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Fractional Gaussian Fields for Modeling and Rendering of SpatiallyCorrelated Media Jie Guo 1* Yanjun Chen 1 Bingyang Hu 1 LingQi Yan 2 Yanwen Guo 1 Yuntao Liu 1 1 State Key Lab for Novel Software 2 University of California, Santa Barbara

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Fractional Gaussian Fields for Modeling and Rendering of Spatially‐Correlated Media

Jie Guo1* Yanjun Chen1 Bingyang Hu1 Ling‐Qi Yan2 Yanwen Guo1 Yuntao Liu1

1State Key Lab for Novel Software

Technology, Nanjing University

2University of California, Santa Barbara

*guojie@nju.edu.cn

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Participating Media

stanschaap.com

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Participating Media

videoblocks.com

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Participating Media

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Participating Media

scattering

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Participating Media

absorption

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Participating Media

Independent Scattering Approximation Classical Radiative Transport Equation

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Related Work

(But, a lot of works have observed correlations in particle distribution)

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Related Work

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Related Work

[Meng et al. 2015] [Müller et al. 2016] [Jarabo et al. 2018] [Bitterli et al. 2018]

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Related Work

[Meng et al. 2015] [Müller et al. 2016] [Jarabo et al. 2018] [Bitterli et al. 2018]

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Our method

 General media  Heterogeneity  Long‐range correlations

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Spatially‐Correlated Media

Uncorrelated Media Correlated Media Uncorrelated Media

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Spatially‐Correlated Media

Correlated Media Uncorrelated Media

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Spatially‐Correlated Media

𝜕

  • 𝑢

𝒚 𝒛 𝜏 𝜏

Transmittance: 𝑈 𝒚, 𝒛 𝑓 𝒚

  • 𝑓 𝒚,

𝜏 𝒚 𝜏 𝒚 𝜏 𝒚 Extinction field:

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Spatially‐Correlated Media

𝜕

  • 𝑢

𝒚 𝒛 𝜏 𝜏

Transmittance: 𝑈 𝒚, 𝒛 𝑓 𝒚

  • 𝑓 𝒚,

We model 𝜏 as a Fractional Gaussian Field (FGF) 𝜏 𝒚 𝜏 𝒚 𝜏 𝒚 Extinction field:

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Fractional Gaussian Fields

Random field A collection of random variables: 𝑌 𝑌 𝑢, 𝜕 , 𝑢 ∈ ℝ, 𝜕 ∈ Ω

𝑢 𝜕

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Fractional Gaussian Fields

𝑌 𝑌 𝑢, 𝜕 , 𝑢 ∈ ℝ, 𝜕 ∈ Ω

𝑢 𝜕 𝑌 𝑢,·

Random field A collection of random variables:

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Fractional Gaussian Fields

𝑌 𝑌 𝑢, 𝜕 , 𝑢 ∈ ℝ, 𝜕 ∈ Ω

𝑢 𝜕 realization 𝑌 ·, 𝜕

Random field A collection of random variables:

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Fractional Gaussian Fields

𝑌 𝑌 𝑢, 𝜕 , 𝑢 ∈ ℝ, 𝜕 ∈ Ω Gaussian (random) field A collection of Gaussian‐distributed random variables.

White Noise

Random field A collection of random variables:

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Fractional Gaussian Fields

Then what is “fractional”?

Let 𝐽 be the integral operator 𝐽 1 𝑦 𝐽 1 𝑦/2 𝐽 𝑦 ? 𝐽 𝑦 ? , 𝑏 0

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Fractional Gaussian Fields

Then what is “fractional”?

Let 𝐽 be the integral operator 𝐽 1 𝑦 𝐽 1 𝑦/2 𝐽 𝑦 ? 𝐽 𝑦 ? , 𝑏 0

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Fractional Gaussian Fields

With the fractional integral operator 𝐽

.[ ]

White Noise

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Fractional Gaussian Fields

.[ ]

White Noise

With the fractional integral operator 𝐽

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Fractional Gaussian Fields

.[ ]

White Noise

With the fractional integral operator 𝐽

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Fractional Gaussian Fields

Fractional Gaussian Fields

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Fractional Gaussian Fields

  • fractional Laplacian

𝑋: white noise

𝑒‐dimensional FGF: 𝑡 0

white noise

1 2 𝑡 3 2 , 𝑒 1

fractional Brownian motion (fBm)

𝑡 1, 𝑒 1

Brownian motion

The FGF family

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Fractional Gaussian Fields

Hurst parameter 𝐼 𝑡

  • 𝑋: white noise

𝑒‐dimensional FGF:

𝐼

Correlation

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Autocovariance Function

  • 𝑋: white noise

𝑒‐dimensional FGF:

Autocovariance function of pink noise: 𝐷 𝐼, 𝑒 Scaling term 𝑇 Power spectral density of white noise

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Autocovariance Function

  • 𝑋: white noise

𝑒‐dimensional FGF:

Autocovariance function of k‐fBm:

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Autocovariance Function

covariance matrix (1D)

Pink noise (H<0) k‐fBm (H>0)

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Autocovariance Function

covariance matrix (1D)

Pink noise k‐fBm Long‐range correlation

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2D Fractional Gaussian Fields

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2D Fractional Gaussian Fields

𝐼

Correlation

Aggregates

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Extinction Field

We model 𝜏 as a FGF :

  • Macro‐scale: Constant

Micro‐scale: FGF

Fixing 𝒚, 𝜏 𝒚 is a random variable with mean 𝜏 and its variance is controlled by the FGF

var𝜏 𝒚 ?

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Extinction Field

We model 𝜏 as a FGF :

The line‐averaged extinction 𝜏 𝒚

𝒚

  • 𝒚,
  • is a

Gaussian‐distributed random variable. var 𝜏 1 𝑢 𝜐 𝒚, 𝑢 𝜐 𝒚, 𝑢

1

𝑢 cov 𝒚, 𝒚 d𝑢d𝑢

  • autocovariance function of the

FGF

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Extinction Field

var 𝜏 1 𝑢 𝜐 𝒚, 𝑢 𝜐 𝒚, 𝑢

1

𝑢 cov 𝒚, 𝒚 d𝑢d𝑢

  • Pink noise

(H<0)

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Extinction Field

var 𝜏 1 𝑢 𝜐 𝒚, 𝑢 𝜐 𝒚, 𝑢

1

𝑢 cov 𝒚, 𝒚 d𝑢d𝑢

  • k‐fBm

(H>0)

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Extinction Field

Numerical verification:

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Extinction Field

Pink noise k‐fBm

One‐point scale‐independence

Numerical verification:

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Extinction Field

The variance of 𝜏 𝒚 :

White noise Pink noise K‐fBm

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Transmittance

Ensemble‐averaged transmittance

Tr 𝑢 𝑓 𝑓𝑞𝑒𝑔 𝜏 d𝜏

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Transmittance

Tr 𝑢 𝑓 𝑓𝑞𝑒𝑔 𝜏 d𝜏

  • Ensemble‐averaged transmittance

characteristic function 𝜒 𝐣𝑢

𝜏 ~ Γ 𝜏 var 𝜏 , 𝜏 var 𝜏

  • Use gamma distribution for non‐negative extinction
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Transmittance

Tr 𝑢 𝑓 𝑓𝑞𝑒𝑔 𝜏 d𝜏

  • Ensemble‐averaged transmittance
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Transmittance

Transparent

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Transmittance

𝐼 0.2 𝐼 0.5 𝐼 0.8

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Rendering Techniques

Energy‐Conserving Volumetric Rendering Equation

Transport kernel:

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Results

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Short‐range correlations Long‐range correlations

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Low density

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Comparing to the GBE [Jarabo et al. 2018]

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Comparing to the Bitterli model [Bitterli et al. 2018]

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Uncorrelated media

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Correlated media

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Conclusion

  • A mathematical tool for physically‐based modeling spatial correlations in

random media.

  • Using k‐th order fBm to generate long‐range correlations.
  • The usage of the non‐exponential transmittance functions in an energy‐

conserving RTE framework.

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Thank you! Q&A