CS 354Global Illumination

Mark KilgardUniversity of TexasApril 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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Daily Quiz1. Multiple choice: determining

if a ray intersects a sphere involves solving a

a) linear equation

b) quadratic equation

c) system of linear equations

d) cubic equation

2. True or False: With ray tracing, the problem of anti-aliasing doesn’t apply.

3. Multiple choice: With distribution ray tracing, one can accomplish a) depth-of-field

b) soft shadows

c) motion blur

d) a. and b. but not c.

e) a., b., and c.

On a sheet of paper• Write your EID, name, and date• Write #1, #2, #3 followed by its answer

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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

dtLtftLtL ireo )(),,,(),,,,(),,,(),,,( nxxxx

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Rendering Equation Parts

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

dtLtftLtL ireo )(),,,(),,,,(),,,(),,,( nxxxx

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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

Physically based BRDF properties Must be non-negative Must be reciprocal

Swap incoming & relected directions generates same ratio Conserves energy

Integrating over entire hemisphere must be ≤ 1

dtdL

tdL

tdE

tdLtf

i

r

i

rr )(),,,(

),,,(

),,,(

),,,(),,,,(

nx

x

x

xx

L = radiance

E = irradiance

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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

Phong model, a.k.a. Reflection Specular

Blinn model, a.k.a. Half-angle Specular

ddiffuse ktf ),,,( x

d

n

sblinn kN

HNktf

)(

),,,( x

d

n

sphong kN

Rktf

)(

),,,( x

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Hemisphere over a Surface

Incident radiance is integrated over the hemisphere

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Rendering Equation Interpretation

Recursive equation

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

dtLtftLtL ireo )(),,,(),,,,(),,,(),,,( nxxxx

outgoing light “sums up” all incoming light

for all directions on hemisphere

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Alternate Rendering Equation

Integrate over all points in the scene Instead of all directions for a hemisphere

Notes G(x,y) returns the visibility (occlusion) between points x & y Integral over all surface points (Γ) in the scene

yyxyxxx yx

y

yx dGtLtftLtL reo ),(),,,(),,,,(),,,(),,,(

outgoing light “sums up” all incoming light

for all surfaces, modulated by visibility

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o

Two Versions ofRendering Equation

dtLtf

tLtL

ir

e

o )(),,,(),,,,(

),,,(),,,( nxx

xx

yyxyx

xx

yx

y

yx dGtLtf

tLtL

r

e

o ),(),,,(),,,,(

),,,(),,,(

Integrate over hemisphere Integrate over all surface points

Occlusion (G) is zero

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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 viaSimple Stochastic Ray Tracing

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o

Two Versions ofRendering Equation (again)

dtLtf

tLtL

ir

e

o )(),,,(),,,,(

),,,(),,,( nxx

xx

yyxyx

xx

yx

y

yx dGtLtf

tLtL

r

e

o ),(),,,(),,,,(

),,,(),,,(

Integrate over hemisphere Integrate over all surface points

Occlusion (G) is zero

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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-emittedSelf-emittedradiosityradiosity

reflectivityreflectivity total total radiosityradiosity

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

i iS S xy

iij dAdAyxK

AF ),(

1

),(),(),( yxVyxGyxK

2

),cos(),cos(),(

xy

yxyxxy

r

NNyxG

yxbetweenvisibilityyxV &),(

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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 SolutionConstant ApproximationConstant Approximation ““true” solutiontrue” solution Quadratic ApproximationQuadratic Approximation

flat smooth

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Iterative Radiosity Solver

1 iteration1 iteration 4 iterations4 iterations 16 iterations16 iterations

64 iterations64 iterations 252 iterations252 iterations

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Participating Media Light doesn’t just interact with surfaces

Gaseous phenomena affects lighting Usually handled statistically

absorption

in-scattering

out-scattering

emission[Jarosz et.al. 2008]

[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

caustics

diffuse interreflection

without diffuseinterreflection

photo mapvisualization

sub-surface scattering

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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