Radiosity Part II Form Factors

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Radiosity Part IIForm Factors

• Alex Telea

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Goal

• Learn ways of computing form factors.

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Recall the Radiosity Equation

• The Fij are the form factors
• Form factors independent of radiosities(depend
  only on scene geometry)

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Form Factors in Detail

• Expanding G, we get
• where Vij is the visibility (0 or 1)

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Alternate Formulation

• We have two integrals to compute

Area integral over object j
Area integral over object i
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Computing the Form Factor
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Computing the Form Factor
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Analytic solutions

• Only feasible for VERY simple scenes
• Visibility is hard to compute analytically!

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Numerical approximation
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Numerical IntegrationThe Nusselt Analog
This gives the form factor FdAiAj
Aj
dAi
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The Nusselt Analog

• Project Aj along its normalAj cos qj
• Project result on sphereAj cos qj / r2
• Project result on unit circleAj cos qj cos qi
  /r2
• Divide by unit circle area
• Aj cos qj cos qi / pr2
• Integrate for all points on Aj

area Aj
qj
r
qi
sphere projection Aj cos qj/r2
second projection Aj cos qj cos qi /r2
unit circle area p
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Hemicube Method

• Approximation of Nusselts analog betweena point
  dAi and a polygon Aj

Polygonal Area (Aj)
Infinitesimal Area (dAi)
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Hemicube

• For convenience, a cube 1 unit high with a top
  face 2 x 2 is used. Side faces are 2 wide by 1
  high.
• Decide on a resolution for the cube. Say 512 by
  512 for the top.

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Compute Delta Form Factors
These are the inner integrals integrand, i.e.
FdAidAj

• Store delta factors in table
• Use the cube symmetry to store less factors

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Hemicube Method

• Scan convert all scene objects onto hemicubes 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 hemicubes base to all
  elements, i.e. FdAiAj for given i and all j

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Hemicube Problem Aliasing
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Method 2 Area Sampling

• Subdivide Aj into small pieces dAj
• 2. For all dAj
• cast ray dAj-dAj to determine V if
  visible compute FdAidAj
• sum up
• FdAiAj FdAidAj
• 3. We have now FdAiAj

Aj
dAj
ray
dAi
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Summary

• Several ways to find form factors
• Hemicube was original method
• Hardware acceleration
• Gives FdAiAj for all j in one pass
• - Aliasing
• Area sampling methods now preferred - Slower
  than hemicube - As accurate as desired since
  adaptive

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Next

• Ok, we have the form factors
• How do we solve the matrix?
• Shooting
• Progressive Radiosity
• Meshing

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References

• Cohen and Wallace, Radiosity and Realistic Image
  Synthesis, Chapter 4.