Advanced Global Illumination

Advanced Global Advanced Global IlluminationIllumination

Brian ChenBrian Chen

Thursday, February 12, 2004Thursday, February 12, 2004

Physical Simulation Physical Simulation of Light Transport:of Light Transport:

AccuracyAccuracyaccount for ALL light pathsaccount for ALL light pathsconservation of energyconservation of energyPredictionPredictionforward renderingforward renderingcalculate light meter readingscalculate light meter readingsAnalysisAnalysisinverse renderinginverse rendering! find surface properties !! find surface properties !Realism?Realism?perceptually necessary?perceptually necessary?

Global IlluminationGlobal Illumination

Global IlluminationGlobal Illumination ““Everything is lit by Everything is lit by Everything ElseEverything Else””

Screen color = Screen color = entire scene’s lightingentire scene’s lighting * surface reflectance * surface reflectance Refinements: Models of area light sources, caustics, Refinements: Models of area light sources, caustics,

soft-shadowing, fog/smoke, photometric calibration, … soft-shadowing, fog/smoke, photometric calibration, …

H. Rushmeier et al., SIGGRAPH`98 Course 05 “A Basic Guide to Global Illumination”H. Rushmeier et al., SIGGRAPH`98 Course 05 “A Basic Guide to Global Illumination”

OverviewOverview

Physics (Optics) conceptsPhysics (Optics) concepts Models of LightModels of Light RadiometryRadiometry BRDFs & Rendering EquationBRDFs & Rendering Equation Monte Carlo MethodsMonte Carlo Methods Photon MappingPhoton Mapping

Optics DefinitionsOptics Definitions Reflection – when light bounces off a surfaceReflection – when light bounces off a surface

Refraction – the bending of light when it goes Refraction – the bending of light when it goes through different mediums (ex. air and glass)through different mediums (ex. air and glass)

Diffraction – the phenomenon where light Diffraction – the phenomenon where light “bends” around obstructing objects“bends” around obstructing objects

Dispersion – when white light splits into its Dispersion – when white light splits into its components colors, result of refractioncomponents colors, result of refraction

Light’s Dual NatureLight’s Dual Nature

Is light a wave? Or is it a beam of particles?Is light a wave? Or is it a beam of particles?

Wave Theory of Light – Thomas Young’s Wave Theory of Light – Thomas Young’s double-slit experimentdouble-slit experiment

Particle Theory of Light – Newton, refraction Particle Theory of Light – Newton, refraction & dispersion& dispersion

Light ModelsLight Models

Models of light are used to capture/explain the Models of light are used to capture/explain the different behaviors of light that stem from its different behaviors of light that stem from its dual naturedual nature

Quantum OpticsQuantum Optics Wave ModelWave Model Geometric OpticsGeometric Optics

Quantum OpticsQuantum Optics

Explains dual wave-particle nature at the Explains dual wave-particle nature at the submicroscopic level of electronssubmicroscopic level of electrons

Way too detailed for purposes of image Way too detailed for purposes of image generation for computer graphics scenesgeneration for computer graphics scenes

Therefore, it is not usedTherefore, it is not used

Wave ModelWave Model

Simplification of Quantum ModelSimplification of Quantum Model Captures effects that occur when light interacts Captures effects that occur when light interacts

with objects of size comparable to the with objects of size comparable to the wavelength of light (diffraction, polarization)wavelength of light (diffraction, polarization)

This model is also ignored, too complex and This model is also ignored, too complex and detaileddetailed

Geometric OpticsGeometric Optics

Simplest and most commonly used light modelSimplest and most commonly used light model Assumes light is emitted, reflected, and Assumes light is emitted, reflected, and

transmitted (refraction)transmitted (refraction) Also assumes:Also assumes:

Light travels in straight lines, no diffractionLight travels in straight lines, no diffraction Light travels instantaneously through a medium Light travels instantaneously through a medium

(travels at infinite speed)(travels at infinite speed) Light is not influenced by gravity, magnetic fieldsLight is not influenced by gravity, magnetic fields

RadiometryRadiometry

Radiometry – area of study involved in the Radiometry – area of study involved in the physical measurement of lightphysical measurement of light

Goal of illumination algorithm is to compute Goal of illumination algorithm is to compute the steady-state distribution of light energy in the steady-state distribution of light energy in a scenea scene

Radiometric QuantitiesRadiometric Quantities

Radiant Power Radiant Power flux (in Watts = Joule/sec), flux (in Watts = Joule/sec), ΦΦ How much total energy flows from/to/through a How much total energy flows from/to/through a

surface per unit timesurface per unit time Irradiance (“ear”)Irradiance (“ear”)

incoming radiant power on a surface per unit incoming radiant power on a surface per unit surface area (Watt/msurface area (Watt/m22))

E = dE = dΦΦ/dA/dA

More Radiometric QuantitiesMore Radiometric Quantities

RadiosityRadiosity Outgoing radiant power per unit surface area Outgoing radiant power per unit surface area

(Watt/m(Watt/m22)) B = dB = dΦΦ/dA/dA

RadianceRadiance Flux per unit projected area per unit solid Flux per unit projected area per unit solid

angle (W/(steradian x angle (W/(steradian x mm22)))) How much power arrives at (or leaves from) a How much power arrives at (or leaves from) a

certain point on a surface, per unit solid angle, certain point on a surface, per unit solid angle, and per unit projected areaand per unit projected area

L(x, L(x, ӨӨ) = d) = d22ΦΦ / (d / (dωω dA cos dA cosӨӨ)) The most important quantity in global The most important quantity in global

illumination because it captures the illumination because it captures the “appearance” of objects“appearance” of objects

Radiance cont’dRadiance cont’d

Radiance: Radiance: The Pointwise Measure of LightThe Pointwise Measure of Light Free-space light power Free-space light power LL ==(energy/time) ==(energy/time)

At At leastleast a 5D scalar function: a 5D scalar function: L(x, y, z, L(x, y, z, , , , …), …) Position (x,y,z), Angle (Position (x,y,z), Angle (,,) and more (t, ) and more (t, , …) , …) Power density units, but tricky…Power density units, but tricky…

SolidAnglereaProjectedAtime

energyL

**

Taken with permission from Jack Tumblin

Yet More RadianceYet More RadianceTricky: think Hemispheres Tricky: think Hemispheres

with a floor:with a floor:

Solid AngleSolid Angle (steradians) (steradians)=d=dωω = fraction of a = fraction of ahemisphere’s area (4hemisphere’s area (4))

dAdA

Projected AreaProjected Area

cos cos dA dA

SolidAnglereaProjectedAtime

energyL

**

Taken with permission from Jack Tumblin

Properties of RadianceProperties of Radiance 1: Radiance is invariant along straight paths and does 1: Radiance is invariant along straight paths and does

not attenuate with distancenot attenuate with distance Radiance leaving point x directed towards point y is equal Radiance leaving point x directed towards point y is equal

to the radiance arriving at point y from the point xto the radiance arriving at point y from the point x 2: Sensors, such as cameras and human eye, are 2: Sensors, such as cameras and human eye, are

sensitive to radiancesensitive to radiance The response of our eyes is proportional to the radiance The response of our eyes is proportional to the radiance

incident upon them.incident upon them. It is now clear that radiance is the quantity that global It is now clear that radiance is the quantity that global

illumination algorithms must compute and display to the illumination algorithms must compute and display to the observerobserver

BRDF (Bidirectional Reflectance BRDF (Bidirectional Reflectance Distribution Function)Distribution Function)

Intuitively the BRDF represents, for each Intuitively the BRDF represents, for each incoming angle, the amount of light that is incoming angle, the amount of light that is scattered in each outgoing anglescattered in each outgoing angle

BRDF at a point x is defined as the ratio of the BRDF at a point x is defined as the ratio of the radiance reflected in an exitant direction (radiance reflected in an exitant direction (ӨӨ), ), and the irradiance incident in a differential and the irradiance incident in a differential solid angle (solid angle (ΨΨ).).

ffrr(x,(x,ΨΨ->->ӨӨ) = dL(x->) = dL(x->ӨӨ) / (L(x<-) / (L(x<-ΨΨ)cos(N)cos(Nxx,,ΨΨ)d)dωωΨΨ)) cos(Ncos(Nxx,,ΨΨ) = cos of angle formed by normal & ) = cos of angle formed by normal &

incident direction vectorincident direction vector

Point-wise Reflectance: Point-wise Reflectance: BRDFBRDFBBidirectional idirectional RReflectance eflectance DDistribution istribution FFunctionunction

((i i , , i i , , r r , , r r , , i i , , r , …r , … )) == (L == (Lrr / L / Lii) ) a scalar (sra scalar (sr-1-1))

Illuminant Illuminant LLii

Reflected Reflected LLrr

InfinitesimalInfinitesimalSolid AngleSolid Angle

BRDF PropertiesBRDF Properties

Value of BRDF remains unchanged if the Value of BRDF remains unchanged if the incident and outgoing directions are incident and outgoing directions are interchangedinterchanged

Law of conservation of energy requires that Law of conservation of energy requires that the total amount of power reflected all the total amount of power reflected all directions must be less than or equal to the directions must be less than or equal to the total amount of power incident on the surfacetotal amount of power incident on the surface

BRDF Examples, DiffuseBRDF Examples, Diffuse

Andrew Glassner et al.. SIGGRAPH`94 Course 18:Andrew Glassner et al.. SIGGRAPH`94 Course 18:““Fundamentals and Overview of Computer Graphics”Fundamentals and Overview of Computer Graphics”

Fr = (x,ΨΨ<-><->ӨӨ) = ) = ρρdd//ππ, constant, constant

ρρdd is the fraction of incident energy that is reflected at a is the fraction of incident energy that is reflected at a

surface, 0-1surface, 0-1

BRDF Examples, SpecularBRDF Examples, Specular


Exitant direction:

R = 2(N dot ΨΨ)N – )N – ΨΨ

For Most SurfacesFor Most Surfaces


Most Materials:Most Materials:Combination ofCombination ofDiffuse & Diffuse & Specular…their Specular…their BRDF is difficult BRDF is difficult to model with to model with formulasformulas

Practical Shading Models (BRDF)Practical Shading Models (BRDF)

Lambert: kLambert: kdd = = ρρd/d/ππ

Phong: fr = kPhong: fr = kss*((R*((R··ӨӨnn)/(N)/(N··ΨΨ)) + k)) + kdd

Blinn-Phong = kBlinn-Phong = kss*((N*((N·H·H)/(N)/(N··ΨΨ)) + k)) + kdd, ,

H=halfway vector between H=halfway vector between ΨΨ & & ӨӨ Modified Blinn-Phong = kModified Blinn-Phong = kss*(N*(N·H)·H)nn + + kkdd Cook-TorranceCook-Torrance WardWard

Lambert BRDFLambert BRDF

Phong BRDFPhong BRDF

Blinn-Phong BRDFBlinn-Phong BRDF

Ward Anisotropic BRDFWard Anisotropic BRDF

Rendering EquationRendering EquationFinally! Putting radiance and the BRDF together to get:

BRDF & Rendering Eq.BRDF & Rendering Eq. For example, suppose that we wish to For example, suppose that we wish to

determine the illumination of a scene determine the illumination of a scene containing containing nn light sources – light source 1 to light sources – light source 1 to light source light source nn. In this case, the local . In this case, the local illumination of a surface is given by,illumination of a surface is given by,

where where LijLij is the intensity of the jth light source is the intensity of the jth light source and and wijwij = ( = (ij,ij,ijij) is the direction to the jth ) is the direction to the jth light source. light source.

n

j

ji

jioo

ji

jio LBRDFL

1

cos),,,(

BRDF & Rendering Eq.BRDF & Rendering Eq.

For a single point light source, the light For a single point light source, the light reflected in the direction of an observer isreflected in the direction of an observer is

This is the general BRDF lighting equation for This is the general BRDF lighting equation for a single point light source a single point light source

iiooiio LBRDFL cos),,,(

Rendering EquationRendering EquationOpportunitiesOpportunities Scalar operations only: Scalar operations only: ()() and and L()L(), indep. of , indep. of , x,y,z, , x,y,z, ,, … … Linearity: Linearity:

Solution = weighted sum of one-light solns.Solution = weighted sum of one-light solns. Many BRDFs Many BRDFs weighted sum of diffuse, specular, gloss terms weighted sum of diffuse, specular, gloss terms

DifficultiesDifficulties Almost Almost nono nontrivial analytic solutions exist; nontrivial analytic solutions exist;

MUST use approximate methods to solveMUST use approximate methods to solve Verification: tough to measure real-world Verification: tough to measure real-world ()() and and L() L() wellwell Notable wavelength-dependent surfaces exist Notable wavelength-dependent surfaces exist

(iridescent insect wings & casing, CD grooves, oil)(iridescent insect wings & casing, CD grooves, oil) BRDF doesn’t capture important subsurface scatteringBRDF doesn’t capture important subsurface scattering

Review 1Review 1Big Ideas:Big Ideas:

Measure Light: RadianceRadiance

Measure Light Attenuation: : BRDFBRDF

Light will ‘bounce around’ endlessly, decaying on each bounce: The Rendering Equation The Rendering Equation (intractable: must approximate)(intractable: must approximate)

Monte Carlo TechniquesMonte Carlo Techniques

Mathematical techniques that use statistical Mathematical techniques that use statistical sampling to simulate phenomena or evaluate sampling to simulate phenomena or evaluate values of functionsvalues of functions

Use a probability distribution function (PDF) Use a probability distribution function (PDF) to generate random samples, p(x)to generate random samples, p(x)

EstimatorsEstimators

Unbiased – when the expected value of the Unbiased – when the expected value of the estimator is exactly the value of the integralestimator is exactly the value of the integral

Biased – when the above property is not trueBiased – when the above property is not true Bias – the difference between the expected Bias – the difference between the expected

value of the estimator and the actual value of value of the estimator and the actual value of the integralthe integral

As number of samples N increases, the As number of samples N increases, the estimate becomes closer estimate becomes closer


The reliability of Monte-Carlo sampling is The reliability of Monte-Carlo sampling is measured by the variance of the estimators. measured by the variance of the estimators. The variance of the estimators is given as The variance of the estimators is given as

where is the primary estimator, while its where is the primary estimator, while its average is a secondary estimatoraverage is a secondary estimator


As a consequence of the above formula, the Monte-As a consequence of the above formula, the Monte-Carlo technique requires a large number of samples to Carlo technique requires a large number of samples to reduce the variance of the primary estimator reduce the variance of the primary estimator

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


For importance sampling we adjust the pdf For importance sampling we adjust the pdf pp to be to be similar in shape to the integrand similar in shape to the integrand ff. If we were able to . If we were able to choose choose pp((xx)= )= CfCf((xx), with a constant factor ), with a constant factor CC= 1/= 1/II, the , the variance of the primary estimator would be zero and a variance of the primary estimator would be zero and a single sample would always give the correct result. single sample would always give the correct result.

Unfortunately, the constant C is determined by the Unfortunately, the constant C is determined by the integral we want to compute and is therefore integral we want to compute and is therefore unavailable. On the other hand, some a priori unavailable. On the other hand, some a priori knowledge about the shape of knowledge about the shape of ff is often available and is often available and can be used to adjust can be used to adjust pp to reduce the variance of the to reduce the variance of the estimators. estimators.

Monte Carlo ExampleMonte Carlo Example

Computing a one-dimensional integralComputing a one-dimensional integral Samples are selected randomly over the domain of Samples are selected randomly over the domain of

the integral to get a close approximation (estimator the integral to get a close approximation (estimator 

 = (1/N) = (1/N) ΣΣ f(xi) / p(xi) f(xi) / p(xi)

Another ExampleAnother Example

Integration over a hemisphereIntegration over a hemisphere Estimate the radiance at a point by integrating the Estimate the radiance at a point by integrating the

contribution of light sources in the scenecontribution of light sources in the scene Light source LLight source L

I = I = ∫∫LLsourcesourcecoscosddωω = = ∫∫0022ππ ∫ ∫0 0

ππ/2 /2 LLsourcesourcecoscossinsinddddφφ

=(1/N)=(1/N)Σ Σ ((LLsourcesource((ωωii) cos) cossinsin) / p() / p(ωωii))

p(p(ωωii) = cos) = cossinsin/ / ππ

=(1/N)=(1/N)Σ Σ LLsourcesource((ωωii))

StepsSteps General:General:

Sampling according to a probability distribution functionSampling according to a probability distribution function Evaluation of the function at that sampleEvaluation of the function at that sample Averaging these appropriately weighted sampled valuesAveraging these appropriately weighted sampled values

Graphics:Graphics: Generate random photon paths from source (lights or Generate random photon paths from source (lights or

pixels)pixels) Set a discrete random length for the pathSet a discrete random length for the path Count how many photons terminate in state i, average Count how many photons terminate in state i, average

radiances for that pointradiances for that point

http://graphics.stanford.edu/courses/cs348b-02/lectures/lecture15/walk014.html

http://graphics.stanford.edu/courses/cs348b-02/lectures/lecture15/walk014.html

Advantages/DisadvantagesAdvantages/Disadvantages

AdvantagesAdvantages Simple: just sample signal/function and average estimatesSimple: just sample signal/function and average estimates Widely applicable: nuclear physics, graphics, high-Widely applicable: nuclear physics, graphics, high-

dimensional integrations of complicated functionsdimensional integrations of complicated functions Can be used with radiosityCan be used with radiosity

DisadvantagesDisadvantages Very slow, take many, many samples to converge to Very slow, take many, many samples to converge to

correct solutioncorrect solution Need 4x samples to decrease error by halfNeed 4x samples to decrease error by half

Monte Carlo ImagesMonte Carlo Images

Monte Carlo ImagesMonte Carlo Images

Photon MappingPhoton Mapping

One of the “fastest” algorithms available for One of the “fastest” algorithms available for global illuminationglobal illumination

Monte-Carlo basedMonte-Carlo based 2-pass algorithm 2-pass algorithm

Photon Mapping, 1Photon Mapping, 1stst pass pass 1st pass1st pass

photons shot from the light into the scenephotons shot from the light into the scene they bounce around interacting with all the types of they bounce around interacting with all the types of

surfaces they encountersurfaces they encounter Clever twistsClever twists

11stst, instead of redoing those same computations over and over, a , instead of redoing those same computations over and over, a few thousand of time for each pixels, the photons are stored only few thousand of time for each pixels, the photons are stored only once in a special data structure called a photon map for later reuseonce in a special data structure called a photon map for later reuse

22ndnd, instead of trying to completely fill the whole scene with , instead of trying to completely fill the whole scene with billions of photons, a few thousands to a million photons are billions of photons, a few thousands to a million photons are sparsely stored and the rest is statistically estimated from the sparsely stored and the rest is statistically estimated from the density of the stored photons.density of the stored photons.

After all the photons have been stored in the map, a After all the photons have been stored in the map, a statistical estimate of the irradiance at each photon statistical estimate of the irradiance at each photon position is computed.position is computed.

Photon Mapping, 2Photon Mapping, 2ndnd Pass Pass

Direct illumination is computed just like Direct illumination is computed just like regular ray tracing, but the indirect regular ray tracing, but the indirect illumination, which comes from the walls and illumination, which comes from the walls and other objects around, is computed from other objects around, is computed from querying the stored photons in the photon mapquerying the stored photons in the photon map

At each secondary hit, the photon map is At each secondary hit, the photon map is queried in order to gather the radiance coming queried in order to gather the radiance coming from the objects around in the environment.from the objects around in the environment.

Photon Mapping ImagesPhoton Mapping Images

SummarySummary

Global illumination’s goal is to calculate the steady-Global illumination’s goal is to calculate the steady-state distribution of light energy in a scenestate distribution of light energy in a scene

BRDFs used to calculate reflection radiancesBRDFs used to calculate reflection radiances Rendering Equation used to calculate the radiance Rendering Equation used to calculate the radiance

sent out from an objectsent out from an object Monte-Carlo techniques used to calculate irradiance Monte-Carlo techniques used to calculate irradiance

at random locations throughout sceneat random locations throughout scene Photon Maps used to accurately calculate caustics, Photon Maps used to accurately calculate caustics,

inter-object reflectionsinter-object reflections

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Advanced Global Illumination