CPSC 441 Computer Graphics: Radiosity

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CPSC 441 Computer Graphics Radiosity

• Jinxiang Chai

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Rendering Illumination Computing

• Direct (local) illumination
• Light directly from light sources
• No shadows

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Local Illumination
Ir kaIa Ii (kd (n.l) ks(h.n)m )
ambient
diffuse
specular
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Local Illumination
Ir kaIa Ii (kd (n.l) ks(h.n)m )
ambient
diffuse
specular
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Local Illumination
Ir kaIa Ii (kd (n.l) ks(h.n)m )
ambient
diffuse
specular
if there are multiple lights there is a sum of
the specular and diffuse components for each
light
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Direct and Indirect Light
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Rendering Illumination Computing

• Direct (local) illumination
• Light directly from light sources
• No shadows
• Indirect (global) illumination
• Transparent, reflective surfaces, and hard
  shadows (Ray tracing)
• Diffuse interreflections, color bleeding, and
  soft shadow (radiosity)

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Radiosity
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Radiometry

• The goal of a global illumination algorithm is to
  compute a steady-state distribution of light in a
  scene
• To compute this distribution, we need an
  understanding of the physical quantities that
  represent light energy
• Radiometry is the basic terminology used to
  describe light

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Photons

• The basic quantity in lighting is the photon
• The energy of a photon with frequency f is
• - ? wavelength
• - c the speed of light
• In vacuum, c 299.792.458m/s
• - h 6.6310-34Js is Plancks constant

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(Spectral) Radiant Energy

• The spectral radiant energy, Q?, in n? photons
  with wavelength ? is
• How to compute the radiant energy, Q, for actual
  light?

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Spectral Energy Distribution

• Actual light radiations contain a rage of
  frequencies

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Light Radiant-energy

• The total energy of light radiation is the sum
  over all photons of all frequencies or
  wavelengths.

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Radiant Flux

• Definition The radiant flux is the amount of
  radiant energy transmitted per unit of time
  (measured in watts)
• Also known radiant power

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Radiant Exitance

• Definition The radiant (luminous) exitance is
  the
• energy per unit area leaving a surface.
• In computer graphics, this quantity is often
• referred to as the radiosity (B)

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Radiant Intensity

• Definition Radiant intensity is the radiant flux
  radiated from a point on a light source into a
  unit solid angle in a particular direction. And
  it is measured in watt per steradian.

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Angles and Solid Angles
l

• Angle
• Solid angle

?
Þ circle has 2p radians
Þ sphere has 4p steradians
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Radiosity

• The radiosity model computes radiant-energy
  interactions between all the surfaces in a scene

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Radiosity Key Idea 1
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Diffuse Surface
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Radiosity Key Idea 2
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Constant Surface Approximation
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Radiosity Equation
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Radiosity Equation
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Radiosity Algorithm
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Energy Conservation Equation
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Energy Conservation Equation
The total rate of radiant energy leaving surface
i per unit square
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Energy Conservation Equation
The rate of energy emitted from surface i per
unit area - zero if surface i is not a light
source
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Energy Conservation Equation
Reflectivity factor
Percent of incident light that is reflected in
all directions
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Energy Conservation Equation
Form factor
Fractional amount of radiant energy from surface
j that reaches surface i
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Compute Form Factors
The form factor specifies the fraction of the
energy leaving one patch and arrives at the
other. In other words, it is an expression of
radiant exchange between two surface patchesl
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Compute Form Factors
Radiant energy reaching Ay from Ax
Radiant energy leaving Ax in all directions
The form factor specifies the fraction of the
energy leaving one patch and arrives at the
other. In other words, it is an expression of
radiant exchange between two surface patchesl
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Form Factor Reciprocity
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Radiosity Equation

• Radiosity for each polygon
• Linear system
• - radiosity of patch I (unknown)
• - emission of patch I (known)
• - reflectivity of patch I (known)
• - form-factor (known)

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Linear System
X
B
A
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Radiosity Algorithm
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Form Factors
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Form Factor How to compute?

• Closed Form
• - anlytical
• Hemicube

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Form Factor Analytical
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Form Factor How to compute?

• Closed Form
• - anlytical
• Hemicube

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Form Factor Nusselt Analog
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Form Factor Nusselt Analog
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Form Factor HemiCube
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Delta Form Factor Top Face
Top of hemicube
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Delta Form Factors Side Faces
Side of hemicube
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The Hemicube in Action
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Form Factors HemiCube
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Form Factors
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Radiosity Algorithm
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How to Solve Linear System?

• Matrix conversion
• Iterative approaches
• - Jacobian (gathering)
• - Gauss-Seidel (gathering)
• - progressive refinement (shooting)

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Matrix Conversion
- Computational cost O(N3) - Very slow for a
large set of polygons
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Iterative Approaches
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Jacobian Iterations

• For all patches i, i1,,N,
• While not converged
• for all patches i1,,N

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Jacobian Iterations

• For all patches i, i1,,N,
• While not converged
• for all patches i1,,N

Update of one patch requires evaluation of N Form
Factors Whats the computational cost?
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Successive Approximation
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Rendering

• - The final Fi's can be used in place of
  intensities in a standard renderer (Gouraud)
• - Radiosities are constant over the extent of a
  patch
• - A standard renderer requires vertex intensities
  (or radiosities)
• - If the radiosities of surrounding patches are
  know, vertex radiosities can be estimated using
  bilinear interpolation

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Vertex Intensity Bilinear Interpolation
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Consolation Room
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Theatre
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Steel Mills
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Radiosity Benefit

• Global illumination method modeling diffuse
  inter-reflection
• Color bleeding a red wall next to a white one
  casts a reddish glow on the white wall near the
  corner
• Soft shadows an area light source casts a
  soft shadow from a polygon
• No ambient term hack, so when you want to look at
  your object in low light, you dont have to
  adjust parameters of the objects just the
  intensities of the lights!
• View independent it assigns a brightness to
  every surface

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Radiosity Limitation

• Radiation is uniform in all directions
• Radiosity is piecewise constant
• usual renderings make this assumption, but
  then interpolate cheaply to fake a nice-looking
  answer
• this introduces quantifiable errors
• No surface is transparent or translucent
• Reflectivity is independent of directions to
  source and destination