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Monte Carlo Integration for Image Synthesis Adapted from Thomas Funkhouser Princeton University C0S 526, Fall 2002 Main Sources Books Realistic Ray Tracing, Peter Shirley Realistic Image Synthesis Using Photon Mapping,

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SLIDE 1

Monte Carlo Integration for Image Synthesis

Adapted from… Thomas Funkhouser Princeton University C0S 526, Fall 2002

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SLIDE 2

Main Sources

  • Books
  • Realistic Ray Tracing, Peter Shirley
  • Realistic Image Synthesis Using Photon Mapping, Henrik Wann Jensen
  • Advanced Global Illumination, Dutre, Bekaert & Bala
  • Theses
  • Robust Monte Carlo Methods for Light Transport Simulation, Eric Veach
  • Mathematical Models and Monte Carlo Methods for Physically Based

Rendering, Eric La Fortune

  • Course Notes
  • Mathematical Models for Computer Graphics, Stanford, Fall 1997
  • State of the Art in Monte Carlo Methods for Realistic Image Synthesis,

Course 29, SIGGRAPH 2001

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SLIDE 3

Outline

  • Motivation
  • Monte Carlo integration
  • Monte Carlo path tracing
  • Variance reduction techniques
  • Sampling techniques
  • Conclusion
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SLIDE 4

Motivation

  • Rendering = integration

Antialiasing Soft shadows Indirect illumination Caustics

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SLIDE 5

Motivation

  • Rendering = integration

Antialiasing Soft shadows Indirect illumination Caustics Surface Eye Pixel x dA e) L(x L

S P

→ =

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SLIDE 6

Motivation

  • Rendering = integration

Antialiasing Soft shadows Indirect illumination Caustics

Surface Eye Light x

x’

dA x x G x x x)V x L( e x x x x f e) (x,x L ) w L(x,

S r e

) , ( ) , ( ) , , ( ′ ′ → ′ → → ′ + → =

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SLIDE 7

Motivation

  • Rendering = integration

Antialiasing Soft shadows Indirect illumination Caustics

dA x x G x x x)V x L( e x x x x f e) (x,x L ) w L(x,

S r e

) , ( ) , ( ) , , ( ′ ′ → ′ → → ′ + → =

  • Herf
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SLIDE 8

Motivation

  • Rendering = integration

Antialiasing Soft shadows Indirect illumination Caustics w d n w ) w (x, L w w x f ) w (x, L ) w (x, L

i r e

  • )

( ) , , (

  • ′

′ ′ + =

  • Ω

Surface Eye Light x ω Surface ω’

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SLIDE 9

Motivation

  • Rendering = integration

Antialiasing Soft shadows Indirect illumination Caustics w d n w ) w (x, L w w x f ) w (x, L ) w (x, L

i r e

  • )

( ) , , (

  • ′

′ ′ + =

  • Ω

Debevec

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SLIDE 10

Motivation

  • Rendering = integration

Antialiasing Soft shadows Indirect illumination Caustics w d n w ) w (x, L w w x f ) w (x, L ) w (x, L

i r e

  • )

( ) , , (

  • ′

′ ′ + =

  • Ω

Diffuse Surface Eye Light x ω Specular Surface ω’

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SLIDE 11

Motivation

  • Rendering = integration

Antialiasing Soft shadows Indirect illumination Caustics w d n w ) w (x, L w w x f ) w (x, L ) w (x, L

i r e

  • )

( ) , , (

  • ′

′ ′ + =

  • Ω

Jensen

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SLIDE 12

Challenge

  • Rendering integrals are difficult to evaluate

Multiple dimensions Discontinuities » Partial occluders » Highlights » Caustics

dA x x G x x x)V x L( e x x x x f e) (x,x L ) w L(x,

S r e

) , ( ) , ( ) , , ( ′ ′ → ′ → → ′ + → =

  • Drettakis
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SLIDE 13

Challenge

  • Rendering integrals are difficult to evaluate

Multiple dimensions Discontinuities » Partial occluders » Highlights » Caustics

dA x x G x x x)V x L( e x x x x f e) (x,x L ) w L(x,

S r e

) , ( ) , ( ) , , ( ′ ′ → ′ → → ′ + → =

  • Jensen
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SLIDE 14

Outline

  • Motivation
  • Monte Carlo integration
  • Monte Carlo path tracing
  • Variance reduction techniques
  • Sampling techniques
  • Conclusion
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SLIDE 15

Integration in 1D

x=1 f(x)

? ) (

1

  • =

dx x f

Slide courtesy of Peter Shirley

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SLIDE 16

We can approximate

x=1 f(x) g(x)

  • =

1 1

) ( ) ( dx x g dx x f

Slide courtesy of Peter Shirley

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SLIDE 17

Or we can average

x=1 f(x) E(f(x))

)) ( ( ) (

1

x f E dx x f =

  • Slide courtesy of

Peter Shirley

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SLIDE 18

Estimating the average

x1 f(x) xN

  • =

=

N i i

x f N dx x f

1 1

) ( 1 ) (

E(f(x))

Slide courtesy of Peter Shirley

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SLIDE 19

Other Domains

x=b f(x) < f >ab x=a

  • =

− =

N i i b a

x f N a b dx x f

1

) ( ) (

Slide courtesy of Peter Shirley

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SLIDE 20

Multidimensional Domains

  • Same ideas apply for integration over …

Pixel areas Surfaces Projected areas Directions Camera apertures Time Paths Surface Eye x

  • =

=

N i i UGLY

x f N dx x f

1

) ( 1 ) (

Pixel

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SLIDE 21

Outline

  • Motivation
  • Monte Carlo integration
  • Monte Carlo path tracing
  • Variance reduction techniques
  • Sampling techniques
  • Conclusion
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SLIDE 22

Monte Carlo Path Tracing

  • Integrate radiance

for each pixel by sampling paths randomly

Diffuse Surface Eye Light x Specular Surface

Pixel w d n w ) w (x, L w w x f ) w (x, L ) w (x, L

i r e

  • )

( ) , , (

  • ′

′ ′ + =

  • Ω
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SLIDE 23

Pixel Sampling

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SLIDE 24

Simple Stochastic Ray Tracing

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SLIDE 25

Radiance Sampling

  • Monte Carlo approximation
  • but RHS has unknowns
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SLIDE 26

Simple Monte Carlo Path Tracer

  • Step 1: Choose a ray (x, y), (u, v), t; weight = 1
  • Step 2: Trace ray to find intersection with nearest surface
  • Step 3: Randomly decide whether to

compute emitted or reflected light

  • Step 3a: If emitted,

return weight * Le

  • Step 3b: If reflected,
  • weight *= reflectance
  • Generate ray in random direction
  • Go to step 2
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SLIDE 27

A Simple Algorithm

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SLIDE 28

Monte Carlo Path Tracing

  • Advantages

Any type of geometry (procedural, curved, ...) Any type of BRDF (specular, glossy, diffuse, ...) Samples all types of paths (L(SD)*E) Accuracy controlled at pixel level Low memory consumption Unbiased - error appears as noise in final image

  • Disadvantages

Slow convergence Noise in final image

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SLIDE 29

Monte Carlo Path Tracing

Big diffuse light source, 20 minutes

Jensen

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SLIDE 30

Monte Carlo Path Tracing

1000 paths/pixel

Jensen

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SLIDE 31

Variance

x1 xN E(f(x))

[ ]

2 1

))] ( ( ) ( [ )) ( ( x f E x f x f E Var

N i i −

=

=

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SLIDE 32

Variance

x1 xN E(f(x))

[ ] [ ]

) ( 1 )) ( ( x f Var N x f E Var =

Variance decreases with 1/N Error decreases with 1/sqrt(N)

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SLIDE 33

Outline

  • Motivation
  • Monte Carlo integration
  • Monte Carlo path tracing
  • Variance reduction techniques
  • Sampling techniques
  • Conclusion
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SLIDE 34

Variance

  • Problem: variance decreases with 1/N

x1 xN E(f(x)) More samples removes noise SLOWLY More samples removes noise SLOWLY

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SLIDE 35

Variance Reduction Techniques

  • Importance sampling
  • Stratified sampling
  • Metropolis sampling
  • Quasi-random
  • =

=

N i i

x f N dx x f

1 1

) ( 1 ) (

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SLIDE 36

Importance Sampling

  • Put more samples where f(x) is bigger

) ( ) ( 1 ) (

1 i i i N i i

x p x f Y Y N dx x f = =

  • =

Ω

x1 xN E(f(x))

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SLIDE 37

Importance Sampling

  • This is still “unbiased”

x1 xN E(f(x))

[ ]

  • Ω

Ω Ω

= = = dx x f dx x p x p x f dx x p x Y Y E

i

) ( ) ( ) ( ) ( ) ( ) (

for all N

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SLIDE 38

Importance Sampling

  • Zero variance if p(x) ~ f(x)

x1 xN E(f(x)) Less variance with better importance sampling

) ( 1 ) ( ) ( ) ( ) ( = = = = Y Var c x p x f Y x cf x p

i i i

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SLIDE 39

Stratified Sampling

  • Estimate subdomains separately

x1 xN Ek(f(x))

Arvo

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SLIDE 40

Stratified Sampling

  • This is still unbiased
  • =

=

= =

M k i i N i i N

F N N x f N F

1 1

1 ) ( 1

x1 xN Ek(f(x))

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SLIDE 41

Stratified Sampling

  • Less overall variance if less variance

in subdomains

[ ] [ ]

  • =

=

M k i i N

F Var N N F Var

1 2

1

x1 xN Ek(f(x))

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SLIDE 42

Outline

  • Motivation
  • Monte Carlo integration
  • Monte Carlo path tracing
  • Variance reduction techniques
  • Sampling techniques
  • Conclusion
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SLIDE 43

Basic Monte Carlo Path Tracer

  • Step 1: Choose a ray (x, y), (u, v), t
  • Step 2: Trace ray to find intersection with nearest surface
  • Step 3: Randomly decide whether to

compute emitted or reflected light

  • Step 3a: If emitted,

return weight * Le

  • Step 3b: If reflected,
  • weight *= reflectance
  • Generate ray in random direction
  • Go to step 2
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SLIDE 44

Sampling Techniques

  • Problem: how do we generate random

points/directions during path tracing?

Non-rectilinear domains Importance (BRDF) Stratified

Surface Eye x

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SLIDE 45

Generating Random Points

  • Uniform distribution:
  • Use random number generator

Probability 1 Ω

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SLIDE 46

Generating Random Points

  • Specific probability distribution:

Function inversion Rejection Metropolis Probability 1 Ω

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SLIDE 47

Generating Random Points

  • Specific probability distribution:

Function inversion Rejection Metropolis Cumulative Probability 1 Ω

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SLIDE 48

Generating Random Points

  • Specific probability distribution:

Function inversion Rejection Metropolis Cumulative Probability 1 Ω

y

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SLIDE 49

Generating Random Points

  • Specific probability distribution:

Function inversion Rejection Metropolis Cumulative Probability 1 Ω

y

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SLIDE 50

Generating Random Points

  • Specific probability distribution:

Function inversion Rejection Metropolis Cumulative Probability 1 Ω

y x

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SLIDE 51

Generating Random Points

  • Specific probability distribution:

Function inversion Rejection Metropolis Cumulative Probability 1 Ω

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SLIDE 52

Generating Random Points

  • Specific probability distribution:

Function inversion Rejection Metropolis Probability 1 Ω

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SLIDE 53

Generating Random Points

  • Specific probability distribution:

Function inversion Rejection Metropolis Probability 1 Ω x x x x x x x x x x

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SLIDE 54

Generating Random Points

  • Specific probability distribution:

Function inversion Rejection Metropolis Probability 1 Ω x x x x x x x x x x

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SLIDE 55

Combining Multiple PDFs

  • Balance heuristic

Use combination of samples generated for each PDF Number of samples for each PDF chosen by weights Near optimal

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SLIDE 56

Monte Carlo Path Tracing Image

2000 samples per pixel, 30 SGIs, 30 hours

Jensen

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SLIDE 57

Monte Carlo Extensions

  • Unbiased

Bidirectional path tracing Metropolis light transport

  • Biased, but consistent

Noise filtering Adaptive sampling Irradiance caching

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SLIDE 58

Monte Carlo Extensions

  • Unbiased

Bidirectional path tracing Metropolis light transport

  • Biased, but consistent

Noise filtering Adaptive sampling Irradiance caching

RenderPark

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SLIDE 59

Monte Carlo Extensions

  • Unbiased

Bidirectional path tracing Metropolis light transport

  • Biased, but consistent

Noise filtering Adaptive sampling Irradiance caching

Heinrich

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SLIDE 60

Monte Carlo Extensions

  • Unbiased

Bidirectional path tracing Metropolis light transport

  • Biased, but consistent

Noise filtering Adaptive sampling Irradiance caching Unfiltered Filtered

Jensen

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SLIDE 61

Monte Carlo Extensions

  • Unbiased

Bidirectional path tracing Metropolis light transport

  • Biased, but consistent

Noise filtering Adaptive sampling Irradiance caching Adaptive Fixed

Ohbuchi Ohbuchi

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SLIDE 62

Monte Carlo Extensions

  • Unbiased

Bidirectional path tracing Metropolis light transport

  • Biased, but consistent

Noise filtering Adaptive sampling Irradiance caching

Jensen

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SLIDE 63

Summary

  • Monte Carlo Integration Methods

Very general Good for complex functions with high dimensionality Converge slowly (but error appears as noise)

  • Conclusion

Preferred method for difficult scenes Noise removal (filtering) and irradiance caching (photon maps) used in practice

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SLIDE 64

More Information

  • Books
  • Realistic Ray Tracing, Peter Shirley
  • Realistic Image Synthesis Using Photon Mapping, Henrik Wann Jensen
  • Theses
  • Robust Monte Carlo Methods for Light Transport Simulation, Eric Veach
  • Mathematical Models and Monte Carlo Methods for Physically Based

Rendering, Eric La Fortune

  • Course Notes
  • Mathematical Models for Computer Graphics, Stanford, Fall 1997
  • State of the Art in Monte Carlo Methods for Realistic Image Synthesis,

Course 29, SIGGRAPH 2001