CS 655 Computer Graphics

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CS 655 Computer Graphics

• The Rendering Equation

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The Rendering Equation

• Developed by Kajiya in 1986
• An attempt to unify rendering so that all
  rendering has a basic model as a basis
• Accounts for all light interactions in an
  environment

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Energy Balance

• Outgoing Incoming Emitted Absorbed
• The total light energy put into the system must
  equal theenergy leaving the system (usually, via
  heat).
• Outgoing Emitted Reflected Transmitted

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Outgoing Energy
wo
H wi n(x) lt 0
N
wir
N surface normal
wo outgoing energy
wir incoming reflected energy
wit incoming transmitted energy
H- wi n(x) gt 0
H half space above object
wit
H- half space below object
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BRDF

• Bidirectional Reflectance Distribution Function
• A measurement of the amount of energy being
  distributed about all directions from a point.

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Diffuse BRDF

• Light is reflected equally in all directions

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Specular BRDF

• Light is reflected only in one direction

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Glossy BRDF

• Light is reflected unequally in many directions
• Several models exist that attempt to represent
  glossy BRDFs.

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Phong Model
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Blinn - Phong Model
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Modified Blinn - Phong Model
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Outgoing Energy
wo
H wi n(x) lt 0
N
wir

• Outgoing Emitted Reflected Transmitted

H- wi n(x) gt 0
wit
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Outgoing Energy
wo
H
N
wir
H-
wit
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The Rendering Equation
Unoccluded two point transfer No participating
media
If occluded, this is 0 If not occluded, this is
the inverse square of the distance between x
and x
Energy emitted from point x that reaches point x
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The Rendering Equation
The intensity of energy originating from
x, coming through point x, and terminating at
point x The BRDF
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The Rendering Equation

• In other words, the transport intensity from x
  to x is the sum of the emitted light from x that
  reaches x, plus all of the light from x that
  eventually gets to x through x
• We can rewrite the equation as
• Where M is the linear integral operator

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Breaking Down the Rendering Equation

• Rearranging terms gives

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Breaking Down the Rendering Equation
Light to x directly from x
Light from light source to x, then to x
Light to x via x scattered twice
Light to x via x scattered three times
etc.
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Applying the Rendering Equation

• Local reflection models
• only the first two terms are used
• the ge term is non-zero only for light sources
• M operates on e rather than g, so shadows are not
  computed

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Applying the Rendering Equation

• Basic ray tracing
• Mo is the sum of the reflection and refraction
  terms
• the geo gives shadows, but only for point light
  sources
• Ambient lighting is accounted for in e
• M is generally approximated by a small sum

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Distributed Ray Tracing

• An extension of the three component Whitted (ray
  tracing) approximation
• M is approximated by a distribution around the
  reflection and transmission delta functions

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Radiosity

• by performing the transformations outlined in the
  paper, we get
• dB(x) is the radiosity of surface element dx
• ro is r(x, x, x) and is constant
• H(x) is the energy incident on the surface
  element dx

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Applying the Rendering Equation

• Extended two pass algorithm by Sillion (1989)
• with
• In the first pass, extended form factors are used
  to compute diffuse to diffuse interaction with
  any number of specular transfers inbetween
• The second pass uses standard ray tracing to
  compute specular transfers

diffuse
specular
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Sampling

• Goal to reduce the variance in the final image
• Sampling uniformly will introduce significant
  variance (noise)
• Five sampling techniques are introduced
• Sequential uniform sampling
• Multidimensional sequential uniform sampling
• Hierarchical integration
• Adaptive sampling hierarchical integration
• Importance sampling

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Importance Sampling the Integral Equation

• Path Tracing
• A ray is sent out, then
• either a reflection or transmission ray is sent
• a shadow ray is sent
• The path tracers time is concentrated in the
  more important samples

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Images from the Original Paper
Left ray tracing, 401 minutes Right using the
integral equation, 533 minutes 40 paths per
pixel
Using the integral equation 1221 minutes