Monte Carlo I

Previous lecture Analytical illumination formula

This lecture Numerical evaluation of illumination Review random variables and probability Monte Carlo integration Sampling from distributions Sampling from shapes Variance and efficiency

Lighting and Soft Shadows

Challenges

Visibility and blockers

Varying light distribution

Complex source geometry

2

( ) ( , )cosi

H

E x L x dω θ ω=∫

Source: Agrawala. Ramamoorthi, Heirich, Moll, 2000

Penumbras and Umbras

Monte Carlo Lighting

1 eye ray per pixel1 shadow ray per eye

ray

Fixed Random

Monte Carlo Algorithms

Advantages Easy to implement Easy to think about (but be careful of statistical bias) Robust when used with complex integrands and

domains (shapes, lights, …) Efficient for high dimensional integrals Efficient solution method for a few selected points

Disadvantages Noisy Slow (many samples needed for convergence)

Random Variables

is chosen by some random process

probability distribution (density) function

X

~ ( )X p x

Discrete Probability Distributions

Discrete events Xi with probability pi

Cumulative PDF (distribution)

Construction of samples

To randomly select an event,

Select Xi if

0ip ≥1

1n

ii

p=

=∑

1i iP U P− < ≤

ip

1

j

j ii

P p=

=∑

Uniform random variable

U

1

0

3X

iP

Continuous Probability Distributions

PDF (density)

CDF (distribution)1

0

( ) Pr( )P x X x= <

Pr( ) ( )

( ) ( )

X p x dx

P P

β

α

α β

β α

≤ ≤ =

= −

∫

( )p x

10Uniform

( ) 0p x ≥

0

( ) ( )x

P x p x dx=∫(1) 1P =

( )P x

Sampling Continuous Distributions

Cumulative probability distribution function

Construction of samplesSolve for X=P-1(U)

Must know:1. The integral of p(x)2. The inverse function P-1(x)

U

X

1

0

( ) Pr( )P x X x= <

Example: Power Function

Assume( ) ( 1) np x n x= +

11 1

0 0

1

1 1

nn x

x dxn n

+

= =+ +∫

1( ) nP x x +=

1 1~ ( ) ( ) nX p x X P U U− +⇒ = =

1 2 1max( , , , , )n nY U U U U += L1

1

1

Pr( ) Pr( )n

n

i

Y x U x x+

+

=

< = < =∏

Trick

Sampling a Circle12 1 1 2 2

2

00 0 0 0 0

2

rA r dr d r dr d

π ππθ θ θ π

⎛ ⎞= = = =⎜ ⎟

⎝ ⎠∫∫ ∫ ∫

( , ) ( ) ( )p r p r pθ θ=

2

( ) 2

( )

p r r

P r r

=

=

12 Uθ π=

rdθ

dr

1( )

21

( )2

p

P

θπ

θ θπ

=

=

1( , ) ( , )

rp r dr d r dr d p rθ θ θ θ

π π= ⇒ =

2r U=

Sampling a Circle

1

2

2 U

r U

θ π=

=1

2

2 U

r U

θ π=

=

RIGHT Equi-ArealWRONG Equi-Areal

Rejection Methods

Algorithm

Pick U1 and U2

Accept U1 if U2 < f(U1)

Wasteful?

1

0

( )

( )

y f x

I f x dx

dx dy<

=

=

∫

∫∫ ( )y f x=

Efficiency = Area / Area of rectangle

Sampling a Circle: Rejection

May be used to pick random 2D directions

Circle techniques may also be applied to the sphere

do {

X=1-2*U1

Y=1-2*U2

}

while(X2 + Y2 > 1)

Monte Carlo Integration

Definite integral

Expectation of f

Random variables

Estimator

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 =

= ∑

Unbiased Estimator

1

1 1

1

1 0

1

1 0

1

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

Over Arbitrary Domains

∫=b

a

dxxfI )( abxp U

b

a −==

1)(

11

)()( =−

==∫b

a

b

a abxpxP∫−=

b

a

dxxpxfabI )()()(

)()( fEabI −=

fabI )( −≈

)(1

)(1

i

N

i

XfN

abI ∑=

−≈ a b

ab −1

Non-Uniform Distributions

∫=b

a

dxxfI )(

∫=b

a

dxxpxp

xfI )(

)(

)(

)(

)()(

xp

xfxg =

∫=b

a

dxxpxgI )()(

)(gEI =

∑=

≈N

i

iXgN

I1

)(1

Direct Lighting – Directional Sampling

2

( ) ( , ) cosH

E x L x dω θ ω=∫

*( ( , ), ) cos 2i i i iY L x x ω ω θ π= −

*( , )x x ωRay intersectionA′

ω

*x

x

θ Sample uniformly byΩω

Direct Lighting – Area Sampling

22

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

AH

E x L x d L x V x x dAx x

θ θω θ ω ω′

′′ ′ ′ ′= =

′−∫ ∫

2

cos cos( , ) ( , ) i i

i o i i i

i

Y L x V x x Ax x

θ θω′

′ ′ ′=′−x

x xω′ ′= −Ray direction

Sample uniformly byA′x′

0( , )

1

visibleV x x

visible

¬⎧′ =⎨⎩

A′

ω′

x′

θθ′

Examples

4 eye rays per pixel1 shadow ray per eye ray

Fixed Random

Examples

4 eye rays per pixel16 shadow rays per eye ray

Uniform grid Stratified random

Examples

4 eye rays per pixel64 shadow rays per eye ray

Uniform grid Stratified random

Examples

4 eye rays per pixel100 shadow rays per eye ray

Uniform grid Stratified random

Examples

4 eye rays per pixel16 shadow rays per eye ray

64 eye rays per pixel1 shadow ray per eye ray

Variance

Definition

Properties

Variance decreases with sample size

2

2 2

2 2

[ ] [( [ ]) ]

[ 2 [ ] [ ] ]

[ ] [ ]

V Y E Y E Y

E Y YE Y E Y

E Y E Y

≡ −

= − +

= −

[ ] [ ]i ii i

V Y V Y=∑ ∑2[ ] [ ]V aY a V Y=

21 1

1 1 1[ ] [ ] [ ]

N N

i ii i

V Y V Y V YN N N= =

= =∑ ∑

Direct Lighting – Directional Sampling

2

( ) ( , ) cosH

E x L x dω θ ω=∫

*( ( , ), ) cos 2i i i iY L x x ω ω θ π= −

*( ( , ), )i i iY L x x ω ω π= −

*( , )x x ωRay intersection

A′

ω

*x

x

θSample uniformly byΩω

Sample uniformly byΩ%ω

Sampling Projected Solid Angle

Generate cosine weighted distribution

Examples

Projected solid angle

4 eye rays per pixel100 shadow rays

Area

4 eye rays per pixel100 shadow rays

Variance Reduction

Efficiency measure

Techniques Importance sampling Sampling patterns: stratified, …

1Efficiency

Variance Cost∝

•

Sampling a Triangle

121 1 1

0 0 00

(1 ) 1(1 )

2 2

u uA dv du u du

− −= = − =− =∫∫ ∫

( , ) 2p u v =

0

0

1

u

v

u v

≥≥+ ≤

u

v

1u v+ =

( ,1 )u u−

)(

),()|(

vp

vupvup ≡

Sampling a Triangle

Here u and v are not independent!

Conditional probability

( , ) 2p u v =

00 0 00 0

0 0

1( | ) ( | )

(1 ) (1 )o ov v v

P v u p v u dv dvu u

= = =− −∫ ∫

1( | )

(1 )p v u

u=

−

0 20 00

( ) 2(1 ) (1 )u

P u u du u= − = −∫

1

0

( ) 2 2(1 )u

p u dv u−

= = −∫0 11u U= −

0 1 2v U U=

∫≡ dvvupup ),()(