The Radiance EquationThe Radiance Equation

MotivationMotivation Photo‑realistic image rendering is particularly difficult to compute

because of the complexity of the physical nature of light. However, the radiosity global illumination methods approximates the physical nature of light and provides the necessary foundation for extremely high quality rendered photo‑realistic images.

Radiosity has become established as the global illumination method for rendering the highest quality, view independent images for virtual environments and captures subtle lighting effects such as colour bleeding. The method is able to correctly compute shadows due to area light sources, producing accurate penumbra and umbra.

MotovationMotovation

 Radiosity is a powerful tool for rendering

photo‑realistic scenes. Once the radiosity of a scene has been calculated, a ‘virtual reality’ walkthrough of the scene is immediately available. However, this comes at a costly price as calculating the radiosity of a scene is anything but trivial.

IntroductionIntroduction

Real-time walkthrough with global illumination– Possible under limited conditions

Radiosity (diffuse surfaces only)

Real-time interaction– Not possible except for special case local

illumination

Why is the problem so hard?

LightLight

Remember visible light is electromagnetic radiation with wavelengths approximately in the range from 400nm to 700nm

400nm 700nm

Light: PhotonsLight: Photons

Light can be viewed as wave or particle phenomenon

Particles are photons– packets of energy which travel in a straight line

in vaccuum with velocity c (300,000m.p.s.)

The problem of how light interacts with surfaces in a volume of space is an example of a transport problem.

Light: Radiant PowerLight: Radiant Power

denotes the radiant energy or flux in a volume V.

The flux is the rate of energy flowing through a surface per unit time (watts).

The energy is proportional to the particle flow, since each photon carries energy.

The flux may be thought of as the flow of photons per unit time.

Light: Flux EquilibriumLight: Flux Equilibrium

Total flux in a volume in dynamic equilibrium– Particles are flowing– Distribution is constant

Conservation of energy

– Total energy input into the volume = total energy that is output by or absorbed by matter within the volume.

Light: EquationLight: Equation

(p,) denotes flux at pV, in direction It is possible to write down an integral equation

for (p,) based on:– Emission+Inscattering = Streaming+Outscattering +

Absorption Complete knowledge of (p,) provides a

complete solution to the graphics rendering problem.

Rendering is about solving for (p,).

Simplifying AssumptionsSimplifying Assumptions

Wavelength independence – No interaction between wavelengths (no fluorescence)

Time invariance– Solution remains valid over time unless scene changes

(no phosphorescence)

Light transports in a vacuum (non-participating medium) – – ‘free space’ – interaction only occurs at the surfaces of

objects

RadianceRadiance

Radiance (L) is the flux that leaves a surface, per unit projected area of the surface, per unit solid angle of direction.

n

dA

L

d = L dA cos d

RadianceRadiance

For computer graphics the basic particle is not the photon and the energy it carries but the ray and its associated radiance.

n

dA

L d

Radiance is constant along a ray.

Radiance: Radiosity, Radiance: Radiosity, IrradianceIrradiance

Radiosity - is the flux per unit area that radiates from a surface, denoted by B.– d = B dA

Irradiance is the flux per unit area that arrives at a surface, denoted by E.

– d = E dA

Radiosity and IrradianceRadiosity and Irradiance

L(p,) is radiance at p in direction E(p,) is irradiance at p in direction E(p,) = (d/dA) = L(p,) cos d

Recall ReflectanceRecall ReflectanceBRDF

– Bi-directional– Reflectance– Distribution– Function

Relates– Reflected

radiance to incoming irradiance

ir

Incident ray

Reflected ray

Illumination hemisphere

f(p, i , r )

Recall Reflectance: BRDFRecall Reflectance: BRDF

Reflected Radiance = BRDFIrradiance Formally:

L(p, r ) = f(p, i , r ) E(p, i ) = f(p, i , r ) L(p, i ) cosi di

In practice BRDF’s hard to specify Rely on ideal types

– Perfectly diffuse reflection– Perfectly specular reflection– Glossy reflection

BRDFs taken as additive mixture of these

The Radiance EquationThe Radiance Equation

Radiance L(p, ) at a point p in direction is the sum of– Emitted radiance Le(p, )

– Total reflected radiance

Radiance = Emitted Radiance + Total Reflected Radiance

The Radiance Equation: The Radiance Equation: ReflectionReflection

Total reflected radiance in direction :

f(p, i , ) L(p, i ) cosi di

Radiance Equation: L(p, ) = Le(p, ) + f(p, i , ) L(p, i ) cosi di

– (Integration over the illumination hemisphere)

The Radiance EquationThe Radiance Equation p is considered to be on a surface, but can be

anywhere, since radiance is constant along a ray, trace back until surface is reached at p’, then– L(p, i ) = L(p’, i )

p*

i

pL(p, )

L(p, ) depends on allL(p*, i) which in turnare recursively defined.

The radiance equation models global illumination.

Traditional Solutions to the Traditional Solutions to the Radiance EquationRadiance Equation

The radiance equation embodies totality of all 2D projections (view).

21

IrradianceIrradiance

Power per unit area incident on a surface.

E = d /dA

Unit: Watt / m2

dA

arriving

22

Radiant ExitanceRadiant Exitance

Power per unit area leaving surface

Also known as radiosity

B = d /dA

Same units as irradiance

just direction changes.dA

leaving

BasicBasic DefinitionsDefinitions

Radiosity: (B) Energy per unit area per unit time. Emission: (E) Energy per unit area per unit time

that the surface emits itself (e. g., light source). Reflectivity: () The fraction of light which is

reflected from a surface. (0 <= Form- Factor: (F) The fraction of the light

leaving one surface which arrives to another. (0<=F<=1)

The Basic Radiosity The Basic Radiosity EquationEquation

We will compute the light emitted from a single differential surface area dAi.

It consists of: 1. Light emitted by dAi. 2. Light reflected by dAi.

– depends on light emitted by other dAj, fraction of it reaches dAi.

The fraction depends on the geometric relationship between dAi and dAj: the formfactor .

Classic Radiosity Algorithm

Mesh Surfaces into Elements

Compute Form FactorsBetween Elements

Solve Linear Systemfor Radiosities

Reconstruct and Display Solution

26

n

jjjijiiiii AFBAEAB

1

Total power leaving an element i

is sum of emitted light by element i

and reflected light.

Reflected light depends on

contribution from every other element j

weighted by geometric coupling

j->iand

reflectivity

The Descrete Radiosity Equation

Surface i

Surface j

It is a purely geometric relationship, independent of viewpoint or surface attributes

The Form Factor:

the fraction of energy leaving one surface that reaches another surface

The Reciprocity The Reciprocity RelationshipRelationship

If we had equal sized emitters and receivers, the fraction of energy emitted by one and received by the other would be identical to the fraction of energy going the other way.

Thus, the formfactors from Ai to Aj and from Aj to Ai are related by the ratios of their areas:

Thus:

The radiosity equation is now:

Patches and ElementsPatches and Elements Patches are used for emitting light. Some patches are

divided into elements, which are used to more accurately compute the received light after the patch solution have been computed.

Next Step: Next Step: Learn ways of computing Learn ways of computing form factorsform factors

Needed to solve the Descrete Radiosity Equation:

Form factors Fij are independent of radiosities(depend only on scene geometry)

ijjiii FBEB

Surface i

Surface j

jdA

idA

j

i r

2

coscos

rdAFdA ji

jj

Between differential areas, the form factor equals:

The overall form factor between i and j is found by integrating

ji

A A

ji

iij dAdA

rAF

i j

2

coscos1

Form Factors in (More) DetailForm Factors in (More) Detail

ji

A A

ijji

iij dAdAV

rAF

i j

2

coscos1

where Vij is the visibility (0 or 1)

ji

A A

ji

iij dAdA

rAF

i j

2

coscos1

We have We have two integralstwo integrals to compute: to compute:

Area integralover surface i

ijijji

AAiij dAdAV

rAF

ji

2

coscos1

Area integralover surface j

Surface i

Surface j

jdA

idA

j

i r

The Nusselt Analog The Nusselt Analog

Integration of the basic form factor equation is difficult even for simple surfaces!

Nusselt developed a geometric analog which allows the simple and accurate calculation of the form factor between a surface and a point on a second surface.

The Nusselt Analog The Nusselt Analog

The "Nusselt analog" involves placing a hemispherical projection body, with unit radius, at a point on a surface.

The second surface is spherically projected onto the projection body, then cylindrically projected onto the base of the hemisphere.

The form factor is, then, the area projected on the base of the hemisphere divided by the area of the base of the hemisphere.

Numerical Integration:Numerical Integration:The Nusselt AnalogThe Nusselt AnalogThis gives the form factor FdAiAj

dAi

Aj

Method 1: HemicubeMethod 1: HemicubeApproximation of Nusselt’s analog between a

point dAi and a polygon Aj

InfinitesimalArea (dAi)

PolygonalArea (Aj)

The Hemi-cubeThe Hemi-cube We compute the delta formfactor of each grid cells F and

store in a table. Project all patches onto the ‘ hemi- cube ’ screen, drawing

a patch- id instead of color. Sum the delta form factors of all grid cells covered by the

patch’s id.

Delta form factor

The Hemicube In Action The Hemicube In Action

The Hemicube In Action The Hemicube In Action

This illustration demonstrates the calculation of form factors between a particular surface on the wall of a room and several surfaces of objects in the room.

Compute the form factors from a point on a surface to all Compute the form factors from a point on a surface to all other surfaces by:other surfaces by:

Projecting all other surfaces onto the hemicube

Storing, at each discrete area, the identifying index of the surface that is closest to the point.

Discrete areas with the indices of the surfaces which are ultimately visible to the point.

From there the form factors between the point and the surfaces are calculated.

For greater accuracy, a large surface would typically be broken into a set of small surfaces before any form factor calculation is performed.

Hemicube MethodHemicube Method

1. Scan convert all scene objects onto hemicube’s 5 faces

2. Use Z buffer to determine visibility term

3. Sum up the delta form factors of the hemicube cells covered by scanned objects

4. Gives form factors from hemicube’s base to all elements,

i.e. FdAiAj for given i and all j

Hemicube AlgorithmsHemicube Algorithms

Advantages+ First practical method+ Use existing rendering systems; Hardware+ Computes row of form factors in O(n)

Disadvantages- Computes differential-finite form factor- Aliasing errors due to sampling- Proximity errors- Visibility errors- Expensive to compute a single form factor

We have found the Radiosity We have found the Radiosity Matrix ElementsMatrix Elements

n

jjjjiiiiii ABFAEAB

1

n

jjijiii BFEB

1

nnnnnnnnn

n

n

E

E

E

B

B

B

FFF

FFF

FFF

2

1

2

1

21

22222212

11121111

1

1

1

jijiji FAFA

i

n

jjijii EBFB

1

Ei

Bi

Radiosity MatrixRadiosity Matrix The "full matrix" radiosity solution calculates the form

factors between each pair of surfaces in the environment, as a set of simultaneous linear equations.

This matrix equation is solved for the "B" values, which can be used as the final intensity (or color) value of each surface.

nnnnnnnnn

n

n

E

E

E

B

B

B

FFF

FFF

FFF

2

1

2

1

21

22222212

11121111

1

1

1

Radiosity MatrixRadiosity Matrix

This method produces a complete solution, at the substantial cost of – first calculating form factors between each pair of surfaces – and then the solution of the matrix equation.

Each of these steps can be quite expensive if the number of surfaces is large: complex environments typically have above ten thousand surfaces, and environments with one million surfaces are not uncommon.

This leads to substantial costs not only in computation time but in storage.

Solve [F][B] = [E]Solve [F][B] = [E]

Direct methods: O(n3)

– Gaussian elimination Goral, Torrance, Greenberg, Battaile, 1984

Iterative methods: O(n2)

Energy conservation¨diagonally dominant ¨ iteration converges

– Gauss-Seidel, Jacobi: Gathering Nishita, Nakamae, 1985 Cohen, Greenberg, 1985

– Southwell: Shooting Cohen, Chen, Wallace, Greenberg, 1988

GatheringGathering

In a sense, the light leaving patch i is determined by gathering in the light from the rest of the environment

n

jijjiii FBEB

1

ijjiji FBBtodueB

n

jjijiii BFEB

1

GatheringGathering

Gathering light through a hemi-cube allows one patch radiosity to be updated.

n

jjijiii BFEB

1

Gathering

Shooting RadiosityShooting Radiosity

Shoot the radiosity of patch i and update the radiosity of all other patches.

ShootingShooting

Shooting light through a single hemi-cube allows the whole environment's radiosity values to be updated simultaneously.

jijijj EBBB

j

iijji A

AFF

For all j

where

Shooting

Progressive Radiosity

Next Accuracy from meshing

Artifacts

Increasing Resolution

Adaptive Meshing

Some Radiosity ResultsSome Radiosity Results

Discontinuity Meshing Discontinuity Meshing Dani Lischinski, Filippo Tampieri

and Donald P. Greenberg created this image for the 1992 paper Discontinuity Meshing for Accurate Radiosity.

It depicts a scene that represents a pathological case for traditional radiosity images, many small shadow casting details.

Notice, in particular, the shadows cast by the windows, and the slats in the chair.