A radiative transfer framework for non-exponential media Benedikt - - PowerPoint PPT Presentation

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A radiative transfer framework for non-exponential media Benedikt - - PowerPoint PPT Presentation

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A radiative transfer framework for non-exponential media Benedikt Bitterli 1 Srinath Ravichandran 1 Thomas Mller 2 3 Magnus Wrenninge 4 Jan Novk 3 Steve Marschner 5 Wojciech Jarosz 1 1 Dartmouth College 2 ETH Zurich 3 Disney Research 4 Pixar

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A radiative transfer framework for non-exponential media

Benedikt Bitterli 1 Srinath Ravichandran 1 Thomas Müller 2 3 Magnus Wrenninge 4 Jan Novák 3 Steve Marschner 5 Wojciech Jarosz 1

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1 Dartmouth College 2 ETH Zurich 3 Disney Research 4 Pixar Animation Studios 5 Cornell University

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SLIDE 2 2

Ross Burgener Jan Novak NASA

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Transmittance

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Transmittance

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SLIDE 6 4

Transmittance

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SLIDE 9 5

E[ ]

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SLIDE 10 6

Transmittance

E[ ]

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SLIDE 11

Radiative Transfer Theory

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Radiative Transfer Theory

  • Radiative Transfer, Chandrasekhar, 1960
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Radiative Transfer Theory

  • Radiative Transfer, Chandrasekhar, 1960
  • Assumes that particle positions are independent
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Transmittance

E[ ]

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SLIDE 15 8

e−τ

Transmittance

E[ ]

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Particle Correlations

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Inter-particle Forces

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Inter-particle Forces

Dusty plasma correlation function experiment, Smith et al. 2004 Interactions in colloidal suspensions, Grier and Behrens, 2001 Fat Particle Structure and Stability of Food Emulsions, Xu et al. 2008 A Model for the Stability of a TiO2 Dispersion, Goicochea, 2013 The electrostatic interaction in colloidal systems with low added electrolyte, Beresford-Smith et al. 1985

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SLIDE 19 11

Inter-particle Forces

Dusty plasma correlation function experiment, Smith et al. 2004 Interactions in colloidal suspensions, Grier and Behrens, 2001 Fat Particle Structure and Stability of Food Emulsions, Xu et al. 2008 A Model for the Stability of a TiO2 Dispersion, Goicochea, 2013 The electrostatic interaction in colloidal systems with low added electrolyte, Beresford-Smith et al. 1985

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SLIDE 20 11

Inter-particle Forces

Dusty plasma correlation function experiment, Smith et al. 2004 Interactions in colloidal suspensions, Grier and Behrens, 2001 Fat Particle Structure and Stability of Food Emulsions, Xu et al. 2008 A Model for the Stability of a TiO2 Dispersion, Goicochea, 2013 The electrostatic interaction in colloidal systems with low added electrolyte, Beresford-Smith et al. 1985

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SLIDE 21 12

Clouds

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SLIDE 22 12

Clouds

Horizontal structure of marine boundary layer clouds from centimeter to kilometer scales, Davis et al. 1999 On the Spatial Distribution of Cloud Particles Kostinski and Jameson, 2000

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Clouds

Horizontal structure of marine boundary layer clouds from centimeter to kilometer scales, Davis et al. 1999 On the Spatial Distribution of Cloud Particles Kostinski and Jameson, 2000

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Clouds

Horizontal structure of marine boundary layer clouds from centimeter to kilometer scales, Davis et al. 1999 On the Spatial Distribution of Cloud Particles Kostinski and Jameson, 2000

Kilometers

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Clouds

Horizontal structure of marine boundary layer clouds from centimeter to kilometer scales, Davis et al. 1999 On the Spatial Distribution of Cloud Particles Kostinski and Jameson, 2000

Kilometers

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SLIDE 27 13

e−τ

Distance Transmittance

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Distance

e−τ

Distance Transmittance

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SLIDE 29 13

Distance Distance

e−τ

Distance Transmittance

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The Rendering Equation

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Li(x, ω) = Tr(x, xs)Lo(xs, ω) + Z s Tr(x, xt)σs(xt)Ls(xt, ω)dt

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The Rendering Equation

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Li(x, ω) = Tr(x, xs)Lo(xs, ω) + Z s Tr(x, xt)σs(xt)Ls(xt, ω)dt

xs x

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SLIDE 32

The Rendering Equation

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Li(x, ω) = Tr(x, xs)Lo(xs, ω) + Z s Tr(x, xt)σs(xt)Ls(xt, ω)dt

x xt xs x

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SLIDE 33

The Rendering Equation

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Li(x, ω) = Tr(x, xs)Lo(xs, ω) + Z s Tr(x, xt)σs(xt)Ls(xt, ω)dt

x xt xs x

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The Rendering Equation

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Li(x, ω) = Tr(x, xs)Lo(xs, ω) + Z s Tr(x, xt)σs(xt)Ls(xt, ω)dt

e−τ(x,xt) e−τ(x,xt)

x xt xs x

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The Rendering Equation

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Li(x, ω) = Tr(x, xs)Lo(xs, ω) + Z s Tr(x, xt)σs(xt)Ls(xt, ω)dt

x xt xs x

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SLIDE 36 16

e−τ

Tr =

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e−τ

Tr = Tr =

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e−τ

Tr = Tr =

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SLIDE 39

Summary

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Summary

  • Classical transport assumes particle independence
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Summary

  • Classical transport assumes particle independence
  • This model is not necessarily accurate
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Summary

  • Classical transport assumes particle independence
  • This model is not necessarily accurate
  • Correlated particles lead to non-exponential transmittance
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Summary

  • Classical transport assumes particle independence
  • This model is not necessarily accurate
  • Correlated particles lead to non-exponential transmittance
  • Non-exponential transmittance breaks classical transport
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Summary

  • Classical transport assumes particle independence
  • This model is not necessarily accurate
  • Correlated particles lead to non-exponential transmittance
  • Non-exponential transmittance breaks classical transport
  • We need a new transport framework
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The Rendering Equation

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Li(x, ω) = Tr(x, xs)Lo(xs, ω) + Z s Tr(x, xt)σs(xt)Ls(xt, ω)dt x xt xs x

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SLIDE 46

The Rendering Equation

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Li(x, ω) = Tr(x, xs)Lo(xs, ω) + Z s Tr(x, xt)σs(xt)Ls(xt, ω)dt x xt xs x

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The Rendering Equation

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Li(x, ω) = Tr(x, xs)Lo(xs, ω) + Z s Tr(x, xt)σt(xt)α(xt)Ls(xt, ω)dt x xt xs x

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SLIDE 48

The Rendering Equation

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Li(x, ω) = Tr(x, xs)Lo(xs, ω) + Z s Tr(x, xt)σt(xt)α(xt)Ls(xt, ω)dt x xt xs x

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SLIDE 49

The Rendering Equation

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Li(x, ω) = Tr(x, xs)Lo(xs, ω) + Z s Tr(x, xt)σt(xt)α(xt)Ls(xt, ω)dt

e−τ(x,xt)

x xt xs x

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SLIDE 50

The Rendering Equation

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Li(x, ω) = Tr(x, xs)Lo(xs, ω) + Z s Tr(x, xt)σt(xt)α(xt)Ls(xt, ω)dt

e−τ(x,xt)

x xt xs x

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SLIDE 51

The Rendering Equation

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Li(x, ω) = Tr(x, xs)Lo(xs, ω) + Z s pdf(x, xt)α(xt)Ls(xt, ω)dt x xt xs x

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Transport Functions

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Transport Functions

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Tr(x, xt)

pdf(x, xt)

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Transport Functions

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Tr(x, xt)

pdf(x, xt)

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Transport Functions

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f

p: “particle”

: “free space”

ff(x, xt) fp(x, xt) pf(x, xt) pp(x, xt)

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ff(x, xt) fp(x, xt) pf(x, xt) pp(x, xt)

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ff(x, xt) fp(x, xt) pf(x, xt) pp(x, xt)

Transmittance

Probability: Starts at 1

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ff(x, xt) fp(x, xt) pf(x, xt) pp(x, xt)

Transmittance

Probability: Starts at 1

Free-flight PDF

Probability Density: Integrates to 1

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ff(x, xt) fp(x, xt) pf(x, xt) pp(x, xt)

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ff(x, xt) fp(x, xt) pf(x, xt) pp(x, xt)

Z Z

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ff(x, xt) fp(x, xt) pf(x, xt) pp(x, xt)

Z

d dt

Z

d dt

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ff(x, xt) fp(x, xt) pf(x, xt) pp(x, xt)

Z

¯ σ

d dt

Z

d dt

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ff(x, xt) fp(x, xt) pf(x, xt) pp(x, xt)

Z

¯ σ

d dt

¯ σ−1

Z

d dt

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Transport Kernel

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) =

⇢

if x ∈ p and xt ∈ p

if x ∈ f and xt ∈ p if x ∈ f and xt ∈ f if x ∈ p and xt ∈ f

ff(x, xt)

fp(x, xt)

pf(x, xt)

pp(x, xt)

T(x, xt)

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A Non-Exponential Rendering Equation

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Li(x, ω) = Tr(x, xs)Lo(xs, ω) + Z s Tr(x, xt)σs(xt)Ls(xt, ω)dt

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A Non-Exponential Rendering Equation

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Li(x, ω) = Tr(x, xs)Lo(xs, ω) + Z s Tr(x, xt)σs(xt)Ls(xt, ω)dt

T(x, xt) T(x, xt)α(xt)

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A Non-Exponential Rendering Equation

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Li(x, ω) = Tr(x, xs)Lo(xs, ω) + Z s Tr(x, xt)σs(xt)Ls(xt, ω)dt

T(x, xt) T(x, xt)α(xt)

This Talk: Rendering Equation

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A Non-Exponential Rendering Equation

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Li(x, ω) = Tr(x, xs)Lo(xs, ω) + Z s Tr(x, xt)σs(xt)Ls(xt, ω)dt

T(x, xt) T(x, xt)α(xt)

This Talk: Rendering Equation In Paper: Path Integral

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A Non-Exponential Rendering Equation

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Li(x, ω) = Tr(x, xs)Lo(xs, ω) + Z s Tr(x, xt)σs(xt)Ls(xt, ω)dt

T(x, xt) T(x, xt)α(xt)

This Talk: Rendering Equation In Paper: Path Integral

Reciprocity, energy conservation, …

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Summary

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Summary

  • In correlated media, transmittance becomes four functions
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Summary

  • In correlated media, transmittance becomes four functions
  • These represent different interactions at the end points
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Summary

  • In correlated media, transmittance becomes four functions
  • These represent different interactions at the end points
  • Given one, all others can be derived
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Summary

  • In correlated media, transmittance becomes four functions
  • These represent different interactions at the end points
  • Given one, all others can be derived
  • This talk: High level overview
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Summary

  • In correlated media, transmittance becomes four functions
  • These represent different interactions at the end points
  • Given one, all others can be derived
  • This talk: High level overview
  • Paper: Rigorous derivation
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Modelling Transmittance

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Modelling Transmittance

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Data Driven

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Modelling Transmittance

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Data Driven

A Data-Driven Reflectance Model, Matusik et al., 2003

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Modelling Transmittance

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Data Driven

A Data-Driven Reflectance Model, Matusik et al., 2003 A Microfacet-based BRDF Generator, Ashikhmin et al., 2000

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e−τ

Transmittance

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Modelling Transmittance

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Phenomenological

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Modelling Transmittance

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Phenomenological

Illumination for Computer Generated Pictures, Bui Tuong Phong, 1975 Models of Light Reflection For Computer Synthesized Pictures James F. Blinn, 1977

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Modelling Transmittance

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Phenomenological

Illumination for Computer Generated Pictures, Bui Tuong Phong, 1975 Models of Light Reflection For Computer Synthesized Pictures James F. Blinn, 1977

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Modelling Transmittance

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Data Driven Phenomenological

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Modelling Transmittance

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Data Driven Phenomenological Statistical Models

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Modelling Transmittance

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Statistical Models

Theory for Off-Specular Reflection From Roughened Surfaces, Torrance and Sparrow, 1966 Microfacet Models for Refraction through Rough Surfaces, Walter et al., 2007

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Statistical Transmittance

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Statistical Transmittance

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Statistical Transmittance

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Statistical Transmittance

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Statistical Transmittance

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Davis and Mineev-Weinstein, 2011

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Results

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Phenomenological Transmittance

(non-physical)

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SLIDE 95 38 0.0 0.5 1.0
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SLIDE 96 39 0.0 0.5 1.0
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SLIDE 98 41 0.0 0.1 0.2
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Davis-Mineev-Weinstein Model

(physically based)

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Limitations

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Limitations

  • Same correlations everywhere
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Limitations

  • Same correlations everywhere
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Limitations

  • Same correlations everywhere
  • Unbiased distance sampling in heterogeneous media only

in special cases

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Future Work

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Future Work

  • Non-exponentiality as a tool for…
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Future Work

  • Non-exponentiality as a tool for…
  • Multi-scattering approximation
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Future Work

  • Non-exponentiality as a tool for…
  • Multi-scattering approximation
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Oz: The Great and Volumetric, Wrenninge et al. 2013

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Future Work

  • Non-exponentiality as a tool for…
  • Multi-scattering approximation
  • Level-of-detail for media
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Future Work

  • Non-exponentiality as a tool for…
  • Multi-scattering approximation
  • Level-of-detail for media
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Related Work

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Related Work

  • Radiation propagation in random media: From positive to

negative correlations in high-frequency fluctuations,
 Davis and Mineev-Weinstein, 2011

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Related Work

  • Radiation propagation in random media: From positive to

negative correlations in high-frequency fluctuations,
 Davis and Mineev-Weinstein, 2011

  • A generalized linear Boltzmann equation for non-classical

particle transport, Larsen and Vasquez, 2011

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Related Work

  • Radiation propagation in random media: From positive to

negative correlations in high-frequency fluctuations,
 Davis and Mineev-Weinstein, 2011

  • A generalized linear Boltzmann equation for non-classical

particle transport, Larsen and Vasquez, 2011

  • A Radiative Transfer Framework for Spatially-Correlated

Materials, Jarabo et al. 2018

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Comparison to Related Work

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Surfaces Heterogeneity Reciprocity Path Integral

Ours Larsen and Vasquez, 2011 Jarabo et al., 2018

≈

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Comparison to Related Work

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Surfaces Heterogeneity Reciprocity Path Integral

Ours Larsen and Vasquez, 2011 Jarabo et al., 2018

≈

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Comparison to Related Work

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Surfaces Heterogeneity Reciprocity Path Integral

Ours Larsen and Vasquez, 2011 Jarabo et al., 2018

≈

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Comparison to Related Work

48

Surfaces Heterogeneity Reciprocity Path Integral

Ours Larsen and Vasquez, 2011 Jarabo et al., 2018

≈

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Comparison to Related Work

48

Surfaces Heterogeneity Reciprocity Path Integral

Ours Larsen and Vasquez, 2011 Jarabo et al., 2018

≈

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SLIDE 123 49

Thank you!

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Thank you!