Chapter 8.

The Rendering Equation

Graphics Programming, 29th Sep.

Graphics and Media Lab.

Seoul National University 2011 Fall

3D Rendering by David Keegan

Goal of This Chapter

• Understand the basic radiometry

• Understand BRDF and reflection models

• Understand the rendering equation

Solid Angle

• A quantity that measures the relative coverage of a hemisphere.

– Unit: sr – steradian

• The whole hemisphere is 2π sr.

(Radiant) Flux F

• Time rate of the flow of radiant energy Q

– Amount of energy arriving at a surface during 1 sec.

– Unit: Watt

Irradiance E

• Flux arriving at a surface location x

– Unit: W/m2

– Positional distribution of flux

surface

(Radiant) Intensity I

• Radiant flux per unit solid angle

– Unit: W/sr

– Directional distribution of flux

Radiance L

• Radiant intensity per unit projected area

– Unit: W/m2sr

– Flux density per unit projected area per unit solid angle

surface

Radiance L

• Radiant intensity per unit projected area

– Unit: W/m2sr

– Irradiance in terms of radiance:

dxLxE cos),()(

Radiance L

• Radiant intensity per unit projected area

– Unit: W/m2sr

– Flux in terms of radiance:

Light Emission Example

• Assume a point light source with power

• What is the irradiance E at a surface?

• What is the radiance L at a surface?

Light Scattering

3D Rendering by Kevin Beason

Light Scattering

3D Rendering by Kevin Beason

Light Scattering

3D Rendering by Kevin Beason

Light Scattering

• Materials interact with light in different ways.

• Different materials have different appearances given the same lighting conditions.

• The reflectance properties of a surface are described by a reflectance function, which models the interaction of light reflecting at a surface.

• The bi-directional reflectance distribution function (BRDF) is the most general expression of reflectance of a material

The BRDF

• The BRDF is defined as the ratio between differential radiance reflected in an exitant direction, and incident irradiance through a differential solid angle

• The BRDF is defined as the ratio between differential radiance reflected in an exitant direction, and incident irradiance through a differential solid angle

The BRDF

incoming

outgoing

Why BRDF is Complicated? • How about

• Differential irradiance dE(p,w’) – dEi(x,w’) Li(x,w’)cosi dw’ – Why do we put d in front of E?

• dLr(x,w) differential exitance radiance due to dEi(x,w’) • dLr(x,w) dEi(x,w’)

– Why? – The relationship is called the “linearity assumption”.

• fr(x,w,w’) fr(x,w’,w) – Reciprocity: the value of the BRDF will remain unchanged if the

incident and exitant directions are interchanged.

ww

w

w

www

dpL

xdL

xdE

xdLxf

ii

r

i

rr

cos),(

),(

),(

),(),,(

),(

),(),,(

w

www

xL

xLxf

i

rr

The Scattering Equation

wwwww

wwwww

dxLpf

dxLpfxL

ir

iirr

)(n ),(),,(

cos ),(),,(),(

• Calculation of the exitant radiance from x toward w, by summing contributions made by all the incident radiances.

Reflectance r

• Why dF instead of F?

• BRDF vs. Reflectance – Both are about a surface point x.

– Reflectance: General (or average) tendency of reflection

– BRDF: Bi-directional reflection distribution

Object Color and Reflectance

• How do we see the color? – The surface absorbs photons of certain range of l

– Is it general or direction specific behavior of the surface?

• Object color is represented with the reflectance! – For red, use the vector reflectance r(1,0,0).

– For yellow, use r(0.5,0.5,0).

– …

Diffuse Reflection

• A diffuse surface reflects light in all directions

– A special case of diffuse reflection is Lambertian: the reflection is uniform over the whole hemisphere.

Rendered using Mitsuba

Lambertian

• For a Lambertian surface, the reflected radiance is constant in all directions regardless of the irradiance.

• This gives a constant BRDF:

)(

)(

)(

),()(,

xE

xL

xE

xLxf

i

r

i

rdr

w

Constant wrt w

Specular Reflection

• Specular reflection happens when light strikes a smooth surface – e.g. Metal, glass, and water

Specular Reflection

• Specular reflection happens when light strikes a smooth surface – e.g. Metal, glass, and water

• The reflected radiance due to specular reflection is:

and the reflection direction is:

Relationship Between Reflectance and Index of Refraction

• From left-to-right, only the index of refraction is varied. – We can observe the reflectance is related with the IOR.

– Why such thing happens?

Image from Maya 2009 Mental Ray Document

The Fresnel Equations

• For smooth homogeneous metals and dielectrics, specular reflectance is given by the following formulae for polarized lights:

Specular Reflectance for Unpolarized Light

• For unpolarized light the specular reflectance becomes:

• The amount of refracted ray is computed as:

)(2

1)( 22

|| rrr rs F

More General Reflection Models

• Phong

• Microfacet models

– Torrance-Sparrow

– Oren-Nayar

• Lafortune

• BSSDF

What’s Next?

• Now, we know how to describe light.

• We also know how it gets reflected.

• From now on, let’s talk about how light travels around!

The Rendering Equation

• Introduced by David Immel et al. and James Kajiya in 1986.

• The rendering equation describes the total amount of light coming from a point x along a particular viewing direction. – Based on the law of conservation of energy.

Image from Wikipedia

The Rendering Equation

• In short, the outgoing radiance Lo is the sum of the emitted radiance Le and the reflected radiance Lr:

• Using BRDF, we can rewrite about eq. as:

• How can we solve the above eq?

– Difficult!

Understanding the Situation

E

D

S

L

D

E: Eye, D: Diffuse, S: Specular

Light -> Eye

E

D

S

L

D

E: Eye, D: Diffuse, S: Specular

Light -> Diffuse -> Eye

E

D

S

L

D

E: Eye, D: Diffuse, S: Specular

Light -> Specular -> Eye

E

D

S

L

D

E: Eye, D: Diffuse, S: Specular

Light -> Specular -> Specular -> Diffuse -> Eye

E

D

S

L

D

E: Eye, D: Diffuse, S: Specular

Light -> Specular -> Specular -> Diffuse -> Diffuse -> Specular -> Diffuse -> Specular -> … -> Eye

E

D

S

L

D

E: Eye, D: Diffuse, S: Specular

Computing Rendering Equation

• You might have felt that solving the rendering eq won’t be a trivial job.

– How can we evaluate this integral numerically?

• Let’s rewrite the eq in more readable (and computable) form.

1. Neumann series expansion

2. Light transport notation

Neumann Series Expansion

• The rendering eq can be represented using the following compact form:

where the integral operator T is:

Neumann Series Expansion

• Then, recursive evaluation of L=Le+TL gives:

E

D

S

L

D

Light Transport Notation

• We can also represent light traversals in general form. – L: a light source

– E: the eye

– S: a specular reflection/refraction

– D: a diffuse reflection

• Using regular expression: – (k)+: one or more of k events

– (k)*: zero or more of k events

– (k)?: zero or one k events

– (k|k’): a k or a k’ events

LE

E

D

S

L

D

LSE

E

D

S

L

D

LSSDE

E

D

S

L

D

L(S|D)+DE

E

D

S

L

D

Ray Tracing Models

• Appel’s Method (1968) – E(D|G)L – aka. Ray casting – Only computes local illuminations

• Whitted’s Method (1980) – ES*(D|G)L – aka. (Recursive) ray tracing – Only computes local illuminations

• Kajiya’s Method (1986) – E[(D|G|S)+(D|G)]L – aka. Path tracing – Fully computes local and global illuminations

* G is glossy reflection

How to Solve the Rendering Equation

• Trace all possible paths, and integrates every contribution

– Called the “path tracing”

– Extremely expensive.

• The next two lectures will teach how to integrate the equation numerically.

– Monte-Carlo integration

E

D

S

L

D