April 19, 2012; CS 354 Computer Graphics; University of Texas at Austin

1. CS 354
Global Illumination
Mark Kilgard
University of Texas
April 19, 2012

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Today’s material
 In-class quiz
 On ray casting & tracing lecture
 Lecture topic
 Project 4
 Global illumination

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My Office Hours
 Tuesday, before class
 Painter (PAI) 5.35
 8:45 a.m. to 9:15
 Thursday, after class
 ACE 6.302
 11:00 a.m. to 12
 Randy’s office hours
 Monday & Wednesday
 11 a.m. to 12:00
 Painter (PAI) 5.33

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Last time, this time
 Last lecture, we discussed
 Ray casting and tracing
 This lecture
 Global illumination
 Projects
 Project 3 due yesterday
 Project 4 on ray tracing on Piazza
 Due May 2, 2012

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On a sheet of paper
Daily Quiz • Write your EID, name, and date
• Write #1, #2, #3 followed by its answer
 Multiple choice: determining
if a ray intersects a sphere  Multiple choice: With
involves solving a distribution ray tracing, one can
accomplish
a) linear equation
a) depth-of-field
b) quadratic equation
b) soft shadows
c) system of linear equations
c) motion blur
d) cubic equation
d) a. and b. but not c.
 True or False: With ray
tracing, the problem of anti- e) a., b., and c.
aliasing doesn’t apply.

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Solving the Rendering Equation
with Ray Tracing
 Ray “tracing” or “shooting”
 Given a point and direction, sample radiance
 Building block for global illumination
algorithms
 Use ray traces to approximate integrals
 Conceptually, solving the rendering equation
 Intractable to shoot every ray
 Instead use Monte Carlo techniques

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Aspects of Modeling Light
 Geometric optics
 Our concern
 Assume light travels instantaneously
 Light travels in straight lines
 Light not influenced by gravity or magnetic fields
 Wave optics
 Diffraction, etc.
 Quantum optics
 Sub-microscopic

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Rendering Equation
 Theory for light-surface interactions
Lo (x, ω , λ , t ) = Le (x, ω , λ , t ) + ∫ f r (x, ω ′, ω , λ , t ) Li (x, ω ′, λ , t ) (−ω ′ • n) dω ′
Ω

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Rendering Equation Parts
Lo (x, ω , λ , t ) = Le (x, ω , λ , t ) + ∫ f r (x, ω ′, ω , λ , t ) Li (x, ω ′, λ , t ) (−ω ′ • n) dω ′
Ω
 Lo = outgoing light from x in direction ω
 x = point on a surface
 ω = normalized outgoing light vector
 λ = wavelength of light
 t = time
 Le = emitted light at x going towards from ω
 n = surface normal at x

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Rendering Equation Integral
 ∫ = integrate over a region
 Ω = region of a hemisphere
 Together: “integrate overall the incoming directions
for a hemisphere at a point x”
 ωˊ = normalized incoming light vector
 Li = incoming light at x coming from ωˊ
 n = surface normal at x
 (-ωˊ • n) = cosine of angle between incoming
light and surface normal
 dωˊ = differential of incoming angle

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Bidirectional Reflectance
Distribution Function
 Ratio of differential outgoing (reflected) radiance
to differential incoming irradiance
dLr (x, ω , λ , t ) dLr (x, ω , λ , t )
f r (x, ω ′, ω , λ , t ) = =
dEi (x, ω ′, λ , t ) dLi (x, ω ′, λ , t ) (−ω ′ • n) dω ′
 Physically based BRDF properties L = radiance
 Must be non-negative E = irradiance
 Must be reciprocal
 Swap incoming & relected directions generates same ratio
 Conserves energy
 Integrating over entire hemisphere must be ≤ 1

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Cosine Weighting for Irradiance
 At shallow angles, incoming light spreads over a
wider area of the surface
 Thus spreading the energy
 Thus dimming the received light

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Measuring BRDFs Empirically
 Gonioreflectometer is device to measure
BRDFs

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BRDF Decomposition
 Broad characterization of BRDF
appearance
Pure diffuse Pure specular Glossy

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Common BRDFs
 Diffuse model, a.k.a. Lambertian
f diffuse (x, Ψ ↔ Θ, λ , t ) = k d
( R • Θ) n
 Phongphong (x, Ψ ↔a.k.a.=Reflection dSpecular
f model, Θ, λ , t ) k s +k
N •Ψ
( N • H )n
f blinn (x, Ψ ↔ Θ, λ , t ) = k s + kd
 Blinn model, a.k.a. Half-angle Specular N •Ψ

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Hemisphere over a Surface
 Incident radiance is integrated over the
hemisphere

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Rendering Equation Interpretation
 Recursive equation
Lo (x, ω , λ , t ) = Le (x, ω , λ , t ) + ∫ f r (x, ω ′, ω , λ , t ) Li (x, ω ′, λ , t ) (−ω ′ • n) dω ′
Ω
outgoing light “sums up” all incoming light
for all directions on hemisphere
 Integral over hemisphere (Ω)
 Limitations
 Treats wavelengths all independent
 Treats time in Newtonian way
 Ignores volumetric and subsurface scattering
 More complex models can incorporate these aspects of light
 Impractical for actual computer graphics rendering

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Alternate Rendering Equation
 Integrate over all points in the scene
 Instead of all directions for a hemisphere
Lo (x, ω , λ , t ) = Le (x, ω , λ , t ) + ∫ f (x, ω
y∈Γ
r yx , ω , λ , t ) L(y, ωyx , λ , t ) G (x, y ) dy
outgoing light “sums up” all incoming light
for all surfaces, modulated by visibility
 Notes
 G(x,y) returns the visibility (occlusion) between points x & y
 Integral over all surface points (Γ) in the scene

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Two Versions of
Rendering Equation
Occlusion (G) is zero
o
Le (x, ω , λ , t ) +
Le (x, ω , λ , t ) +
Lo (x, ω , λ , t ) =
∫
Ω
f r (x, ω ′, ω , λ , t ) Li (x, ω ′, λ , t ) (−ω ′ • n) dω ′ Lo (x, ω , λ , t ) =
∫ f r (x, ωyx , ω , λ , t ) L(y, ωyx , λ , t ) G (x, y ) dy
y∈Γ
Integrate over hemisphere Integrate over all surface points

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Discrete Integral Approximation
 Uniform sampling
 Shown in 1D, but we need to integrate over
hemisphere (solid angles)

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Hemisphere Sampling
 Vary Euler angles

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Monte Carlo Integrations
 Randomly sample the rendering equation
integrals
 Sample space based on probability
distribution function
 Compute estimated variance to know when to
stop
 Importance sampling
 Try to weight probability function to match
variance

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Ray Casting for Initial Samples

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Path Tracing via
Simple Stochastic Ray Tracing

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Two Versions of
Rendering Equation (again) Occlusion (G) is zero
o
Le (x, ω , λ , t ) +
Le (x, ω , λ , t ) +
Lo (x, ω , λ , t ) =
∫
Ω
f r (x, ω ′, ω , λ , t ) Li (x, ω ′, λ , t ) (−ω ′ • n) dω ′ Lo (x, ω , λ , t ) =
∫ f r (x, ωyx , ω , λ , t ) L(y, ωyx , λ , t ) G (x, y ) dy
y∈Γ
Integrate over hemisphere Integrate over all surface points

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Sufficient Shadow Ray Sampling

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Light Tracing
 Trace rays from the light
 Contribution rays accumulate image samples

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Light Tracing Needs Lots of
Samples

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How many rays to shoot?
 Two approaches
 Fixed number of ray evaluations
 Limits the depth of recursion
 Means some paths can never be explored
 Implies bias in result, since favors short paths
 Russian Roulette approach
 Roll dice to see if ray tracing should continue
 Reduces bias in rendering result
 Allows “obscure” paths to be explored

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Radiosity History
 Used for thermal engineering since 1960s
 Introduced to computer graphics in 1984
 SIGGRAPH paper: “Modeling the Interaction
of Light Between Diffuse Surfaces”
 Goral, Torrance, Greenberg, Battaile (Cornell)
 Reduces to solving a linear system of
equations
 In simple formulation, just works for diffuse
surfaces
 OK for building interiors

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Radiosity Example

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Scene Diced into Uniform Patches

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Radiosity Method
Self-emitted reflectivity total
radiosity radiosity
Form factor
Approximation to general Rendering Equation

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Form Factors
 Form factor between i,j is Fij
 Tells how much of patch A is visible from patch B
 Indicates how much radiance can be transferred
1
Fij = ∫ ∫ K ( x, y ) dAy dAx
Ai Si Si
K ( x , y ) = G ( x, y ) V ( x, y )
cos(Θ xy , N x ) cos(−Θ xy , N y )
G ( x, y ) =
π rxy
2
V ( x, y ) =< visibility between x & y >

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Form Factor Visualized
 Mutual visibility of two patches
 Don’t have to be same size

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Differential Patches and
Foreshortening
 Cosine fall-off applies
 And assuming Lambertian surfaces
foreshortening

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Hemi-cube Approach to Sampling
Form Factors
 Think about drawing hemi-cube at every point
 What you see is an input for radiosity
 Known as “instant radiosity”

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Interpolation of Radiosity Solution
Constant Approximation “true” solution Quadratic Approximation
flat smooth

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Iterative Radiosity Solver
1 iteration 4 iterations 16 iterations
64 iterations 252 iterations

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Participating Media
 Light doesn’t just interact with surfaces
 Gaseous phenomena affects lighting
 Usually handled statistically
emission
in-scattering [Jarosz et.al. 2008]
absorption
out-scattering [FogShop 2007]

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Participating Media Assumptions
 Simpler
 Single scattering
 Uniform
 Ignore shadowing
 More complex
 Multiple scatterings
 Non-uniform
 Account for shadowing

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Photon Mapping
 Two-pass global illumination algorithm
 Developed by Henrick Jensen (1996)
 Two passes
 Randomly distribute photons around the scene
 Called “photon map construction”
 Render treating photons as mini-light sources
 Capable of efficiently generating otherwise very
expensive effects
 Caustics
 Diffuse inter-reflections, such as color bleed
 Sub-surface scattering

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Photon Mapping Examples
without diffuse photo map
caustics interreflection visualization
sub-surface scattering
diffuse interreflection

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Next Class
 Next lecture
 Acceleration structures
 How do we make the ray trace operation faster?
 Reading
 Chapter 5, 297-298
 Chapter 11, 569-577
 Project 4
 Project 4 is a simple ray tracer
 Due Wednesday, May 2, 2012