Advanced Global Illumination

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Advanced Global Illumination

• Brian Chen
• Thursday, February 12, 2004

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Global Illumination
Physical Simulation of Light Transport Accuracy
account for ALL light pathsconservation of
energy Predictionforward renderingcalculate
light meter readings Analysisinverse rendering!
find surface properties ! Realism?perceptually
necessary?
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Global Illumination

• Everything is lit by Everything Else
• Screen color entire scenes lighting surface
  reflectance
• Refinements Models of area light sources,
  caustics, soft-shadowing, fog/smoke,
  photometric calibration,

H. Rushmeier et al., SIGGRAPH98 Course 05 A
Basic Guide to Global Illumination
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Overview

• Physics (Optics) concepts
• Models of Light
• Radiometry
• BRDFs Rendering Equation
• Monte Carlo Methods
• Photon Mapping

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Optics Definitions

• Reflection when light bounces off a surface
• Refraction the bending of light when it goes
  through different mediums (ex. air and glass)
• Diffraction the phenomenon where light bends
  around obstructing objects
• Dispersion when white light splits into its
  components colors, result of refraction

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Lights Dual Nature

• Is light a wave? Or is it a beam of particles?
• Wave Theory of Light Thomas Youngs double-slit
  experiment
• Particle Theory of Light Newton, refraction
  dispersion

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Light Models

• Models of light are used to capture/explain the
  different behaviors of light that stem from its
  dual nature
• Quantum Optics
• Wave Model
• Geometric Optics

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Quantum Optics

• Explains dual wave-particle nature at the
  submicroscopic level of electrons
• Way too detailed for purposes of image generation
  for computer graphics scenes
• Therefore, it is not used

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Wave Model

• Simplification of Quantum Model
• Captures effects that occur when light interacts
  with objects of size comparable to the wavelength
  of light (diffraction, polarization)
• This model is also ignored, too complex and
  detailed

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Geometric Optics

• Simplest and most commonly used light model
• Assumes light is emitted, reflected, and
  transmitted (refraction)
• Also assumes
• Light travels in straight lines, no diffraction
• Light travels instantaneously through a medium
  (travels at infinite speed)
• Light is not influenced by gravity, magnetic
  fields

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Radiometry

• Radiometry area of study involved in the
  physical measurement of light
• Goal of illumination algorithm is to compute the
  steady-state distribution of light energy in a
  scene

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Radiometric Quantities

• Radiant Power
• flux (in Watts Joule/sec), F
• How much total energy flows from/to/through a
  surface per unit time
• Irradiance (ear)
• incoming radiant power on a surface per unit
  surface area (Watt/m2)
• E dF/dA

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More Radiometric Quantities

• Radiosity
• Outgoing radiant power per unit surface area
  (Watt/m2)
• B dF/dA

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Radiance

• Flux per unit projected area per unit solid angle
  (W/(steradian x m2))
• How much power arrives at (or leaves from) a
  certain point on a surface, per unit solid angle,
  and per unit projected area
• L(x, ?) d2F / (d? dA cos?)
• The most important quantity in global
  illumination because it captures the appearance
  of objects

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Radiance contd

• Radiance The Pointwise Measure of Light
• Free-space light power L (energy/time)
• At least a 5D scalar function L(x, y, z, ?, ?,
  )
• Position (x,y,z), Angle (?,?) and more (t, ?, )
• Power density units, but tricky

Taken with permission from Jack Tumblin
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Yet More Radiance

• Tricky think Hemispheres
• with a floor

Solid Angle (steradians) d? fraction of
a hemispheres area (4?)
Projected Area
cos ? dA
?
dA
Taken with permission from Jack Tumblin
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Properties of Radiance

• 1 Radiance is invariant along straight paths and
  does not attenuate with distance
• Radiance leaving point x directed towards point y
  is equal to the radiance arriving at point y from
  the point x
• 2 Sensors, such as cameras and human eye, are
  sensitive to radiance
• The response of our eyes is proportional to the
  radiance incident upon them.
• It is now clear that radiance is the quantity
  that global illumination algorithms must compute
  and display to the observer

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BRDF (Bidirectional Reflectance Distribution
Function)

• Intuitively the BRDF represents, for each
  incoming angle, the amount of light that is
  scattered in each outgoing angle
• BRDF at a point x is defined as the ratio of the
  radiance reflected in an exitant direction (?),
  and the irradiance incident in a differential
  solid angle (?).
• fr(x,?-gt?) dL(x-gt?) / (L(xlt-?)cos(Nx,?)d??)
• cos(Nx,?) cos of angle formed by normal
  incident direction vector

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Point-wise Reflectance BRDF
Bidirectional Reflectance Distribution Function
?(?i , ?i , ?r , ?r , ?i , ?r , ) (Lr /
Li) a scalar (sr-1)
Illuminant Li
Reflected Lr
Infinitesimal Solid Angle
?
?
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BRDF Properties

• Value of BRDF remains unchanged if the incident
  and outgoing directions are interchanged
• Law of conservation of energy requires that the
  total amount of power reflected all directions
  must be less than or equal to the total amount of
  power incident on the surface

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BRDF Examples, Diffuse
Fr (x,?lt-gt?) ?d/p, constant ?d is the
fraction of incident energy that is reflected at
a surface, 0-1
Andrew Glassner et al.. SIGGRAPH94 Course
18 Fundamentals and Overview of Computer
Graphics
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BRDF Examples, Specular
Exitant direction R 2(N dot ?)N ?
Andrew Glassner et al.. SIGGRAPH94 Course
18 Fundamentals and Overview of Computer
Graphics
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For Most Surfaces
Most Materials Combination of Diffuse
Speculartheir BRDF is difficult to model with
formulas
Andrew Glassner et al.. SIGGRAPH94 Course
18 Fundamentals and Overview of Computer
Graphics
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Practical Shading Models (BRDF)

• Lambert kd ?d/p
• Phong fr ks((R?n)/(N?)) kd
• Blinn-Phong ks((NH)/(N?)) kd, Hhalfway
  vector between ? ?
• Modified Blinn-Phong ks(NH)n kd
• Cook-Torrance
• Ward

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Lambert BRDF
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Phong BRDF
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Blinn-Phong BRDF
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Ward Anisotropic BRDF
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Rendering Equation
Finally! Putting radiance and the BRDF together
to get
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BRDF Rendering Eq.

• For example, suppose that we wish to determine
  the illumination of a scene containing n light
  sources light source 1 to light source n. In
  this case, the local illumination of a surface is
  given by,
• where Lij is the intensity of the jth light
  source and wij (?ij,?ij) is the direction to
  the jth light source.

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BRDF Rendering Eq.

• For a single point light source, the light
  reflected in the direction of an observer is
• This is the general BRDF lighting equation for a
  single point light source

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Rendering Equation

• Opportunities
• Scalar operations only ?() and L(), indep. of ?,
  x,y,z, ?,?
• Linearity
• Solution weighted sum of one-light solns.
• Many BRDFs ? weighted sum of diffuse, specular,
  gloss terms
• Difficulties
• Almost no nontrivial analytic solutions exist
  MUST use approximate methods to solve
• Verification tough to measure real-world ?() and
  L() well
• Notable wavelength-dependent surfaces exist
  (iridescent insect wings casing, CD grooves,
  oil)
• BRDF doesnt capture important subsurface
  scattering

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Review 1

• Big Ideas
• Measure Light Radiance
• Measure Light Attenuation BRDF
• Light will bounce around endlessly, decaying on
  each bounce The Rendering Equation
  (intractable must
  approximate)

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Monte Carlo Techniques

• Mathematical techniques that use statistical
  sampling to simulate phenomena or evaluate values
  of functions
• Use a probability distribution function (PDF) to
  generate random samples, p(x)

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Estimators

• Unbiased when the expected value of the
  estimator is exactly the value of the integral
• Biased when the above property is not true
• Bias the difference between the expected value
  of the estimator and the actual value of the
  integral
• As number of samples N increases, the estimate
  becomes closer

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Monte Carlo Techniques

• The reliability of Monte-Carlo sampling is
  measured by the variance of the estimators. The
  variance of the estimators is given as
•  
• where is the primary estimator, while
  its average is a secondary estimator

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Monte Carlo Techniques

• As a consequence of the above formula, the
  Monte-Carlo technique requires a large number of
  samples to reduce the variance of the primary
  estimator
• The error of the
  approximation is itself a random variable with
  zero mean, i.e. the estimator is unbiased. The
  standard deviation of this estimator is reduced
  only proportional to which is known as
  the diminishing return of the Monte-Carlo
  technique, i.e. the number of samples must be
  quadrupled to reduce the standard deviation by
  one half.

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Monte Carlo Techniques

• For importance sampling we adjust the pdf p to be
  similar in shape to the integrand f. If we were
  able to choose p(x) Cf(x), with a constant
  factor C 1/I, the variance of the primary
  estimator would be zero and a single sample would
  always give the correct result.
• Unfortunately, the constant C is determined by
  the integral we want to compute and is therefore
  unavailable. On the other hand, some a priori
  knowledge about the shape of f is often available
  and can be used to adjust p to reduce the
  variance of the estimators.

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Monte Carlo Example

• Computing a one-dimensional integral
• Samples are selected randomly over the domain of
  the integral to get a close approximation
  (estimator ltIgt
• ltIgt (1/N) S f(xi) / p(xi)

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Another Example

• Integration over a hemisphere
• Estimate the radiance at a point by integrating
  the contribution of light sources in the scene
• Light source L
• I ?Lsourcecos?d?? ?02p ?0 p/2
  Lsourcecos?sin?d?df
• ltIgt(1/N)S (Lsource(?i) cos?sin?) / p(?i)
• p(?i) cos?sin?/ p
• ltIgt(1/N)S Lsource(?i)

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Steps

• General
• Sampling according to a probability distribution
  function
• Evaluation of the function at that sample
• Averaging these appropriately weighted sampled
  values
• Graphics
• Generate random photon paths from source (lights
  or pixels)
• Set a discrete random length for the path
• Count how many photons terminate in state i,
  average radiances for that point

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http//graphics.stanford.edu/courses/cs348b-02/lec
tures/lecture15/walk014.html
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http//graphics.stanford.edu/courses/cs348b-02/lec
tures/lecture15/walk014.html
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Advantages/Disadvantages

• Advantages
• Simple just sample signal/function and average
  estimates
• Widely applicable nuclear physics, graphics,
  high-dimensional integrations of complicated
  functions
• Can be used with radiosity
• Disadvantages
• Very slow, take many, many samples to converge to
  correct solution
• Need 4x samples to decrease error by half

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Monte Carlo Images
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Monte Carlo Images
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Photon Mapping

• One of the fastest algorithms available for
  global illumination
• Monte-Carlo based
• 2-pass algorithm

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Photon Mapping, 1st pass

• 1st pass
• photons shot from the light into the scene
• they bounce around interacting with all the types
  of surfaces they encounter
• Clever twists
• 1st, instead of redoing those same computations
  over and over, a few thousand of time for each
  pixels, the photons are stored only once in a
  special data structure called a photon map for
  later reuse
• 2nd, instead of trying to completely fill the
  whole scene with billions of photons, a few
  thousands to a million photons are sparsely
  stored and the rest is statistically estimated
  from the density of the stored photons.
• After all the photons have been stored in the
  map, a statistical estimate of the irradiance at
  each photon position is computed.

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Photon Mapping, 2nd Pass

• Direct illumination is computed just like regular
  ray tracing, but the indirect illumination, which
  comes from the walls and other objects around, is
  computed from querying the stored photons in the
  photon map
• At each secondary hit, the photon map is queried
  in order to gather the radiance coming from the
  objects around in the environment.

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Photon Mapping Images
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Summary

• Global illuminations goal is to calculate the
  steady-state distribution of light energy in a
  scene
• BRDFs used to calculate reflection radiances
• Rendering Equation used to calculate the radiance
  sent out from an object
• Monte-Carlo techniques used to calculate
  irradiance at random locations throughout scene
• Photon Maps used to accurately calculate
  caustics, inter-object reflections