Biased Monte Carlo Ray Tracing Filtering, Irradiance Caching, and - - PowerPoint PPT Presentation

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Biased Monte Carlo Ray Tracing Filtering, Irradiance Caching, and - - PowerPoint PPT Presentation

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Biased Monte Carlo Ray Tracing Filtering, Irradiance Caching, and Photon Mapping Henrik Wann Jensen Stanford University May 23, 2002 Unbiased and Consistent Unbiased estimator: E { X } = . . . Consistent estimator: N E

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SLIDE 1

Biased Monte Carlo Ray Tracing

Filtering, Irradiance Caching, and Photon Mapping

→ Henrik Wann Jensen

Stanford University May 23, 2002

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SLIDE 2

Unbiased and Consistent

Unbiased estimator: E{X} =

  • . . .

Consistent estimator: lim

N→∞ E{X} →

  • . . .
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SLIDE 3

Unbiased and Consistent

Unbiased estimator: 1 N

N

  • i=1

f(ξi) Consistent estimator: 1 N + 1

N

  • i=1

f(ξi)

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SLIDE 4

Unbiased Methods

  • Variance (noise) is the only error
  • This error can be analyzed using the variance (i.e.

95% of samples are within 2% of the correct result)

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SLIDE 5

Path Tracing (Unbiased)

10 paths/pixel

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SLIDE 6

Path Tracing (Unbiased)

10 paths/pixel

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SLIDE 7

Path Tracing (Unbiased)

100 paths/pixel

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SLIDE 8

How Can We Remove This Noise

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SLIDE 9

The World is Diffuse!

Arnold Rendering

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SLIDE 10

The World is Diffuse!

Arnold Rendering

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SLIDE 11

The World is Diffuse!

Arnold Rendering

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SLIDE 12

Noise Reduction/Removal

  • More samples (slow convergence, σ ∝ 1/

√ N)

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SLIDE 13

Noise Reduction/Removal

  • More samples (slow convergence, σ ∝ 1/

√ N)

  • Better sampling (stratified, importance, qmc etc.)
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SLIDE 14

Noise Reduction/Removal

  • More samples (slow convergence, σ ∝ 1/

√ N)

  • Better sampling (stratified, importance, qmc etc.)
  • Adaptive sampling
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SLIDE 15

Noise Reduction/Removal

  • More samples (slow convergence, σ ∝ 1/

√ N)

  • Better sampling (stratified, importance, qmc etc.)
  • Adaptive sampling
  • Filtering
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SLIDE 16

Noise Reduction/Removal

  • More samples (slow convergence, σ ∝ 1/

√ N)

  • Better sampling (stratified, importance, qmc etc.)
  • Adaptive sampling
  • Filtering
  • Caching and interpolation
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SLIDE 17

Stratified Sampling

Latin Hypercube: 10 paths/pixel

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SLIDE 18

Quasi Monte-Carlo

Halton-Sequence: 10 paths/pixel

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SLIDE 19

Fixed (Random) Sequence

10 paths/pixel

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SLIDE 20

Filtering: Idea

  • Noise is high frequency
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SLIDE 21

Filtering: Idea

  • Noise is high frequency
  • Remove high frequency content
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SLIDE 22

Unfiltered Image

10 paths/pixel

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SLIDE 23

3x3 Lowpass Filter

10 paths/pixel

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SLIDE 24

Unfiltered Image

10 paths/pixel

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SLIDE 25

3x3 Median Filter

10 paths/pixel

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SLIDE 26

Energy Preserving Filters

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SLIDE 27

Energy Preserving Filters

  • Distribute noisy energy over several pixels
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SLIDE 28

Energy Preserving Filters

  • Distribute noisy energy over several pixels
  • Adaptive filter width

[Rushmeier and Ward 94]

  • Diffusion style filters

[McCool99]

  • Splatting style filters

[Suykens and Willems 00]

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SLIDE 29

Problems With Filtering

  • Everything is filtered (blurred)

⋆ Textures ⋆ Highlights ⋆ Caustics ⋆ . . .

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SLIDE 30

Caching Techniques

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SLIDE 31

Caching Techniques

Irradiance caching : Compute irradiance at selected points and inter- polate. Photon mapping : Trace ‘‘photons’’ from the lights and store them in a photon map, that can be used during rendering.

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SLIDE 32

Box: Direct Illlumination

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SLIDE 33

Box: Global Illlumination

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SLIDE 34

Box: Indirect Irradiance

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SLIDE 35

Irradiance Caching: Idea

‘‘A Ray Tracing Solution for Diffuse Interreflection’’. Greg Ward, Francis Rubinstein and Robert Clear:

  • Proc. SIGGRAPH’88.

Idea: Irradiance changes slowly → interpolate.

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SLIDE 36

Irradiance Sampling

E(x) =

  • 2π

L′(x, ω′) cos θ dω′

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SLIDE 37

Irradiance Sampling

E(x) =

  • 2π

L′(x, ω′) cos θ dω′ = 2π π/2 L′(x, θ, φ) cos θ sin θ dθ dφ

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SLIDE 38

Irradiance Sampling

E(x) =

  • 2π

L′(x, ω′) cos θ dω′ = 2π π/2 L′(x, θ, φ) cos θ sin θ dθ dφ ≈ π TP

T

  • t=1

P

  • p=1

L′(θt, φp) θt = sin−1

  • t−ξ

T

  • and φp = 2πp−ψ

P

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SLIDE 39

Irradiance Change

ǫ(x) ≤

  • ∂E

∂x (x − x0) + ∂E ∂θ (θ − θ0)

  • position
  • rotation
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SLIDE 40

Irradiance Change

ǫ(x) ≤

  • ∂E

∂x (x − x0) + ∂E ∂θ (θ − θ0)

  • position
  • rotation

≤ E0 4 π ||x − x0|| xavg +

  • 2 − 2

N(x) · N(x0)

  • position
  • rotation
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SLIDE 41

Irradiance Interpolation

w(x) = 1 ǫ(x) ≈ 1

||x−x0|| xavg +

  • 1 −

N(x) · N(x0) Ei(x) =

  • i

wi(x)E(xi)

  • i

wi(x)

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SLIDE 42

Irradiance Caching Algorithm

Find all irradiance samples with w(x) > q if (samples found) interpolate else compute new irradiance sample

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SLIDE 43

Box: Irradiance Caching

1000 sample rays, w>10

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SLIDE 44

Box: Irradiance Cache Positions

1000 sample rays, w>10

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SLIDE 45

Box: Irradiance Caching

1000 sample rays, w>20

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SLIDE 46

Box: Irradiance Cache Positions

1000 sample rays, w>20

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SLIDE 47

Box: Irradiance Caching

5000 sample rays, w>10

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SLIDE 48

Box: Irradiance Cache Positions

5000 sample rays, w>10

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SLIDE 49

Caustics

Pathtracing – 1000 paths/pixel

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SLIDE 50

A simple test scene

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SLIDE 51

Rendering

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SLIDE 52

Photon Tracing

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SLIDE 53

Photons

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SLIDE 54

Radiance Estimate

L(x, ω) =

  • Ω

fr(x, ω′, ω)L′(x, ω′) cos θ′ dω

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SLIDE 55

Radiance Estimate

L(x, ω) =

  • Ω

fr(x, ω′, ω)L′(x, ω′) cos θ′ dω =

  • Ω

fr(x, ω′, ω) dΦ2(x, ω′) dω cos θ′dA cos θ′dω

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SLIDE 56

Radiance Estimate

L(x, ω) =

  • Ω

fr(x, ω′, ω)L′(x, ω′) cos θ′ dω =

  • Ω

fr(x, ω′, ω) dΦ2(x, ω′) dω cos θ′dA cos θ′dω =

  • Ω

fr(x, ω′, ω)dΦ2(x, ω′) dA

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SLIDE 57

Radiance Estimate

L(x, ω) =

  • Ω

fr(x, ω′, ω)L′(x, ω′) cos θ′ dω =

  • Ω

fr(x, ω′, ω) dΦ2(x, ω′) dω cos θ′dA cos θ′dω =

  • Ω

fr(x, ω′, ω)dΦ2(x, ω′) dA ≈

n

  • p=1

fr(x, ω′

p,

ω)∆Φp(x, ω′

p)

πr2

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SLIDE 58

Radiance Estimate

L

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SLIDE 59

The photon map datastructure

The photons are stored in a left balanced kd-tree struct photon = { float position[3]; rgbe power; // power packed as 4 bytes char phi, theta; // incoming direction short flags; }

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SLIDE 60

Rendering: Caustics

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SLIDE 61

Caustic from a Glass Sphere

Photon Mapping: 10000 photons / 50 photons in radiance estimate

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SLIDE 62

Caustic from a Glass Sphere

Path Tracing: 1000 paths/pixel

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SLIDE 63

Sphereflake Caustic

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SLIDE 64

Reflection Inside A Metal Ring

50000 photons / 50 photons in radiance estimate

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SLIDE 65

Caustics On Glossy Surfaces

340000 photons / ≈ 100 photons in radiance estimate

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SLIDE 66

HDR environment illumination

Using lightprobe from www.debevec.org

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SLIDE 67

Cognac Glass

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SLIDE 68

Cube Caustic

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SLIDE 69

Global Illumination

100000 photons / 50 photons in radiance estimate

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SLIDE 70

Global Illumination

500000 photons / 500 photons in radiance estimate

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SLIDE 71

Fast estimate

200 photons / 50 photons in radiance estimate

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SLIDE 72

Indirect illumination

10000 photons / 500 photons in radiance estimate

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SLIDE 73

Global Illumination

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SLIDE 74

Global Illumination

global photon map caustics photon map

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SLIDE 75

Photon tracing

  • Photon emission
  • Photon scattering
  • Photon storing
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SLIDE 76

Photon emission

Given Φ Watt lightbulb. Emit N photons. Each photon has the power Φ

N Watt.

  • Photon power depends on the number of emitted

photons. Not on the number of photons in the photon map.

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SLIDE 77

What is a photon?

  • Flux (power) - not radiance!
  • Collection of physical photons

⋆ A fraction of the light source power ⋆ Several wavelengths combined into one entity

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SLIDE 78

Diffuse point light

Generate random direction Emit photon in that direction // Find random direction do { x = 2.0*random()-1.0; y = 2.0*random()-1.0; z = 2.0*random()-1.0; } while ( (x*x + y*y + z*z) > 1.0 );

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SLIDE 79

Example: Diffuse square light

  • Generate random position p on square
  • Generate diffuse direction d
  • Emit photon from p in direction d

// Generate diffuse direction u = random(); v = 2*π*random(); d = vector( cos(v)√u, sin(v)√u, √1 − u );

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SLIDE 80

Surface interactions

The photon is

  • Stored (at diffuse surfaces) and
  • Absorbed (A) or
  • Reflected (R) or
  • Transmitted (T)

A + R + T = 1.0

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SLIDE 81

Photon scattering

The simple way: Given incoming photon with power Φp Reflect photon with the power R ∗ Φp Transmit photon with the power T ∗ Φp

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SLIDE 82

Photon scattering

The simple way: Given incoming photon with power Φp Reflect photon with the power R ∗ Φp Transmit photon with the power T ∗ Φp

  • Risk: Too many low-powered photons - wasteful!
  • When do we stop (systematic bias)?
  • Photons with similar power is a good thing.
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SLIDE 83

Russian Roulette

  • Statistical technique
  • Known from Monte Carlo particle physics
  • Introduced to graphics by Arvo and Kirk in 1990
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SLIDE 84

Russian Roulette

Probability of termination: p

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SLIDE 85

Russian Roulette

Probability of termination: p E{X}

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SLIDE 86

Russian Roulette

Probability of termination: p E{X} = p · 0

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SLIDE 87

Russian Roulette

Probability of termination: p E{X} = p · 0 + (1 − p)

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SLIDE 88

Russian Roulette

Probability of termination: p E{X} = p · 0 + (1 − p) · E{X} 1 − p

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SLIDE 89

Russian Roulette

Probability of termination: p E{X} = p · 0 + (1 − p) · E{X} 1 − p = E{X}

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SLIDE 90

Russian Roulette

Probability of termination: p E{X} = p · 0 + (1 − p) · E{X} 1 − p = E{X} Terminate un-important photons and still get the correct result.

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SLIDE 91

Russian Roulette Example

Surface reflectance: R = 0.5 Incoming photon: Φp = 2 W r = random(); if ( r < 0.5 ) reflect photon with power 2 W else photon is absorbed

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SLIDE 92

Russian Roulette Intuition

Surface reflectance: R = 0.5 200 incoming photons with power: Φp = 2 Watt Reflect 100 photons with power 2 Watt instead of 200 photons with power 1 Watt.

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SLIDE 93

Russian Roulette

  • Very important!
  • Use to eliminate un-important photons
  • Gives photons with similar power :)
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SLIDE 94

Sampling a BRDF

fr(x, ωi, ωo) = w1fr,1(x, ωi, ωo) + w2fr,2(x, ωi, ωo)

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SLIDE 95

Sampling a BRDF

fr(x, ωi, ωo) = w1 · fr,d + w2 · fr,s r = random()·(w1 + w2); if ( r < w1 ) reflect diffuse photon else reflect specular

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SLIDE 96

Rendering

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SLIDE 97

Direct Illumination

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SLIDE 98

Specular Reflection

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SLIDE 99

Caustics

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SLIDE 100

Indirect Illumination

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SLIDE 101

Rendering Equation Solution

Lr(x, ω) =

  • Ωx

fr(x, ω′, ω)Li(x, ω′) cos θi dω′

i

=

  • Ωx

fr(x, ω′, ω)Li,l(x, ω′) cos θi dω′

i +

  • Ωx

fr,s(x, ω′, ω)(Li,c(x, ω′) + Li,d(x, ω′)) cos θi dω′

i +

  • Ωx

fr,d(x, ω′, ω)Li,c(x, ω′) cos θi dω′

i +

  • Ωx

fr,d(x, ω′, ω)Li,d(x, ω′) cos θi dω′

i .

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SLIDE 102

Features

  • Photon tracing is unbiased

⋆ Radiance estimate is biased but consistent ⋆ The reconstruction error is local

  • Illumination representation is decoupled from the

geometry

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SLIDE 103

Box

200000 global photons, 50000 caustic photons

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SLIDE 104

Box: Global Photons

200000 global photons

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SLIDE 105

Fractal Box

200000 global photons, 50000 caustic photons

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SLIDE 106

Cornell Box

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SLIDE 107

Indirect Illumination

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SLIDE 108

Little Matterhorn

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SLIDE 109

Mies house (swimmingpool)

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SLIDE 110

Mies house (3pm)

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SLIDE 111

Mies house (6pm)

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SLIDE 112

More Information

Realistic Image Synthesis Using Photon Mapping Realistic Image Synthesis Using Photon Mapping

Realistic Image Synthesis Using Photon Mapping

Realistic Image Synthesis Using Photon Mapping

Foreword by Pat Hanrahan

Realistic Image Synthesis Using Photon Mapping

Henrik Wann Jensen Jensen

Foreword by Pat Hanrahan

The creation of realistic three-dimensional images is central to

computer graphics. Photon mapping, an extension of ray tracing, makes it possible to efficiently simulate global illumination in complex scenes. Photo mapping can simulate caustics (focused light, such as shimmering waves at the bottom of a swimming pool), diffuse inter-reflections (e.g., the `bleeding' of colored light from a red wall onto a white floor, giving the floor a reddish tint), and participating media (e.g., clouds or smoke). This book is a practical guide to photon mapping; it provides both the theory and the practical insight necessary to implement photon mapping and simulate all types of direct and indirect illumination efficiently. A K PETERS LTD.

A K PETERS

Henrik Wann Jensen

http://graphics.stanford.edu/˜henrik henrik@graphics.stanford.edu