne

ss

Mercury vapor lampExamples of light emission

10-2

10-1

1

10

102

103

104

105

106

107

10 100 1000 10000

Wavelength

Inte

nsity

20,000K

10,000K

3000K2000

K1000K

500K

VisibleSpectrum

6000K

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Modeling luminaires• Spectral distribution

– Determined by physics of source– Generally tabulated, often RGB used

• Spatial distribution– Modeled as point or simple area light– Also light probes create high dynamic range inputs

• Directional distribution– Often shaped by reflectors– Tabulated when necessary, cosine lobe is common

approximation


Directional distributions

Lambertian cosine-power arbitrary

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Lighting w/ Environment Maps• High lighting complexity

• Rich: captures real world


Image-based lighting• Acquiring lighting information of real

scenes– Image-based techniques

• Use light probe

• Varying exposure

lecture 3: rendering equation - cornell university · 1 lecture 3: rendering equation cs 6620,...

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