monte carlo techniques for rendering -...

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Monte Carlo Techniques for Rendering

CS 517 Fall 2002Computer ScienceCornell University

© Kavita Bala, Computer Science, Cornell University

Announcements• No Lecture on Thursday• Instead, attend Steven Gortler, Harvard

– Upson Hall B17, 4:15-5:15 (refreshments earlier)– Geometry Images, reconstruction of 3D Geometry

• Interesting talk by Jim Gray– 2:30-3:30, Wednesday, 255 Olin Hall– Can Databases Help Science?

• Therefore, Office hours this week– by appointment

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MC Advantages• Convergence rate of O( )

• Simple– Sampling– Point evaluation– Can use black boxes

• General– Works for high dimensions– Deals with discontinuities, crazy functions,…

N1


Octree Properties• Front to back traversal• Problem: Same object in multiple cells

– Could repeatedly intersectUse mailboxes

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Intersection Acceleration• Intersect ray with root: p = root.intersect(ray)

– If no intersection, done• Find p in tree (node j = root.find(p))• Test ray against elements in node j

– If intersection found, done– Else find exit point (q) from node j, p = q, goto 2


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

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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()()(


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)(

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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)


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


Stochastic Ray Tracing

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

(branching factor)

• Branching factor = 1: path tracing

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

• Shoot # indirect rays, sampled over hemisphere

• Terminate recursion using Russian Roulette


Algorithm

?=L ?=L

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Algorithm

?0

==

r

e

LL

?0

==

r

e

LL


Algorithm

?=inL ?=inL

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Algorithm

?=inL ?=inL


Algorithm

?=L ?=L

10


Algorithm

?0

==

r

e

LL

?0

==

r

e

LL


Algorithm

?=inL ?=inL

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Algorithm

?=L ?=L


Algorithm

?234.1

==

r

e

LL

?234.1

==

r

e

LL

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Algorithm


Path Tracing

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Performance/Error

• Want better quality with smaller number of samples – Fewer samples/better performance

• Well-distributed samples– Stratified sampling

• 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

• Typically importance sample then stratify

Other types of Sampling

Random sampling Regular sampling


Quasi Monte Carlo

Use of low discrepancy sequences(these are not random numbers)

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Quasi Monte Carlo


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

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Performance/Error

• Want better quality with smaller number of samples – Fewer samples/better performance

• Well-distributed samples– Stratified sampling

• Faster convergence– Importance sampling: next-event estimation


Path Tracing

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

Sample hemisphere

• Importance Sampling: compute direct illumination separately!

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Direct Illumination• Paths of length 1 only, between receiver

and light source


Direct Lighting Global Illumination

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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


Next Event Estimation

indirectdirecte LLLxL ++=Θ→ )(

∫∫ΩΩ

⋅⋅+⋅⋅+=xx

rre ffL coscos

• So … sample direct and indirect with separate MC integration

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Algorithm

?=L ?=L


Algorithm

?0

==

r

e

LL

?0

==

r

e

LL

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Algorithm

??0

===

i

d

e

LLL

??0

===

i

d

e

LLL


Algorithm

?=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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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


∫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


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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),(coscos

)( 2 yxVisr

LAreaxEyx

yxsourcesource

θθ=

Applied to direct illumination

sourceAreayp 1)( =


∑=

=N

ii

yx

yxsourcesource yxVisrN

LAreaxEi

i

12 ),(coscos

)(θθ

More points ...

1 shadow ray 9 shadow rays

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∑=

=N

ii

yx

yxsourcesource yxVisrN

LAreaxEi

i

12 ),(coscos

)(θθ

Even more points ...

36 shadow rays 100 shadow rays


Parameters• How many paths (“shadow-rays”)?

– Total?– Per light source? (~intensity, importance, …)

• How to distribute paths within light source?– Distance from point x– Uniform

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Direct paths• Different path generators produce different

estimators and different error characteristics• Direct illumination general algorithm:

compute_radiance (point, direction)est_rad = 0;for (i=0; i<n; i++)

p = generate_path;est_rad += energy_transfer(p) / probability(p);

est_rad = est_rad / n;return(est_rad);


Direct Paths: Using Area Form

1 path / source 9 paths / source 36 paths / source

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Alternative direct paths• Shoot paths at random over hemisphere;

check if they hit light source

– paths not used efficiently– noise in image

– might work if light source occupies large portion on hemisphere


Alternative direct paths

1 paths / point 16 paths / point 256 paths / point

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Alternative direct paths• Pick random point on random surface;

check if on light source and visible to target point

– paths not used efficiently– noise in image

– might work for large surface light sources


Direct path generators

Hemisphere sampling

- Le can be 0- no visibility in

estimator

Surface sampling

- Le can be 0- 1 visibility term in

estimator

Light source sampling

- Le non-zero- 1 visibility term in

estimator

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Stochastic Ray Tracing• Sample area of light source for direct term

• Sample hemisphere with random rays for indirect term

• Optimizations:– Stratified sampling– Importance sampling– Combine multiple probability density functions

into a single PDF


Hemisphere integration

xΘ

nxΨ

∑=

=Θ→N

i i

ir

ypyxGLf

NxL

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

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• Uniform sampling over the hemisphere

)2/(1)( π=Θp

Sampling strategies

∫Ω

Ψ⋅Ψ⋅Θ↔Ψ⋅Ψ←=Θ→

x

dnfxLxL xr ω),cos()()()(


• Sampling according to the cosine factor

πθ /cos)( =Θp

Sampling strategies

∫Ω


x


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• Sampling according to the BRDF

)(~)( Ψ↔ΘΘ rfp

Sampling strategies

∫Ω


x



• Sampling according to the BRDF times the cosine

θcos)(~)( Ψ↔ΘΘ rfp

Sampling strategies

∫Ω


x


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Multi-Importance-Sampling

∫Ω

Ψ⋅Ψ⋅Ψ←⋅Θ↔Ψ=Θ→

x

dnxLfxL xr ω),cos()()()(

BRDF IrradianceRadiance


Comparison

With importance sampling(brdf on sphere)

Without importance sampling(brdf on sphere)

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Stochastic Ray Tracing• Sample area of light source for direct term

• Sample hemisphere with random rays for indirect term

• Optimizations:– Stratified sampling– Importance sampling– Combine multiple probability density functions

into a single PDF


Indirect Illumination• Paths of length > 1

• Many different path generators possible

• Efficiency dependent on:– BRDFs along the path– Visibility function– ...

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Indirect paths - surface sampling• Simple generator (path length = 2):

– select point on light source– select random point on surfaces

– per path:2 visibility checks


Indirect paths - surface sampling• Indirect illumination (path length 2):

• 2 visibility values (which might be 0) cause noise

yzrA A

r dAdAxyGzfyzGzfyLxLsource

),(),(),(),()()( 2211 Θ↔Ψ−Ψ↔Ψ−Ψ→=Θ→ ∫ ∫

∑=

Θ↔Ψ−Ψ↔Ψ−Ψ→=Θ→

N

i iziy

iiiriiiiirii

zpypxyGzfyzGzfyL

NxL

1

2211

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

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Indirect paths - source shooting• Shoot ray from light source, find hit location• Connect hit point to receiver

– per path:1 ray intersection1 visibility check


Indirect paths - receiver gathering• Shoot ray from receiver point, find hit location• Connect hit point to random point on light source

– per path:1 ray intersection 1 visibility check

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Indirect paths

Source shooting

- 1 visibility term- 1 ray intersection

Receiver gathering

- 1 visibility term- 1 ray intersection

Surface sampling

- 2 visibility terms;can be 0


More variants ...• Shoot ray from receiver point, find hit

location• Shoot ray from hit point, check if on light

source– per path:

2 ray intersectionsLe might be zero

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Indirect paths• Same principles apply to paths of length > 2

– generate multiple surface points– generate multiple bounces from light sources

and connect to receiver– generate multiple bounces from receiver and

connect to light sources– …

• Estimator and noise characteristics change with path generator


Indirect paths– General algorithm:

compute_radiance (point, direction)est_rad = 0;for (i=0; i<n; i++)

p = generate_indirect_path;est_rad += energy_transfer(p) / probability(p);

est_rad = est_rad / n;return(est_rad);

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