Radiosity Part I Mathematical Foundations

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Radiosity Part IMathematical Foundations

• COMP238

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Goal

• To obtain a mathematical basis for radiosity.
• Next, the practical details.
• Computing form factors
• Solving the matrix
• Meshing

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Radiosity Equation

• From last time
• where

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Where are we?

• We have an expression relating radiosity at a
  patch to radiosity at ALL other patches
• But no method to solve for the values

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Radiosity Method

• Subdivide the model into elements.
• Select locations (nodes) on elements at which to
  solve for radiosity.
• Select basis functions to approximate radiosity
  across the element, based on values at nodes.
  Most common is to assume constant value of
  radiosity across the element, so a single node is
  placed in the middle.
• Select finite error metric. This will result in a
  set of linear equations.

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Radiosity Method (cont.)

• Compute coefficients of linear system. These are
  based on the geometric relationships between
  elements, called the form factors.
• Solve the system of linear equations.
• Reconstruct the radiosity function (often just
  assign radiosity values to vertices).
• Render often Gouraud interpolation of radiosity
  values at vertices.

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Basis Functions

• Approximate radiosity using finite set of basis
  functions and radiosities at the nodes.
• Choose functions with support only within
  element.
• Most common basis function is constant across
  element.

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Example in 1D
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2D Example
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Nodes

• Constant most common
• Lischinski used quadratic

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Nodes not for Display!

• Nodes used for solution not necessarily where
  color stored for display.
• Usually node values interpolated to per-vertex
  color.
• We used quadratic interpolation.

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Error

• Would like to minimize error
• But we dont know the solution!

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Residual

• Can express error as a residual
• that we want to minimize.
• Note that and r are defined over all
  space but the is computed using a finite
  number of points ( ).

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Reducing Residual

• Cant in general make residual zero
• Basis functions cant match real
• Lower dimensionality of residual computation
• Can use similar method to basis functions

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Weighting Residual

• Weight the residual with function that has only
  local support
• where

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Point Collocation

• Simplest way to deal with residuals. Measure only
  at the nodes.
• Note that this is defined only at the nodes.

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Galerkin Formulation

• Instead of measuring residual only at nodes,
• measure over element.
• Specifically, Galerkin method uses same basis
  function for measuring residual as for computing
  node radiosity

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Residual

• Substituting basis function

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Expanding and Grouping

• We get something of form with

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Constant Basis Function

• Suppose we use constant basis
• Look at each of the terms

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• Basis function has value 1 within the element,
  and 0 elsewhere
• So this integral is only valid when ij, and is
  then the area of the element

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• Integral only valid over elements i and j
• Assume reflectivity constant over element

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• Emissivity assumed constant over element,
  therefore

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Now we have

• Dividing by Ai and putting into form KBE
• or

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Form Factor

• Express as
• where
• Fij is known as the Form Factor.

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Rearranging

• gives the more familiar form

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Intuitive Interpretation

• Add area term
• Use reciprocity of form factors
• to get
• which can be interpreted as

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Total power leaving an element i
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Total power leaving an element i
Is sum of emitted light
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Total power leaving an element i
and reflected light
is sum of emitted light
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Total power leaving an element i
and reflected light
is sum of emitted light
Reflected light depends on contribution from
every other element
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Total power leaving an element i
and reflected light.
weighted by geometric relationship, area, and
reflectivities.
is sum of emitted light
Reflected light depends on contribution from
every other element
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What do we have?

• A mathematical basis for radiosity.
• We have a matrix equation of the form
• to solve.

Note diagonal
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Next

• How do we compute form factors?
• Hemicube method
• How do we solve the matrix?
• Shooting
• Progressive Radiosity

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Form Factor

• Expanding, we get
• where Vij is the visibility

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Alternative

• Area/Hemisphere integral

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Computing the Form Factor
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Computing the Form Factor
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Closed form

• Only feasible for simple cases
• Visibility is hard
• Polygon-to-polygon solution by Schroeder and
  Hanrahan

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Numerical approximation
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Nusselt Analog
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Why?

• Imagine a unit hemisphere centered around patch
  (or node) i.
• Projection onto sphere mechanically computes the
  term
• Projection of solid angle due to patch j.

next
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Solid Angle
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Area on Base
Also, ? is area of unit circle, so division is
appropriate, resulting in
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Hemicube

• Also obeys Nusselts analog

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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

• Store in table.
• Note the symmetry

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Specifically

• Scan convert all primitives onto 5 faces
• Z buffer as usual
• Keep an item buffer

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Aliasing
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Other Problems

• Sampling is not even
• Must render complete dataset
• Should cull
• Could you use levels of detail?

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Monte Carlo

• Sample by casting rays to estimate Nusselts
  analog.
• Distribute the rays to get a good sampling of the
  sphere

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Area
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Area Sampling

• Subdivide the primitive j into small pieces and
  cast a ray to the center of each area to
  determine visibility

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Summary

• Many ways to find form factors
• Hemicube most common
• Hardware acceleration
• Monte Carlo methods also used

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Next

• How do we solve the matrix?
• Shooting
• Progressive Radiosity
• Meshing

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References

• Cohen and Wallace, Radiosity and Realistic Image
  Synthesis, Chapters 3-4.

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Unused Slides
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Point Collocation
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Expanding

• Expanding
• Grouping (note Bj independent of x)

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Matrix Form

• is in form
• or where

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Whats the Significance?

• We are computing the radiosities only at the
  nodes.
• However, depending on basis function we may have
  to consider values over element.

Nj may be dependent on values over element j
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Constant Basis Function

• is the Kronecker delta function, 1
  if i j, 0 otherwise.
• Now
• where