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.

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Rendering Equation

• Recall
• We want to simplify enough to solve

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Radiosity Assumptions

• 1.  Opaque surfaces
• 2.  Vacuum
• 3. Purely diffuse surfaces
• Solve in object space
• Solution represented in object space
• View independent render as tris w/ vertex color

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Other Surfaces

• Lets relate incoming radiance to other surfaces
• where
• and is 0 or 1.

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Radiance at x from x

• So now our rendering equation is (no emitter)
• Next, lets make our integral over surfaces
  instead of solid angles

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Solid Angle to Area

• Recall that
• so

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Geometry Term

• For simplicity, define
• Therefore

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Diffuse Assumption

• All surfaces diffuse, so replace BRDF with a
  constant
• Also angles are now irrelevant, so

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Convert to Radiosities

• so L B / ?, and

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Radiosity Equation

• For convenience subsume the ? into G(). Also, add
  the emissive term back to get
• where

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Where are we?

• We have an expression relating radiosity at a
  point to radiosity at ALL other points
• 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. This often
  involves assigning radiosity values to vertices.
• Render often done using 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!

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

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Error

• We can express the 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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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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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

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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 again 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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References

• Cohen and Wallace, Radiosity and Realistic Image
  Synthesis, Chapter 3.