Monte Carlo 1: Integration

CS348b Lecture 6

Monte Carlo 1: Integration

Previous lecture: Analytical illumination formula

This lecture: Monte Carlo Integration

￭ Review random variables and probability

￭ Sampling from distributions

￭ Sampling from shapes

￭Numerical calculation of illumination

Pat Hanrahan / Matt Pharr, Spring 2021

Irradiance from the Environment

CS348b Lecture 6 Pat Hanrahan / Matt Pharr, Spring 2021

2

( ) ( , )cosiH

E x L x dω θ ω= ∫

ω ω θ ωΦ =2 ( , ) ( , )cosi id x L x dA d

( , ) ( , )cosidE x L x dω ω θ ω= ω( , )iL x

dA

θdω

Irradiance from a Uniform Area Source


2

( ) cos

cosH

E x L d

L d

L

θ ω

θ ωΩ

=

=

= Ω

∫

∫!

A

Ω!

Ω

Direct Illumination

Constant

Uniform Triangle Source






Consider 1 Edge


γ θ γ= = •! !

cos EA N N

Area of sector

θ

Projected area of sector

N̂�N̂E

�

Lambert’s Formula


1 1

n n

i i ii iA γ

= =

= •∑ ∑ N N! !

3

1i

iA

=∑

1A2A−

3A−

Penumbras and Umbras


CS348b Lecture 6

Lighting and Soft Shadows

Challenges

1 Occluders

- Complex geometry

- Number of occluders

2 Non-uniform light sources


2

( ) ( , )cosiH


Source: Agrawala. Ramamoorthi, Heirich, Moll, 2000

Monte Carlo Illumination Calculation


1 shadow ray per eye ray

Center Random

CS348b Lecture 6

Monte Carlo Algorithms

Advantages

￭ Easy to implement

￭ Easy to think about (but be careful of subtleties)

￭ Robust when used with complex integrands (lights, BRDFs) and domains (shapes)

￭ Efficient for high dimensional integrals

￭ Efficient when only need solution at a few points

Disadvantages

￭ Noisy

￭ Slow (many samples needed for convergence)


CS348b Lecture 6

Random Variables

is a random variable

A random variable takes on different values

(represents a distribution of potential values)

probability distribution function (PDF)


X

~ ( )X p x

CS348b Lecture 6

Discrete Probability Distributions

Discrete values

with probability


0ip ≥

11

n

iip

=

=∑

ip

ix

ipix

CS348b Lecture 6


Cumulative PDF


=

=∑1

j

j ii

P p

1

0

jP

≤ ≤0 1iP

=1nP

jP

CS348b Lecture 6


Construction of samples

To randomly select an event,

Select if


U

1

0

Uniform random variable

3x

ix<latexit sha1_base64="yRBqiuPNTVsPiSCrR70z+Hod6JE=">AAAB/nicbVDLSsNAFJ3UV62vqLhyM1gEN5akiLpwUXDTZQT7gCaEyfSmHTp5MDMRSyj4K25cKOLW73Dn3zhts9DWAxcO59zLvfcEKWdSWda3UVpZXVvfKG9WtrZ3dvfM/YO2TDJBoUUTnohuQCRwFkNLMcWhmwogUcChE4xup37nAYRkSXyvxil4ERnELGSUKC355pHj5+zcnuAb7D4y7HLAjs98s2rVrBnwMrELUkUFHN/8cvsJzSKIFeVEyp5tpcrLiVCMcphU3ExCSuiIDKCnaUwikF4+O3+CT7XSx2EidMUKz9TfEzmJpBxHge6MiBrKRW8q/uf1MhVeezmL00xBTOeLwoxjleBpFrjPBFDFx5oQKpi+FdMhEYQqnVhFh2AvvrxM2vWafVmr311UG80ijjI6RifoDNnoCjVQEzmohSjK0TN6RW/Gk/FivBsf89aSUcwcoj8wPn8A8A+UNg==</latexit>

Pi�1 < ⇠ Pi

CS348b Lecture 6

Continuous Probability Distributions

PDF

CDF


1

0

( ) Pr( )P x X x= <

Pr( ) ( )

( ) ( )

X p x dx

P P

β

α

α β

β α

≤ ≤ =

= −

∫

( )p x

10

Uniform

( ) 0p x ≥

0

( ) ( )x

P x p x dx= ∫(1) 1P =

( )P x

CS348b Lecture 6

Sampling Continuous Distributions

Cumulative probability distribution function

Construction of samples

Solve for

Must know the formula for:

1. The integral of

2. The inverse function


X

1

0

( ) Pr( )P x X x= <

p(x)

P�1(x)

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X = P�1(⇠)<latexit sha1_base64="MqtDJzWRv1yXB1W2anb4Ko1ap9c=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR9Vjw0mNF+wFtKJvtpl262YTdiVhCf4IXD4p49Rd589+4bXPQ1gcDj/dmmJkXJFIYdN1vZ219Y3Nru7BT3N3bPzgsHR23TJxqxpsslrHuBNRwKRRvokDJO4nmNAokbwfj25nffuTaiFg94CThfkSHSoSCUbTSfe9J9Etlt+LOQVaJl5My5Gj0S1+9QczSiCtkkhrT9dwE/YxqFEzyabGXGp5QNqZD3rVU0YgbP5ufOiXnVhmQMNa2FJK5+nsio5ExkyiwnRHFkVn2ZuJ/XjfF8MbPhEpS5IotFoWpJBiT2d9kIDRnKCeWUKaFvZWwEdWUoU2naEPwll9eJa1qxbuqVO8uy7V6HkcBTuEMLsCDa6hBHRrQBAZDeIZXeHOk8+K8Ox+L1jUnnzmBP3A+fwBf/Y3i</latexit>

⇠

Sampling a Circle


12 1 1 2 22

00 0 0 0 0

2rA r dr d r dr d

π ππ

θ θ θ π# $

= = = =% &' (

∫ ∫ ∫ ∫

( , ) ( ) ( )p r p r pθ θ=rdθ

dr

1( , ) ( , ) rp r dr d r dr d p rθ θ θ θπ π

= ⇒ =

1( )21( )2

p

P

θπ

θ θπ

=

=

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✓ = 2⇡⇠1

2

( ) 2( )p r rP r r

=

=

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r =p⇠2

Sampling a Circle


RIGHT = Equi-ArealWRONG ≠ Equi-Areal

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r =p⇠2

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r = ⇠2

<latexit sha1_base64="QhPUEPNmTE6hSUJfFXkbVTcmmV8=">AAAB/XicbVDLSsNAFJ3UV62v+Ni5GSyCq5IUUTdCwU2XFewDmhAm00k7dDIJMzdiLcVfceNCEbf+hzv/xmmbhbYeuHA4517uvSdMBdfgON9WYWV1bX2juFna2t7Z3bP3D1o6yRRlTZqIRHVCopngkjWBg2CdVDESh4K1w+HN1G/fM6V5Iu9glDI/Jn3JI04JGCmwjzwYMCD4Glexl3LsPfDADeyyU3FmwMvEzUkZ5WgE9pfXS2gWMwlUEK27rpOCPyYKOBVsUvIyzVJCh6TPuoZKEjPtj2fXT/CpUXo4SpQpCXim/p4Yk1jrURyazpjAQC96U/E/r5tBdOWPuUwzYJLOF0WZwJDgaRS4xxWjIEaGEKq4uRXTAVGEggmsZEJwF19eJq1qxb2oVG/Py7V6HkcRHaMTdIZcdIlqqI4aqIkoekTP6BW9WU/Wi/VufcxbC1Y+c4j+wPr8AVgfk+M=</latexit>

✓ = 2⇡⇠1<latexit sha1_base64="QhPUEPNmTE6hSUJfFXkbVTcmmV8=">AAAB/XicbVDLSsNAFJ3UV62v+Ni5GSyCq5IUUTdCwU2XFewDmhAm00k7dDIJMzdiLcVfceNCEbf+hzv/xmmbhbYeuHA4517uvSdMBdfgON9WYWV1bX2juFna2t7Z3bP3D1o6yRRlTZqIRHVCopngkjWBg2CdVDESh4K1w+HN1G/fM6V5Iu9glDI/Jn3JI04JGCmwjzwYMCD4Glexl3LsPfDADeyyU3FmwMvEzUkZ5WgE9pfXS2gWMwlUEK27rpOCPyYKOBVsUvIyzVJCh6TPuoZKEjPtj2fXT/CpUXo4SpQpCXim/p4Yk1jrURyazpjAQC96U/E/r5tBdOWPuUwzYJLOF0WZwJDgaRS4xxWjIEaGEKq4uRXTAVGEggmsZEJwF19eJq1qxb2oVG/Py7V6HkcRHaMTdIZcdIlqqI4aqIkoekTP6BW9WU/Wi/VufcxbC1Y+c4j+wPr8AVgfk+M=</latexit>

✓ = 2⇡⇠1

classShape{public://ShapeInterface…virtual~Shape();virtualBounds3fObjectBound()const=0;virtualBounds3fWorldBound()const;virtualboolIntersect(constRay&ray,Float*tHit,SurfaceInteraction*isect,booltestAlphaTexture=true)const=0;…virtualFloatArea()const=0;…//Sampleapointonthesurfaceoftheshape//andreturnthePDFwithrespecttoareaonthesurface.virtualInteractionSample(constPoint2f&u,Float*pdf)const=0;virtualFloatPdf(constInteraction&)const{return1/Area();}..};

Rejection Sampling


Area of circle / Area of square

Efficiency?

do{

X=Uniform(-1,1)

Y=Uniform(-1,1)

}while(X*X+Y*Y>1);

Sampling 2D Directions


do{

X=Uniform(-1,1);

Y=Uniform(-1,1);

}while(X*X+Y*Y>1);

R=sqrt(X*X+Y*Y)

dx=X/R

dy=Y/R

Computing Area of a Circle


A=0

for(i=0;i<N;i++){

X=Uniform(-1,1);

Y=Uniform(-1,1);

if(X*X+Y*Y<1)

A+=1;

}

A=4*A/N

Definite integral

Expectation of

Random variables

Estimator

Monte Carlo Integration

1

0

( ) ( )I f f x dx≡ ∫1

0

[ ] ( ) ( )E f f x p x dx≡ ∫

~ ( )iX p x( )i iY f X=

1

1 N

N ii

F YN =

= ∑

f


Unbiased Estimator


1

1 11

1 01

1 01

0

1[ ] [ ]

1 1[ ] [ ( )]

1 ( ) ( )

1 ( )

( )

N

N ii

N N

i ii i

N

i

N

i

E F E YN

E Y E f XN N

f x p x dxN

f x dxN

f x dx

=

= =

=

=

=

= =

=

=

=

∑

∑ ∑

∑∫

∑∫

∫

[ ] ( )NE F I f=

Assume uniform probability distribution for now

[ ] [ ]i ii i

E Y E Y=∑ ∑[ ] [ ]E aY aE Y=

Properties

Direct Lighting: Hemispherical Integral


dA

θdω

2

( ) ( , ) cosH


ω( , )iL x

Direct Lighting: Solid Angle Sampling


2

( ) 1H

p dω ω =∫

Sample hemisphere uniformly

dA

θdω

ω( , )iL x

Direct Lighting: Solid Angle Sampling


Estimator

dA

θdω

ω( , )iL x

Direct Lighting: Hemisphere Sampling


Hemisphere

16 shadow rays

Direct Lighting: Area Integral


A!x!

x

θθ "

0 blocked( , )

1 visibleV x x

!" = #

$Radiance

Visibility

Integral

Direct Lighting: Area Integral


A!x!

x

θθ "

Direct Lighting: Area Sampling


A!x!

x

θθ "

Sample shape uniformly by area



A!

ω"

x!

x

θθ " Estimator



Area

16 shadow rays

Random Sampling Introduces Noise


1 shadow ray per eye ray

Center Random

Quality Improves with More Rays


16 shadow rays1 shadow ray

Area Area

Why is Area Better than Hemisphere?


Hemisphere

16 shadow rays

Area

16 shadow rays

Documents

Monte Carlo 1: Integration