Rendering Equation and Radiosity

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Rendering Equationand Radiosity
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Direct Illumination vs. Global Illumination

• reflected, scattered and focused light (not
  discreet).
• physical-based light transport calculations
  modeled around bidirectional reflective
  distribution functions (BRDFs).

• discreet light source.
• efficient lighting calculations based on light
  and surface vectors (i.e. fast cheats).

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Indirect Illumination
Color Bleeding
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Contact Shadows
Notice the surface just under the sphere. The
shadow gets much darker where the direct
illumination as well as most of the indirect
illumination is occluded. That dark contact
shadow helps enormously in sitting the sphere
in scene. Contact shadows are difficult to fake,
even with area lights.
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Caustics

• Focused and reflected light, or caustics, are
  another feature of the real world that we lack in
  direct illumination.

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Global illumination rendered images

• Caustics are a striking and unique feature of
  Global Illumination.

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Local Illumination (Phong Model)

• Problems with Empirical Models
• What are ka, ks, kd and nshiny? Are they
  measurable quantities?
• What are the coefficients for copper?
• How does the incoming light at a point relate to
  the outgoing light?
• Is energy conserved?
• Just what is light intensity?
• Is my picture accurate?

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What We Want

• A model that uses physical properties that can be
  looked up in the CRC Handbook of Chemistry and
  Physics (indices of refraction, reflectivity,
  conductivity, etc.)
• Parameters that that have clear physical
  analogies (how rough or polished a surface is)
• Models that are predictive (the simulation
  attempts to model the real scene)
• Models that conserve energy
• Complex surface substructures (crystals,
  amorphous materials, boundary-layer behavior)
• If it was easy... everyone would do it.

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Radiance

• Radiance Electromagnetic energy flux, the amount
  of energy traveling
• at some point x
• in a specified direction  
• per unit time
• per unit area perpendicular to the direction
• per unit solid angle
• for a specified wavelength  
• denoted by

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What happens after reflection?

• The amount of reflected radiance is proportional
  to the incident radiance. They are related by
  Bidirectional Reflectance Distribution Function
  (BRDF)

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What does BRDF look like?

• Bidirectional Reflectance Distribution Function
  (BRDF)
• Is a surface property
• Relates energy in to energy out
• Depends on incoming and outgoing directions

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BRDF

• BRDF really just means the way light bounces off
  of something.
• Specular reflection some of the light bounces
  right off of the surface along the angle of
  reflection without really changing color much.
• Diffuse reflection some of the light gets
  refracted into the plastic and bounced around
  between red particles of pigment. Most of the
  green and the blue light is absorbed and only the
  red light makes its way back out of the surface.
  The red light bounced back is scattered every
  which way with fairly equal probability.

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Energy Balance Equation

• The total light leaving a point is given by the
  sum of two major terms
• Emitted from the point
• Incoming light from other sources reflected at
  the point

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Solving the rendering equation

• L is the radiance from a point on a surface in a
  given direction ?
• E is the emitted radiance from a point E is
  non-zero only if x is emissive
• V is the visibility term 1 when the surfaces
  are unobstructed along the direction ?, 0
  otherwise
• G is the geometry term, which depends on the
  geometric relationship between the two surfaces x
  and x

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Photorealistic Lighting

• Photorealistic lighting requires solving the
  equation!
• Not possible in the general case with todays
  technology
• Light transport is concerned with the incoming
  light part of the equation
• Notice the chicken and egg problem
• To know how much light leaves a point, you need
  to know how much light reaches it
• To know how much light reaches a point, you need
  to know light leaves every other point
• Reflectance modeling is concerned with the BRDF
• Hard because BRDFs are high dimensional functions
  that tend to change as surfaces change over time

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Light Emitted from a Surface

• Radiance (L) Power per unit area per unit solid
  angle
• Measured in W/m2sr
• dA is projected area perpendicular to given
  direction
• Radiosity (B) Radiance integrated over all
  directions
• Power from per unit area, measured in W/m2

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Radiosity Concept

• Radiosity of each surface depends on radiosity of
  all other surfaces
• Treat global illumination as a linear system
• Need constant BRDF (diffuse)
• Solve rendering equation as a matrix problem
• Process
• Mesh into patches
• Calculate form factors
• Solve radiosity
• Display patches

Cornell Program of Computer Graphics
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Continuous Radiosity Equation
For an environment composed of diffuse surfaces,
we have the basic radiosity relationship
reflectivity


x
B
E
B
r
G(x,x)V(x,x)
x
x
x
x
x
Form factor

• G geometry term
• V visibility term
• No analytical solution, even for simple
  configurations

x
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Discrete Radiosity Equation
For an environment that has been discretized into
n patches, over which the radiosity is constant,
(i.e. both B and E are constant across a patch),
we have the basic radiosity relationship
reflectivity
n
å


r
B
F
E
B
A
j
ij
i
i
i
j
j1
Form factor

• discrete representation
• iterative solution
• costly geometric/visibility calculations

A
i
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The Radiosity Matrix
Such an equation exists for each patch, and in a
closed environment, a set of n simultaneous equati
ons in n unknown Bi values is obtained
A solution yields a single radiosity value Bi for
each patch in the environment a
view- independent solution. The Bi values can be
used in a standard renderer and a particular view
of the environment constructed from the radiosity
solution.
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Intuition
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Examples
Factory simulation. Program of Computer
Graphics, Cornell University. 30,000 patches.
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Examples
Museum simulation. Program of Computer Graphics,
Cornell University. 50,000 patches. Note
indirect lighting from ceiling.
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Remaining Hard Problems

• Reflective Diffraction Effects
• thin films
• Caustics
• oil on water
• CDs
• Anisotropy
• brushed metals
• strands pulled materials
• satin and velvet cloths
• Need better rendering algorithms.