1. The Radiance Equation

2. Motivation
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.

3. Motovation
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.

4. Introduction 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?

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

6. Light: 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.

7. Light: 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.

8. Light: 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.

9. Light: Equation (p,) denotes flux at pV, 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,).

10. Simplifying 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

11. Radiance
Radiance (L) is the flux that leaves a surface, per unit projected area of the surface, per unit solid angle of direction. n
d = L dA cos d L  dA

12. Radiance
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.

13. Radiance: Radiosity, Irradiance
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

14. Radiosity 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

15. Recall Reflectance BRDF Relates Bi-directional Reflectance
Distribution
Function
Relates
Reflected radiance to incoming irradiance i r
Incident ray
Reflected ray
Illumination hemisphere
f(p, i , r )

16. Recall Reflectance: BRDF
Reflected Radiance = BRDFIrradiance
Formally:
L(p, r ) = f(p, i , r ) E(p, i )
= f(p, i , r ) L(p, i ) cosi di
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

17. The 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

18. The Radiance Equation: Reflection
Total reflected radiance in direction :
 f(p, i ,  ) L(p, i ) cosi di
Radiance Equation:
L(p,  ) = Le(p,  ) +  f(p, i ,  ) L(p, i ) cosi di
(Integration over the illumination hemisphere)

19. The Radiance Equation L(p,  )
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 )
L(p, ) depends on all L(p*, i) which in turn are recursively defined. p* i
L(p,  ) p
The radiance equation models global illumination.

20. Traditional Solutions to the Radiance Equation
The radiance equation embodies totality of all 2D projections (view).

21. Irradiance Power per unit area incident on a surface. E = d /dA
Unit: Watt / m2
arriving dA

22. Radiant Exitance Power per unit area leaving surface
Also known as radiosity
B = d /dA
Same units as irradiance
just direction changes.
leaving dA

23. Basic Definitions 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: (r) The fraction of light which is reflected from a surface. (0 <= r <=1)
Form- Factor: (F) The fraction of the light leaving one surface which arrives to another. (0<=F<=1)

24. The Basic Radiosity Equation
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 .

25. Mesh Surfaces into Elements Reconstruct and Display Solution
Classic Radiosity Algorithm
Mesh Surfaces into Elements
Compute Form Factors
Between Elements
Solve Linear System
for Radiosities
Reconstruct and Display Solution

26. The Descrete Radiosity Equation
Total power leaving an element i
and reflected light.
weighted by geometric coupling
j->i
and reflectivity
is sum of emitted light by element i
Reflected light depends on contribution from every other element j

27. The Form Factor:
the fraction of energy leaving one surface that reaches another surface
It is a purely geometric relationship, independent of viewpoint or surface attributes
Surface j
Surface i

28. The Reciprocity Relationship
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:

29. Patches 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.

30. Next Step: Learn ways of computing form factors
Needed to solve the Descrete Radiosity Equation:
Form factors Fij are independent of radiosities (depend only on scene geometry)

31. The overall form factor between i and j is found by integrating
Between differential areas, the form factor equals:
The overall form factor between i and j is found by integrating
Surface j
Surface i

32. Form Factors in (More) Detail
where Vij is the visibility (0 or 1)

33. We have two integrals to compute:
Surface j
Area integral
over surface i
Area integral
over surface j
Surface i

34. 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.

35. 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.

36. Numerical Integration: The Nusselt Analog
This gives the form factor FdAiAj Aj
dAi

37. Method 1: Hemicube
Approximation of Nusselt’s analog between a point dAi and a polygon Aj
Polygonal
Area (Aj)
Infinitesimal
Area (dAi)

38. The Hemi-cube
We compute the delta formfactor of each grid cells DF 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

39. The Hemicube In Action

40. 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.

41. Projecting all other surfaces onto the hemicube
Compute the form factors from a point on a surface to all 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.

42. 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.

43. Hemicube Method Scan convert all scene objects onto hemicube’s 5 faces
Use Z buffer to determine visibility term
Sum up the delta form factors of the hemicube cells covered by scanned objects
Gives form factors from hemicube’s base to all elements, i.e. FdAiAj for given i and all j

44. Hemicube 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

45. We have found the Radiosity Matrix ElementsEi Bi

46. Radiosity 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.

47. Radiosity 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.

48. Solve [F][B] = [E] Direct methods: O(n3) Iterative methods: O(n2)
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

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

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

51. Gathering

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

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

54. Shooting

55. Progressive Radiosity

56. Next Accuracy from meshing

57. Artifacts

58. Increasing Resolution

59. Adaptive Meshing

60. Some Radiosity Results

62. 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.