accurate translucent material rendering under spherical gaussian lights

PG 2011

Pacific Graphics 2011

The

19th

Pac

ific

Conf

eren

ce o

n Co

mpu

ter G

raph

ics

and

Appl

icati

ons

(Pac

ific

Gra

phic

s 20

11) w

ill b

e he

ld o

n Se

ptem

ber 2

1 to

23,

201

1 in

Kao

hsiu

ng, T

aiw

an. Accurate Translucent

Material Rendering under Spherical Gaussian LightsLing-Qi Yan1, Yahan Zhou2, Kun Xu1, Rui Wang2

1 Tsinghua University2 University of Massachusetts

PG 2012

Pacific Graphics 2012

Pacific Graphics 2011

2 | Kaohsiung, Taiwan

• Motivation• Natural illumination• Accurate rendering

• Goal• Analytic solution

• Challenge• Light integration complexity

Introduction

Accurate Translucent Material Rendering under Spherical Gaussian Lights

Pacific Graphics 2012

Pacific Graphics 2011

3 | Kaohsiung, Taiwan

• Translucent Material Rendering

• BSSRDF representation

• Multiple Scattering & Single Scattering

Background

Accurate Translucent Material Rendering under Spherical Gaussian Lights

BRDF BSSRDF

Pacific Graphics 2012

Pacific Graphics 2011

4 | Kaohsiung, Taiwan

• Environment Lighting

• Natural illumination

• Light modeling methods

• Spherical Harmonics

• Wavelets

• SRBFs

Background

Accurate Translucent Material Rendering under Spherical Gaussian Lights

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Spherical HarmonicsWaveletsSRBFs

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5 | Kaohsiung, Taiwan

• SRBF (Spherical Radial Basis Function)

• Typically Spherical Gaussian (SG)

• Useful Properties

• Closed under multiplication

• Has analytic solution under spherical integration

• Widely used in rendering• Environment lighting [Tsai and Shih 2006]• Light Transport [Green 2007]• BRDF [Wang 2009]

Background

Accurate Translucent Material Rendering under Spherical Gaussian Lights

Pacific Graphics 2012

Pacific Graphics 2011

6 | Kaohsiung, Taiwan

• Motivation• Natural illumination• Accurate rendering

• Goal• Analytic solution

• Challenge• Light integration complexity

Introduction

Accurate Translucent Material Rendering under Spherical Gaussian Lights

Pacific Graphics 2012

Use SG lights!

Pacific Graphics 2011

7 | Kaohsiung, Taiwan

Related Works

Accurate Translucent Material Rendering under Spherical Gaussian Lights

Jensen et al. 2001 d’Eon et al. 2011

• Translucent Material Rendering

Pacific Graphics 2012

• Limitations:

• Vertical scattering path

[d’Eon et al, 2011] ground truth

𝐼𝑛𝑐𝑖𝑒𝑛𝑡 : 45 °

𝐼𝑛𝑐𝑖𝑒𝑛𝑡 : 80 °

Pacific Graphics 2011

8 | Kaohsiung, Taiwan

• Motivation• Natural illumination• Accurate rendering

• Goal• Analytic solution

• Challenge• Light integration complexity

Introduction

Accurate Translucent Material Rendering under Spherical Gaussian Lights

Pacific Graphics 2012

Use SG lights

Account for oblique scattering path

Pacific Graphics 2011

9 | Kaohsiung, Taiwan

Related Works

Accurate Translucent Material Rendering under Spherical Gaussian Lights

Jensen et al. 2001 d’Eon et al. 2011

• Translucent Material Rendering

Pacific Graphics 2012

• Limitations:

• Vertical scattering path

• Unable to handle area lights

(require sampling)

Pacific Graphics 2011

10 | Kaohsiung, Taiwan

Related Works

Accurate Translucent Material Rendering under Spherical Gaussian Lights

• Translucent Rendering under Environment

Lighting

• Wang et al. 2005

• Xu et al. 2007

• ……

Based on pre-computation!

Pacific Graphics 2012

Pacific Graphics 2011

11 | Kaohsiung, Taiwan

• Motivation• Natural illumination• Accurate rendering

• Goal• Analytic solution

• Challenge• Light integration complexity

Introduction

Accurate Translucent Material Rendering under Spherical Gaussian Lights

Pacific Graphics 2012

Use SG lights

Account for oblique scattering path

No extra numerical integration or scene-dependent precomputation

Pacific Graphics 2011

12 | Kaohsiung, Taiwan

• Main Contribution

• An extended BSSRDF model

• under Spherical Gaussian light

• account for oblique scattering path

• include multiple and single

scattering

Overview

Accurate Translucent Material Rendering under Spherical Gaussian Lights

Pacific Graphics 2012

Pacific Graphics 2011

13 | Kaohsiung, Taiwan

Approximating Multiple Scattering

Accurate Translucent Material Rendering under Spherical Gaussian Lights

𝐿𝐷 (𝑥𝑜 ,𝑜 )=∫Ω

𝐺 ( 𝑖; 𝑖 𝑗 ,𝜆 𝑗 )(𝐹∫0

∞

𝑄 (𝑠 ) 𝑅 (𝑑 )𝑑𝑠)𝑑𝑖

• Multiple Scattering

𝑥𝑜

𝐿𝐷 (𝑥𝑜 ,𝑜 )=∫Ω

𝐺 ( 𝑖′ ; 𝑖 𝑗 ′ ,𝜆 𝑗 ′ )(𝐹∫0

∞

𝑄 (𝑠 )𝑅 (𝑑 )𝑑𝑠)𝑑𝑖 ′

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𝑖 𝑗𝜆 𝑗

𝑖 𝑗 ′𝜆 𝑗 ′

𝑖𝑖𝑖

𝑖𝑖

𝑖 ′𝑖 ′ 𝑖 ′ 𝑖 ′

𝑖 ′

𝑜

𝑠 𝑑𝑥𝑖

Pacific Graphics 2011

14 | Kaohsiung, Taiwan

Approximating Multiple Scattering

Accurate Translucent Material Rendering under Spherical Gaussian Lights

𝐿𝐷 (𝑥𝑜 ,𝑜 )=∫Ω

𝐺 ( 𝑖′ ; 𝑖 𝑗 ′ ,𝜆 𝑗 ′ )(𝐹∫0

∞

𝑄 (𝑠 )𝑅 (𝑑 )𝑑𝑠)𝑑𝑖 ′

• Define as a combination of fluence term and flux term [d’Eon et al. 2011]

• Multiple Scattering

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Pacific Graphics 2011

15 | Kaohsiung, Taiwan

Approximating Multiple Scattering

Accurate Translucent Material Rendering under Spherical Gaussian Lights

𝐿𝐷 (𝑥𝑜 ,𝑜 )=∫Ω

𝐺 ( 𝑖′ ; 𝑖 𝑗 ′ ,𝜆 𝑗 ′ )(𝐹∫0

∞

𝑄 (𝑠 ) (𝐶𝜙 𝜙 (𝑑 )+𝐶𝐸(−𝐷 (𝑛⋅𝛻𝜙 ) (𝑑)))𝑑𝑠)𝑑𝑖 ′

• Define as a combination of fluence term and flux term [d’Eon et al. 2011]

• Multiple Scattering

𝐿𝐷 (𝑥𝑜 ,𝑜 )=∫Ω

𝐺 ( 𝑖′ ; 𝑖 𝑗 ′ ,𝜆 𝑗 ′ )(𝐹∫0

∞

𝑄 (𝑠 )𝑅 (𝑑 )𝑑𝑠)𝑑𝑖 ′

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Pacific Graphics 2011

16 | Kaohsiung, Taiwan

• Diffusion Function

Approximating Multiple Scattering

Accurate Translucent Material Rendering under Spherical Gaussian Lights

𝜙 (𝑑 )= 14𝜋𝐷

⋅ 𝑒− √𝜎 𝑎/𝐷𝑑

𝑑

𝑑

𝜙(𝑑)𝑥𝑜𝑥𝑖 𝑟

𝑠 𝑑

𝑖 ′

𝑖2

𝑑=√𝑠2+𝑟2−2 𝑠𝑟 (𝑖′ ⋅𝑖2)

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𝐿𝐷 (𝑥𝑜 ,𝑜 )=𝐹 𝐶𝜙𝐿𝜙+𝐹𝐷𝐶𝐸𝐿𝐸

• Multiple Scattering

Approximating Multiple Scattering

Accurate Translucent Material Rendering under Spherical Gaussian Lights

𝐿𝜙=∫Ω∫

0

∞

𝐺 (𝑖′ ; 𝑖 𝑗′ ,𝜆 𝑗′ )𝑄 (𝑠 )𝜙 (𝑑 )𝑑𝑠𝑑𝑖 ′

𝐿𝐸=∫Ω∫0

∞

𝐺 ( 𝑖′ ;𝑖 𝑗′ ,𝜆 𝑗′ )𝑄 (𝑠)(−𝑛⋅𝛻𝜙) (𝑑 )𝑑𝑠𝑑𝑖 ′

• Fluence Integral

• Flux Integral

Pacific Graphics 2012

Pacific Graphics 2011

18 | Kaohsiung, TaiwanAccurate Translucent Material Rendering under Spherical Gaussian Lights

𝐿𝜙=∫Ω∫

0

∞

𝐺 (𝑖′ ; 𝑖 𝑗′ ,𝜆 𝑗′ )𝑄 (𝑠 )𝜙 (𝑑 )𝑑𝑠𝑑𝑖 ′

• Fluence Integral

Approximating Multiple Scattering

¿∫0

∞

𝑄 (𝑠 )(∫Ω 𝐺 (𝑖′ ; 𝑖 𝑗′ ,𝜆 𝑗′ ) ⋅ 𝜙 (𝑑) 𝑑𝑖 ′)𝑑𝑠

• Change integral order so that• Inner integral: Spherical• Outer integral: Linear

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Pacific Graphics 2011

19 | Kaohsiung, TaiwanAccurate Translucent Material Rendering under Spherical Gaussian Lights

𝐿𝜙=∫Ω∫

0

∞

𝐺 (𝑖′ ; 𝑖 𝑗′ ,𝜆 𝑗′ )𝑄 (𝑠 )𝜙 (𝑑 )𝑑𝑠𝑑𝑖 ′

• Fluence Integral

Approximating Multiple Scattering

¿∫0

∞

𝑄 (𝑠 )(∫Ω 𝐺 (𝑖′ ; 𝑖 𝑗′ ,𝜆 𝑗′ ) ⋅ 𝜙 (𝑑) 𝑑𝑖 ′)𝑑𝑠

• Our key insight• Can be represented by spherical

functions?• YES

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Pacific Graphics 2011

20 | Kaohsiung, Taiwan

• Diffusion Function

Approximating Multiple Scattering

Accurate Translucent Material Rendering under Spherical Gaussian Lights

𝜙 (𝑑 )= 14𝜋𝐷

⋅ 𝑒− √𝜎 𝑎/𝐷𝑑

𝑑

• Approximate withsum of Gaussians

𝜙 (𝑑 )≈∑𝑘

𝑎𝑘𝑔 (𝑑 ;0 ,𝜆𝑘 )𝑑

𝜙(𝑑)

Pacific Graphics 2012

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21 | Kaohsiung, TaiwanAccurate Translucent Material Rendering under Spherical Gaussian Lights

• Fluence Integral

Approximating Multiple Scattering

𝐿𝜙=∫0

∞

𝑄 (𝑠 )(∫Ω 𝐺 ( 𝑖′ ; 𝑖 𝑗′ ,𝜆 𝑗′ ) ⋅𝑒𝜆𝑘𝑑

2

𝑑𝑖 ′)𝑑𝑠𝑑=√𝑠2+𝑟2−2 𝑠𝑟 (𝑖′ ⋅𝑖2)

¿∫0

∞

𝑄 (𝑠 )𝑒−𝜆𝑘 (𝑠−𝑟 )2(∫Ω 𝐺 (𝑖′ ; 𝑖 𝑗′ ,𝜆 𝑗′ )⋅𝑒2𝑠𝑟 𝜆𝑘(𝑖

′⋅ 𝑖2−1 )𝑑𝑖 ′)𝑑𝑠Linear Integration Part Spherical Integration Part

¿∫0

∞

𝑄 (𝑠 )𝑒−𝜆𝑘 (𝑠−𝑟 )2(∫Ω 𝐺 (𝑖′ ; 𝑖 𝑗′ ,𝜆 𝑗′ )⋅𝐺(𝑖′ ; 𝑖2 ,2𝑠𝑟 𝜆𝑘)𝑑𝑖 ′)𝑑𝑠

Product-integral of two SGs!

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22 | Kaohsiung, TaiwanAccurate Translucent Material Rendering under Spherical Gaussian Lights

𝐿𝜙≈∫0

∞

𝑄 (𝑠 )𝑒−𝜆𝑘 (𝑠−𝑟 )2𝑁 𝑑𝑠

• Fluence Integral

Approximating Multiple Scattering

• inner integral

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where

• Variable:

• Parameters:

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23 | Kaohsiung, TaiwanAccurate Translucent Material Rendering under Spherical Gaussian Lights

𝐿𝜙≈∫0

∞

𝑄 (𝑠 )𝑒−𝜆𝑘 (𝑠−𝑟 )2𝑁 𝑑𝑠

• Fluence Integral

Approximating Multiple Scattering

• inner integral

Pre-fit into a 2D table of and

Now has analytical solution!

𝑠

𝑁

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24 | Kaohsiung, Taiwan

• directional derivative

Accurate Translucent Material Rendering under Spherical Gaussian Lights

𝐿𝐸=∫Ω∫0

∞

𝐺 ( 𝑖′ ;𝑖 𝑗′ ,𝜆 𝑗′ )𝑄 (𝑠)(−𝑛⋅𝛻𝜙) (𝑑 )𝑑𝑠𝑑𝑖

• Flux Integral

Approximating Multiple Scattering

Additional term!

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25 | Kaohsiung, Taiwan

• inner integral

Accurate Translucent Material Rendering under Spherical Gaussian Lights

𝐿𝐸=2𝜆𝑘∫0

∞

𝑄 (𝑠)𝑒−𝜆𝑘 (𝑠 −𝑟 )2 𝑀𝑑𝑠

• Flux Integral

Approximating Multiple Scattering

Exponential attenuation!

Pre-fit into a 2D table

Now has analytical solution!

𝐿𝐸=∫Ω∫0

∞

𝐺 ( 𝑖′ ;𝑖 𝑗′ ,𝜆 𝑗′ )𝑄 (𝑠)(−𝑛⋅𝛻𝜙) (𝑑 )𝑑𝑠𝑑𝑖

𝑠

𝑀 𝑖

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The outer integral: Sample along the refracted outgoing directionThe inner integral: To be analytically approximated!

• Single Scattering

Approximating Single Scattering

Accurate Translucent Material Rendering under Spherical Gaussian Lights

𝐿(1) (𝑥𝑜 ,𝑜 )=𝜎𝑠∫0

∞

𝐹 𝑡𝐸(𝑠 ′)∫Ω

𝑝𝐺 (𝑖′ ; 𝑖 𝑗′ ,𝜆 𝑗′ )𝐸 (𝑠)𝑉 (𝑖 ′)𝑑𝑖 ′ 𝑑𝑠 ′

Fresnel transmittance termAttenuation term

𝐽 (𝑠 ′)=∫Ω

𝑝 (𝑖′ ,𝑜 ′)𝐺 (𝑖 ′ ; 𝑖 𝑗′ ,𝜆 𝑗′ )𝐸 (𝑠)𝑉 (𝑖 ′)𝑑𝑖 ′

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𝑥𝑜

𝑜

𝑠 ′

scattering point

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27 | Kaohsiung, Taiwan

• Single Scattering

Approximating Single Scattering

Accurate Translucent Material Rendering under Spherical Gaussian Lights

𝐽 (𝑠 ′)=∫Ω

𝑝 (𝑖′ ,𝑜 ′)𝐺 (𝑖 ′ ; 𝑖 𝑗′ ,𝜆 𝑗′ )𝐸 (𝑠)𝑉 (𝑖 ′)𝑑𝑖 ′

Phase functionRefracted SG lightAttenuation termVisibility term

Use soft shadow technique to approximate!

𝐽 (𝑠 ′ )=𝐸⋅𝑉∫Ω

𝑝 (𝑖′ ,𝑜 ′)𝐺 (𝑖′ ; 𝑖 𝑗′ ,𝜆 𝑗′ )𝑑𝑖 ′

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𝑥𝑜Use SG center to approximate!

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28 | Kaohsiung, Taiwan

• Single Scattering

Approximating Single Scattering

Accurate Translucent Material Rendering under Spherical Gaussian Lights

𝐽 (𝑠 ′ )=𝐸⋅𝑉∫Ω

𝑝 (𝑖′ ,𝑜 ′)𝐺 (𝑖′ ; 𝑖 𝑗′ ,𝜆 𝑗′ )𝑑𝑖 ′

• For Eddington phase function

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𝐽 (𝑠 ′ )=𝐸⋅𝑉 (∫Ω 𝐺 (𝑖′ ; 𝑖 𝑗′ ,𝜆 𝑗′ )𝑑𝑖 ′+3𝑔∫

Ω

(𝑖′ ⋅𝑜′)𝐺 ( 𝑖′ ;𝑖 𝑗′ ,𝜆 𝑗′ )𝑑𝑖 ′ )

Both are analytical!

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29 | Kaohsiung, Taiwan

Results

Accurate Translucent Material Rendering under Spherical Gaussian Lights

Pacific Graphics 2012

Pacific Graphics 2011

30 | Kaohsiung, Taiwan

Conclusion

Accurate Translucent Material Rendering under Spherical Gaussian Lights

• An extended BSSRDF model

• under Spherical Gaussian light

• accounts for oblique scattering path

• including multiple and single scattering

Pacific Graphics 2012

Pacific Graphics 2011

31 | Kaohsiung, Taiwan

• Heterogeneous translucent materials

• Participating media

• ……

Future Works

Accurate Translucent Material Rendering under Spherical Gaussian Lights

Pacific Graphics 2012

Pacific Graphics 2011

32 | Kaohsiung, Taiwan

Thank you!Any questions?

The End

Accurate Translucent Material Rendering under Spherical Gaussian Lights

Pacific Graphics 2012