Return%20to%20Assumptions

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Return to Assumptions

• Recall the two fundamental assumptions of the
  standard radiosity approach
• Constant radiosity over each element (today)
• Perfectly diffuse surfaces (next week)
• We wish to more accurately represent the
  radiosity at every point
• Regardless of the quality of the solution (form
  factors, subdivision, etc.), if the correct
  radiosity cannot be represented, it wont be
  computed

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Better Radiosity Representations

• Two approaches
• Work around the constant radiosity assumption by
  constructing meshes where elements really should
  have constant (or bilinear) radiosity
• Use more expressive representations for radiosity
  (work under different assumptions)

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Meshing Artifacts

• Where does poor meshing really show?
• Find the (many) artifacts in the following image

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Typical Meshing

• Lays down an orthogonal grid in some orientation
• May subdivide, but the principle directions
  remain unchanged
• Require subdivision down to pixel resolution to
  remove artifacts

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Staircase Effect

• Most evident at shadow boundaries
• Elements are dark if they have more than half
  shadow, light otherwise
• Solution
• Subdivide the mesh to sub-pixel resolution
  (costly)
• Blur the boundaries (introduces other problems)
• Align element boundaries with the shadow
  boundaries (discontinuity meshing, later)

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Light and Shadow Leaks

• Occur when the mesh does not respect geometric
  boundaries
• Interior walls meeting the floor
• The wall appears to float
• Shadows leaking out from under objects
• No reasonable perceptual interpretation
• Solutions
• Ensure that there are element boundaries at
  geometric intersections (hard to do without user
  help)
• Discontinuity meshing

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Interpolation Artifacts

• Have to fill in radiosity between vertices, or
  blur element boundaries
• Interpolation schemes are frequently poorly
  implemented, particularly those in hardware
• Solutions
• Meshes that make interpolation easier
• More expensive interpolation

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Preprocessing the Mesh

• Seek to remove inconsistencies before starting
  the solution
• Ensure invariants are met
• Objects are solid
• Objects dont intersect
• This is very hard to do automatically, it should
  be part of the modeling process
• But, commercial modelers are generally poor in
  this regard

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Fixing Interpolation

• Pre-process to remove concave polygons
• Replace them with triangles - not too hard
• Enforce subdivision constraints
• Neighboring patches cannot differ by more than
  one subdivision level
• Post-process the remove T-vertices
• Subdivide mismatched neighbors and interpolate
  new vertices correctly

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Smoother Interpolants

• Bi-linear (hardware accelerated) interpolation
  uses only vertices of given polygon
• Build higher order interpolants
• Must make sure not to use a high-order
  interpolant across a low order discontinuity
• Gather to a texture or cloud of points
• Gather to pixels in the image (slow)

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Discontinuity Meshing

• Identify expected discontinuities and mesh around
  them
• Sharp boundaries due to point light sources or
  object contact
• Derivative discontinuities due to area sources
  and multi-object shadows
• Related to aspect graphs in computer vision
• Places where the set of visible things changes

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Two Types of Discontinuities

• Assume polygonal environment
• Vertex-Edge events
• Discontinuities where the plane defined by a
  vertex and an edge intersects other objects
• Edge-Edge-Edge
• Higher order discontinuities at places where
  three edges appear to meet at a point
• Generally ignored

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Meshing With Discontinuities

• Construct VE planes
• Intersect them with surfaces
• Mesh the resulting edges
• Constrained triangulation is a difficult problem
• Mesh must be able to store different radiosity
  values at one point

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Using Discontinuity Meshes

• Very high number of possible discontinuities
  O(n6) for n vertices
• Only find discontinuities due to bright light
  sources
• Try to only find visible discontinuities
• Research topic Integrate into hierarchical
  scheme

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Better Radiosity Representations

• Standard approach Each point takes on the value
  of the patch on which it lies
• Finite Element Approach The radiosity at each
  point is given by a linear combination of basis
  functions evaluated at that point

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Finite Element Formulation

• Note the similarity to splines a set of weights
  multiply a set of basis functions to give a value
• Choose a set of basis functions that can capture
  the desired behavior
• Find the coefficients, Bj, that give the best
  solution

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Galerkin Method

• Find the set of weights that minimize the
  variation of the found solution from the true
  solution
• In other words Find the closest expressible
  solution to the true one
• The standard radiosity equation, with accurate
  form factors, is a Galerkin method with constant
  basis functions of finite support (supported by
  each patch)

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Point-Collocation Method

• Find the set of weights that zero the error at a
  fixed set of points
• The hemicube algorithm implements a
  point-collocation method for the radiosity
  equation
• Not as accurate as the Galerkin method
• Only locally accurate, as opposed to globally
  optimal

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Alternate Bases

• Linear basis functions
• Wavelets
• Multi-resolution representation
• Behaves like hierarchical radiosity, but without
  redundant information