Lecture 8: Monte Carlo Rendering - cs. · PDF fileLecture 8: Monte Carlo Rendering ... • Problem for higher dimensions ... N-Rooks Sampling - 9 rays not stratified stratified N-Rooks

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Lecture 8: Monte Carlo Rendering

Chapters 4 and 5 in Advanced GI

Fall 2004Kavita Bala

Computer ScienceCornell University

© Kavita Bala, Computer Science, Cornell University

Homework• HW 1out, due Oct 5

• Assignments done separately– Might revisit this policy for later assignments

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Rendering Equation

∫Ω

Ψ⋅Ψ⋅Ψ←⋅Θ↔Ψ+Θ→=Θ→x

dnxLfxLxL xre ω),cos()()()()(

∫Ω

⋅⋅+=x

re fL cos

function to integrate over allincoming directions over thehemisphere around x

Value we want


How to compute?

L(x→Θ) = ?

Check for Le(x→Θ)

Now add Lr(x→Θ) = L=?

∫Ω

Ψ⋅Ψ⋅Ψ←⋅Θ↔Ψx

dnxLf xr ω),cos()()(

3


How to compute?• Monte Carlo!

• Generate random directions on hemisphere Ωx, using pdf p(Ψ)

∫Ω

Ψ⋅Ψ⋅Ψ←⋅Θ↔Ψ=Θ→x

dnxLfxL xr ω),cos()()()(

∑= Ψ

Ψ⋅Ψ←⋅Θ↔Ψ=Θ→

N

i i

xiiir

pnxLf

NxL

1 )(),cos()()(1)(


How to compute?

• evaluate L(x←Ψi)?

• Radiance is invariant along straight paths

• vp(x, Ψi) = first visible point

• L(x←Ψi) = L(vp(x, Ψi) → Ψi)

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How to compute? Recursion ...

• Recursion ….

• Each additional bounce adds one more level of indirect light

• Handles ALL light transport

• “Stochastic Ray Tracing”


PPyf )/(

Russian Roulette

Integral

Estimator

Variance

0 1

)(xf

P

∫∫∫ ===P

dyPPyfPdx

PxfdxxfI

0

1

0

1

0

)/()()(

>

≤=.0

,)(

Px

PxPxf

Ii

ii

roulette if

if

σσ >roulette

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Russian Roulette

• Pick some ‘absorption probability’ α– probability 1-α that ray will bounce– estimated radiance becomes L/ (1-α)

• E.g. α = 0.9– only 1 chance in 10 that ray is reflected– estimated radiance of that ray is multiplied by 10– instead of shooting 10 rays, we shoot only 1, but

count the contribution of this one 10 times


Algorithm so far ...• Shoot viewing ray through each pixel

• Shoot # indirect rays, sampled over hemisphere

• Terminate recursion using Russian Roulette

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Pixel Anti-Aliasing

• Compute radiance only at center of pixel: jaggies

• Simple box filter:

• … evaluate using MC

∫=Pixel

dxxLL )(


Stochastic Ray Tracing

• Parameters?– # starting rays per pixel– # random rays for each surface point

(branching factor)

• Path Tracing– Branching factor == 1

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Path tracing

1 ray / pixel 10 rays / pixel 100 rays / pixel

• Pixel sampling + light source sampling folded into one method


Comparison

1 centered viewing ray100 random shadow rays per

viewing ray

100 random viewing rays1 random shadow ray per

viewing ray

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Algorithm so far ...• Shoot # viewing rays through each pixel

• Shoot # indirect rays, sampled over hemisphere– Path tracing shoots only 1 indirect ray

• Terminate recursion using Russian Roulette


Algorithm

?=L ?=L ?0

==

r

e

LL

?0

==

r

e

LL ?=inL ?=inL ?=inL ?=inL

?0

==

r

e

LL

?0

==

r

e

LL

?=L ?=L?=inL ?=inL

?=L ?=L

?234.1

==

r

e

LL

?234.1

==

r

e

LL

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Performance/Error

• Want better quality with smaller number of samples – Fewer samples/better performance– Stratified sampling– Quasi Monte Carlo: well-distributed samples

• Faster convergence– Importance sampling: next-event estimation


0 1

)(xf

Stratified Sampling

• Samples could be arbitrarily close

• Split integral in subparts

• Estimator

• Variance:

∑=

=N

i i

istrat xp

xfN

I1 )(

)(1

secσσ ≤strat

∫∫ ++=NXX

dxxfdxxfI )()(1

K

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Numerical example


Stratified Sampling

9 shadow raysnot stratified

9 shadow raysstratified

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Stratified Sampling




Stratified Sampling



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•• • •

• • ••

• • • •

• • • •

→ N2 samples

2 Dimensions

• Problem for higher dimensions

• Sample points can still be arbitrarily close to each other


Higher Dimensions• Stratified grid sampling:

→ Nd samples

• N-rooks sampling:

→ N samples

•• • •

• • ••

• • • •

• • • •

••

•

•

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N-Rooks Sampling - 9 rays

notstratified

stratified N-Rooks


N-Rooks Sampling - 36 rays

notstratified

stratified N-Rooks

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• How does it relate to regular sampling

Other types of Sampling

Random sampling Regular sampling


Quasi Monte Carlo• Eliminates randomness to find well-

distributed samples• Samples are determinisitic but “appear”

random

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Quasi-Monte Carlo (QMC)• Notions of variance, expected value don’t

apply

• Introduce the notion of discrepancy– Discrepancy mimics variance– E.g., subset of unit interval [0,x]

Of N samples, n are in subsetDiscrepancy: |x-n/N|

– Mainly: “it looks random”


Quasi Monte Carlo

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Example: Hammersley• Radical inverse φp(i) for primes p• Reflect digits (base p) about decimal point

φ2(i): 1110102 → 0.010111

• For N samples, a Hammersley point – (i/N, φ2(i))

• For more dimensions:– Xi =(i/N, φ2(i), φ3(i), φ5(i), φ7(i), φ11(i), ….)


Halton• Radical inverse function

– Inverts bits around decimal point

• Sample: – Where b1, b2, b3 are primes

1)()(

)(

−−∑

∑=Φ

=

j

jjb

j

jj

biai

biai

),...)(),(),((321iiix bbbi ΦΦΦ=

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Quasi Monte Carlo

• Converges as fast as stratified sampling– Does not require knowledge about how many

samples will be used

• Using QMC, directions evenly spaced no matter how many samples are used

• Samples properly stratified-> better than pure MC


Performance/Error

• Want better quality with smaller number of samples – Fewer samples/better performance– Stratified sampling– Quasi Monte Carlo: well-distributed samples

• Faster convergence– Importance sampling: next-event estimation

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Path Tracing

1 sample/pixel 16 samples/pixel 256 samples/pixel

Sample hemisphere

• Importance Sampling: compute direct illumination separately!


Direct Illumination• Paths of length 1 only, between receiver

and light source

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Direct Illumination Global Illumination


Next Event Estimation

Radiance from light sources + radiance from other surfaces

∫Ω

Ψ⋅Ψ⋅Ψ←⋅Θ↔Ψ+Θ→=Θ→x

dnxLfxLxL xre ω),cos()()()()(

∫Ω

⋅⋅+=x

re fL cos

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Next Event Estimation

indirectdirecte LLLxL ++=Θ→ )(

∫∫ΩΩ

⋅⋅+⋅⋅+=xx

rre ffL coscos

• So … sample direct and indirect with separate MC integration


Algorithm

?=L ?=L?0

==

r

e

LL

?0

==

r

e

LL

??0

===

i

d

e

LLL

??0

===

i

d

e

LLL

?=L ?=L

?345.2

0

=≈=

i

d

e

LLL

?345.2

0

=≈=

i

d

e

LLL

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Algorithm


Algorithm

→a variant of path tracing

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Comparison

Without N.E.E. With N.E.E.

16 samples/pixel


Rays per pixel

1 sample/pixel

4 samples/pixel

16 samples/pixel

256 samples/pixel

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How to sample direct illumination• Sampling a single light source

• Sampling for many lights


Direct Illumination

∫ ⋅⋅Ψ→⋅Θ↔Ψ−=Θ→sourceA

yr dAyxGyLxfxL ),()(),()(

x

y

rxy

Vis(x,y)=?

Ψ

Θ

nx

ny

2

),(),cos(),cos(),(

xy

yx

ryxVisnn

yxGΨΘ

=

xΘ

nxΨ

hemisphere integration area integration

−Ψ

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Two forms of the RE• Hemisphere integration

• Area integration (over polygons from set A)

∫ ⋅⋅⋅

⋅Ψ−→⋅Θ↔Ψ+Θ→=Θ→A

yxy

yxre dAyxV

ryLfxLxL ),(

coscos)()()()( 2

θθ

∫Ω

Ψ⋅⋅Ψ←⋅Θ↔Ψ+Θ→=Θ→x

dxLfxLxL xre ωθcos)()()()(


∫Ω

Ψ⋅⋅Ψ←⋅Θ↔Ψ=Θ→x

dxLfxL xrdirect ωθcos)()()(

Rendering Equation

dAy

Ψωd

Ψ

x

2

cos

xy

yy

rdA

dθ

ω =Ψ

),( Ψ= xvpy

)),(()( Ψ−→Ψ=Ψ← xvpLxL

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Rendering Equation: visible surfaces

∫Ω

Ψ⋅⋅Ψ←⋅Θ↔Ψ=Θ→x

dxLfxL xrdirect ωθcos)()()(

=Θ→ )(xLdirect

),( Ψ= xvpy

Integration domain = surface points y on lights

Coordinate transform

∫lights all on y

)( Ψ−→⋅ yL)( Θ↔Ψrf yxy

yx dA

ryxV ⋅⋅ 2

coscos),(

θθ


RE over light sources

∫ ⋅⋅Ψ−→⋅Θ↔Ψ−=Θ→sourceA

ysourcer dAyxGyLxfxL ),()(),()(

xΘ

nx

ny

2

),(coscos),(

xy

yx

ryxVis

yxGθθ

=

area integration

y

Ψ

rxy

Vis(x,y)=?

−Ψ

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Estimator for direct lighting• Pick a point on the light’s surface with pdf

• For N samples, direct light at point x is:

∑=

=N

i i

iyx

yxsourcer

yp

yxVisr

Lf

NxE i

i

1

2

)(

),(coscos

1)(

θθ

)(yp


Generating direct paths• Pick surface points yi on light source• Evaluate direct illumination integral

x

∑=

=Θ→N

i i

ir

ypyxGLf

NxL

1 )(),((...)(...)1)(

Θ

y

rxy

θx

θy

Vis(x,y)=?

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PDF for sampling light• Uniform

• Pick a point uniformly over light’s area– Can stratify samples

• Estimator:

sourceAreayp 1)( =

∑=

=N

ii

yx

yxsourcer

source yxVisr

LfN

AreaxEi

i

12 ),(coscos

)(θθ


∑=

=N

ii

yx

yxsourcer

source yxVisr

LfN

AreaxEi

i

12 ),(coscos

)(θθ

More points ...

1 shadow ray 9 shadow rays

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Even more points ...

36 shadow rays 100 shadow rays

∑=

=N

ii

yx

yxsourcer

source yxVisr

LfN

AreaxEi

i

12 ),(coscos

)(θθ


Different pdfs• Uniform

• Solid angle sampling– Removes cosine and distance from integrand– Better when significant foreshortening

sourceAreayp 1)( =

∑=

=N

i i

iyx

yxsourcer

yp

yxVisr

Lf

NxE i

i

1

2

)(

),(coscos

1)(

θθ

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Parameters• How to distribute paths within light source?

– Uniform– Solid angle– What about light distribution?

• How many paths (“shadow-rays”)?– Total?– Per light source? (~intensity, importance, …)


Scenes with many lights• Many lights in scenes: M lights

• How to handle many lights?

• Formulation 1: M integrals, one per light– Same solution technique as earlier for each

light

∫∑ ⋅⋅Ψ−→⋅Θ↔Ψ−=Θ→= sourceA

ysourcer

M

idAyxGyLxfxL ),()(),()(

1

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Lighting: point sources

lights

eye

x


Scenes with many lights• Various choices:

– Shadow rays per light source– Distribution of shadow rays within a light

source

• Total # rays = M N– Where, M = #lights, N = #rays per source

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Antialiasing: pixel

lights

eye

pixel90 rays

• Anti-aliasing: k M N


Formulation over all lights• When M is large, each direct lighting

sample is very expensive

• We would like to importance sample the lights

• Instead of M integrals

• Formulation over 1 integration domain∫ ⋅⋅Ψ−→⋅Θ↔Ψ−=Θ→lightsallA

ysourcer dAyxGyLxfxL ),()(),()(

∫∑ ⋅⋅Ψ−→⋅Θ↔Ψ−=Θ→= sourceA

ysourcer

M

idAyxGyLxfxL ),()(),()(

1

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Why?• Do not need a minimum of M rays/sample• Can use only one ray/sample

• Still need N samples, but 1 ray/sample

• Ray is distributed over the whole integration domain– Can importance sample the lights


Anti-aliasing

lights

eye

pixel 10 rays 10 rays

• Can piggy-back on the anti-aliasing of pixel

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How to sample the lights?• A discrete pdf pL(ki) picks the light ki

• A surface point is then picked with pdfp(yi|ki)

• Estimator with N samples:∑

=

=N

i iiiL

isourcer

kypkpyxGLf

NxE

1 )|()(),(1)(


Strategies for picking light

– Uniform

– Area

– Power

MkpL

1)( =

∑=

k

kL P

Pkp )(

∑=

k

kL A

Akp )(

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Example for 2 lights• Light 0 has power 1, Light 1 has power 2

• Using power for pdf: – pL(L0) = 1/3, pL(L1) = 2/3

• Overall pdf )(32)(

31)(

10ypypyp LL +=

0.33

L0 L1


Example for 2 lights• Pick a random value:

• If

• Sample Light 0 and compute estimate e0

• Overall estimate is

31

0 <ξ

310e

0ξ

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Example for 2 lights• If

• Sample Light 1 and compute estimate e1

• Overall estimate is

131

0 <≤ ξ

321e


How to sample light?• Once light is picked, can pick two random

numbers according to pL0(y), pL1(y)

• To decrease variance we should reuse

• But, already used information in to pick the light

0ξ

0ξ

21,ξξ

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Example for 2 lights• Rescale

• Use to pick samples on light 1

0.33

L0 L1

5333.00 =ξ

(0.533-0.333)3/2 = .3

0ξ

),( 10/ ξξ

)31(

23

00/ −= ξξ

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Lecture 8: Monte Carlo Rendering - cs. · PDF fileLecture 8: Monte Carlo Rendering ... • Problem for higher dimensions ... N-Rooks Sampling - 9 rays not stratified stratified N-Rooks