CS 655 Computer Graphics

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CS 655 Computer Graphics

• Radiosity

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Radiosity

• Problems with ray tracing
• Doesnt handle diffuse object intereactions
• Doesnt accurately model nature (?)
• Idea behind radiosity
• Model light interactions between every object in
  the scene
• Object A will receive some light from Object B.
  Of that light
• some will be absorbed
• some will be reflected back to B
• some will be reflected to other objects

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Radiosity Method

• Assume a closed environment
• Split the scene into patches
• Each patch has a constant radiosity
• The radiosity of a patch is the total rate of
  energy leaving that patch
• Emitted Reflected
• Determine the energy transmitted between all
  patches in the environment

B
A
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Radiosity Method

• Attempts to handle color bleeding
• Handles diffuse surface interaction well
• Eliminates the ray tracing ambient term
• The idea comes from the heat transfer literature
• Argument most ray tracing scenes are contrived
• Radiosity models real-world scenes better (??)

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Radiosity Image
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Radiosity Image
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Radiosity

• Very time consuming 10X ray tracing
• Not as amenable to speed-up techniques
• View independent
• Pro Can move the view anywhere in the
  environment and not have to recompute
• Con No specularity
• Objects are perfect (Lambertian) diffusors,
  reflectors, or emitters
• Incident light reflects equally in all directions
• The accuracy of the solution depends on the
  initial discretization of the scene

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More Images
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More Images
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And yet More Images
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Ray tracing vs. Radiosity

• Ray tracing
• Handles specularity well
• Amenable to curved objects
• Transparencies
• Radiosity
• Handles diffuse lighting well
• Color bleeding
• View independent
• The argument for radiosity

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Actual sculpture
Original sculpture vertical slabs White on the
front side, colored on the back Positioned offset
and behind one another Lighting is from behind
the sculpture
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Ray Tracing of the Sculpture
Ray traced image of the sculpture white light in
the scene
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Radiosity Solution
Radiosity image of the sculpture, white light
behind the sculpture
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Computing Radiosity

• We need to compute the radiosity of each patch
• To do this, we need to know
• The emittance of the patch
• How much energy is received by the patch
• How much energy is absorbed by the patch
• The emittance is given as a patch material
  property
• The absorption is given as a patch material
  property
• We need to somehow compute the energy received by
  each patch

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Form Factors

• We will do this by computing form factors
• A form factor specifies the amount of energy
  leaving one patch that makes it to another patch
• Dependent on
• The areas of the patches
• The orientations of the patches
• The radiosities of the patches

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Computing Radiosities
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Computing Radiosities

• In a closed environment, energy transport will be
  in equilibrium
• We divide the environment into N patches, with
  the assumption that the radiosity is constant
  across a patch
• This gives

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Radiosity Computation

• Due to the manner in which form factors are
  defined and computed, the following relationship
  exists
• so,
• hence,

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Radiosity Computation

• This gives the radiosity for a single patch Bi
• Each patch has a similar equation, so in matrix
  form we get

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Observations

• The Eis will only be nonzero for emitting
  surfaces
• This equation is actually valid only for a single
  wavelength
• The Eis and the ris are both wavelength
  dependent, thus the equation must be solved for
  each color band
• FNN is 0 for planar and convex surfaces
• none of the light leaving this surface will
  strike this surface directly

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Radiosity Computation

• ri gt 0 for all i
• Thus, the matrix is diagonally dominant
• Hence, it is guaranteed to converge using
  Gauss-Sidel
• The system is solved using Gauss-Sidel, and the
  radiosities are stored
• Then, the environment can be rendered using any
  standard rendering technique
• Radiosities are computed per patch
• To get smoothly shaded objects, radiosities of
  neighboring patches are combined to produce
  vertex radiosities

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Computing Vertex Radiosities
B2
B3
Bv
Bv ¼ (B1 B2 B3 B4)
B1
B4

• Vertex radiosities are then used for shading

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Steps in Radiosity

• Discretize the environment
• Compute form factors
• Solve the set of equations for the Bs
• Render
• Most of the computation time is in the form
  factor calculation
• An order of magnitude greater than solving the
  set of equations and rendering
• The quality of the image depends on the size and
  number of patches
• Subdivide areas of large change
• If a patch has small change over it, leave the
  patch large

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Substructuring

• An approach at getting better results in areas of
  large change without significantly increasing
  computation time
• Avoid the N2 problem of inserting new patches
• Idea
• Divide areas of large change into smaller
  elements
• Compute patch radiosities
• Compute element radiosities assuming that all
  other patches remain undivided

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Substructuring
Patch
Element
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Element Radiosities

• Assume patch Pi is subdivided into mi elements,
  pq, with 1 q mi, then the radiosity of an
  element is given by
• To compute this, we must first compute
  element-to-patch form factors and patch
  radiosities

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Element-to-patch Form Factors

• Element-to-patch form factors are computed the
  same way as patch-to-patch form factors (more on
  computing form factors later)
• The total number of elements in the system is
• so this generates an M x N matrix for the form
  factors
• Note form factors are computed element to
  patch, not element to element

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Patch to Patch Form Factors

• Once the element-to-patch form factors have been
  computed, we then compute the patch-to-patch form
  factors
• This gives an N x N matrix of form factors

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Patch Radiosities

• Once the patch-to-patch form factors are
  computed, we solve the N x N matrix using
  Guass-Sidel to compute patch radiosities

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Element Radiosities

• Once the patch radiosities have been computed, we
  use them, along with the element-to-patch form
  factors, to compute element radiosities

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Substructuring Summary
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Table no Adaptive Subdivision
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Table some Adaptive Subdivision

• 145 patches

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Table more Adaptive Subdivision

• 1021 patches

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Table even more Adaptive Subdivision

• 1306 patches

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Another Adaptive Subdivision Image
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Progressive Refinement Radiosity

• Radiosity is very time consuming, with no results
  until the final solution is achieved
• Is it possible to get partial results earlier,
  that are not the full solution, but are a good
  approximation?
• Progressive refinement radiosity attempts to do
  this
• Idea
• Get lower quality images quickly
• Iteratively refine the image, getting better
  results with more time

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Progressive Refinement

• In standard radiosity, for each iteration we
  solve one row of the matrix, which gives the
  radiosity of one patch
• i.e., given the outgoing radiosities for all of
  the patches we know, we compute the radiosity for
  one new patch
• Continue this process until we have the
  radiosities for all patches

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Progressive Refinement

• Instead of this approach, what if we updated all
  patches from one patch, rather than updating one
  patch from all others?

Standard Radiosity
Progressive Refinement
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Progressive Refinement

• The radiosity algorithm is reordered to allow for
  this
• One patch shoots out its energy
• All patches are updated to account for this
• The radiosity of patch i due to patch j is given
  by
• so

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Progressive Refinement

• We can use this relationship for all patches Bj
• A patch Bi shoots energy and all patches Bj are
  updated
• Then, another patch shoots its light, and so
  forth until convergence
• Order the patches so that most of the light is
  shot in the first few iterations
• Emitting patches first
• Patches that receive the most light from the
  light sources next
• etc.
• This gives a dark to light progression
• You can view the image early on later
  iterations dont change the image significantly

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Progressive Refinement
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More Progressive Refinement
1 iteration 2 iterations 3 iterations
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Ambient Light

• An ambient term can be introduced to reduce the
  large jumps between iterations
• As the solution nears convergence, the ambient
  term decreases, then finally goes away
• The ambient term is calculated from approximate
  form factors

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Ambient Light

• Then an average reflectivity of all patches is
  computed by
• Since light can be reflected any number of times,
  we define an overall reflection factor m as
• so

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Ambient Light

• The ambient light is then estimated with
• Where DBj is the difference in estimates of Bj
  between the current and last iteration

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Progressive Refinement with Ambient
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Progressive Refinement Summary

• Find the patch with the greatest radiosity or
  emitted energy
• Evaluate a column from the matrix, i.e., the form
  factors from this patch to all other patches
• Update the radiosities of all receiving patches
• Reduce the ambient term as a function of the DBs

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Radiosity Terminology

• Gathering
• Standard radiosity at each iteration, one
  patchs radiosity is updated by accounting for
  contributions from all other patches
• Shooting
• the technique used in progressive refinement in
  which at each iteration, one patch shoots its
  energy and all patches that receive energy from
  that one are updated
• Shooting and ambient
• An ambient term is introduced to lessen the
  variation in the image from iteration to
  iteration. As the solution converges, the
  ambient term goes away

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Radiosity Summary

• Handles diffuse reflections well
• View independent
• No specularity
• Can move viewpoint anywhere in the environment
  and not have to recompute the radiosities
• Time consuming
• gt 90 in computing form factors
• lt 10 solving the system of equations
• 0 in rendering