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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.