Introduction to Global Illumination

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Introduction to Global Illumination

• Overview
• Radiometry
• The rendering equation
• Monte Carlo

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Image synthesis
Deterministic and/or stochastic simulation.
scene description
surface radiance
3
Stages of light transport
luminaire
blocker
blocker
Direct illumination
Indirect illumination
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An example of global illumination
Lischinski, Tampieri, and Greenberg 1993
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Photo-realistic rendering
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Types of surface scattering
diffuse
directional diffuse
specular
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Directional dependence
highly directional
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Defining radiance
Classical Definition
Measure-Theoretic
u
?
d?
dA
Radiant energy defines a measure on R3 x S2.
r
f (r,u) cos? dA d? power crossing surface
The associated density function is radiance.
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Definition of radiance
?
radiance
x
is a scalar density function
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Definition of radiance
d?
dA
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Power from radiance
Integrate over solid angle...
and surface
dA
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Irradiance
power per unit area
dA
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Irradiance
weighted integral over solid angle
dA
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Simulating reflected light
irradiance
d?
x
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Simulating reflected light
irradiance
radiance
d?
df
x
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df ????d?
?
d?
df
x
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df ???(?,?) d?
d?
df
?
?
x
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df ???(?,?) d?
d?
df
x
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df ???(?,?) d?
BRDF sr -1
d?
df
x
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df ?(?,?) fcos? d?
f
df
?
d?
x
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f ???(?,?) fcos? d?
?
f
x
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f
x
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Formulating a balance equation
light leaving a surface
reflected light
emitted light


Easy
Hard
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Classical formulation
Balance equation in terms of radiance Polyak,
1960
measure on sphere
solid angle
source term
r
u
?
u
r
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Classical formulation
Important features of the classical formulation
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Two linear operators
f (r,u)
k?(ru u)
( K f ) (r,u) ?
?
cosine weighted measure
( G f ) (r,u) ?? f (r,u)
implicit function
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Linear operatorsfor global illumination
surface radiance
field radiance
surface radiance
K
G
Field Radiance Operator
Local Reflection Operator
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Another way to writethe rendering equation
f s KG f
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Operator norms
1) First law of thermodynamics
2) Second law of thermodynamics
3) Constancy of radiance along rays
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Irradiance
weighted integral over solid angle
dA
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A vector form of irradiance
Integrate vectors over solid angle
vector irradiance or light vector
r
?
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Lamberts formula for irradiance
polygonal Lambertian luminaire
Vector Irradiance
?i
M
P
?
??r?
?
?
?
i
i
2
?
?i
?
r
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Ideal diffuse reflection
Compute using Lamberts formula
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Ideal diffuse reflection
?
Boundary integral
Eye
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Ideal specular reflection
Compute using ray tracing
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Ideal specular reflection
?
Eye
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Glossy reflection
Use extended Lamberts formula
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Glossy reflection
?
Numerical quadrature
Eye
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Glossy reflection
?
Monte Carlo
Eye
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Boundary integral for glossy reflection
?
boundary integral
Eye
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Applications of directional scattering
glossy reflection
luminaire
directional emission
glossy transmission
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A range of glossy reflections

10th-order moment
45th-order moment
400th-order moment
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Comparison with Monte Carlo
order 65
order 300
order 1000
Region used for comparison
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Comparison with Monte Carlo
order 65
order 300
order 1000
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Monte Carlo integration
luminaire
blocker
estimate irradiance
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Advantages of Monte Carlo

• Arbitrarily complex environments
• Arbitrary reflectance functions
• Small memory requirements
• Easily to distribute
• Relatively easy to implement

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Monte Carlo sampling methods
Phong distribution
Hemisphere
Polygon
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Light-ray tracing
luminaire
Rays represent photons that deposit energy on
surfaces. No inverse-square law here!
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Path tracing
eye
At each scattering event, estimate indirect
irradiance with a single ray continue
recursively.
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Bidirectional path tracing
luminaire
eye
Simultaneously follow paths from the light and
the eye, looking for points that can see each
other.
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Metropolis path tracing
luminaire
eye
Start with a path from eye to luminaire, then
integrate over others by perturbing to nearby
paths.
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A Taxonomy of Errors
Radiance Function Space
Exact Equation
Perturbations
Perturbed Equation
Discretization
Discrete Equation
Computation
Approximation
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Features of surface illumination
luminaire
blocker
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An example of meshing
A simple environment
The underlying mesh
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Classical balance equation
radiance
f(x,?) s(x,?)
?
??(x,???) f(x,?) cos? d?
A point on a distant visible surface
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The change is a pullback
The 2-form on the sphere is pulled back to the
surface
x
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Change of variables
cos?
d? dA
r 2
differential solid angle
differential area
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Kajiyas rendering equation
I(x,x) g(x,x)
?
e(x,x) ??(x,x,x) I(x,x) dx
S
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Kajiyas rendering equation
I(x,x) g(x,x)
?
e(x,x) ??(x,x,x) I(x,x) dx
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Power from transport intensity
Integrate over two surfaces
dx
dx
source
receiver
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Radiance transport intensity
radiance
transport intensity
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Radiance transport intensity
radiance
transport intensity
invariant along lines in free space
obeys inverse square law
defined everywhere
defined only at surfaces
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Another way to writethe rendering equation
f s M f
Radiance
Source
Transport Operator
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The formal solutionto the rendering equation
-1
f ( I - M ) s
Identity operator
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The Neumann series
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Lp-norms for radiance functions


???
???
f p
f (r,u) p
d
m
s2
M
cosine weighted measure
The collection of all functions with finite
Lp-norm is a Banach space
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Significance of the Lp-norms
symbol
meaning
units
total power
watts
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The L1-norm of K
?
K
?
max
max
k
(
r

u
'
?
u
)
d
?
(
u
)
1
'
r
u
?
maximal directional-hemispherical reflectance
over all r and u'
u'
d?
r
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The L -norm of K
?
?
'
?
max
max
k
(
r

u
?
u
)
d
?
(
u
)
'
r
u
?
maximal hemispherical-directional reflectance
over all r and u'
u'
d?
r
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The Lp-norms of K
energy conservation
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The G operator
An enclosure.
Surface radiance function
Equivalent flow through fictitious boundary
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Hilbert adjoint operators
K K
G G
Since
and
it follows that
M I - KG
M I - GK