accurate translucent material rendering under spherical gaussian lights

PG 2011Pacific Graphics 2011

The

19th

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ter G

raph

ics a

nd A

pplic

ation

s (P

acifi

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011)

will

be

held

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Sept

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to 2

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in K

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, Tai

wan

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

1 Tsinghua University2 University of Massachusetts

PG 2012Pacific 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




• Translucent Material Rendering• BSSRDF representation

• Multiple Scattering & Single Scattering

Background


BRDF BSSRDF




• Environment Lighting• Natural illumination• Light modeling methods

• Spherical Harmonics

• Wavelets

• SRBFs

Background



Spherical HarmonicsWaveletsSRBFs



• 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








Introduction



Use SG lights!



Related Works


Jensen et al. 2001 d’Eon et al. 2011

• Translucent Material Rendering


• Limitations:• Vertical scattering path

[d’Eon et al, 2011] ground truth

𝐼𝑛𝑐𝑖𝑒𝑛𝑡 : 45 °

𝐼𝑛𝑐𝑖𝑒𝑛𝑡 : 80 °






Introduction



Use SG lights

Account for oblique scattering path



Related Works


Jensen et al. 2001 d’Eon et al. 2011

• Translucent Material Rendering


• Limitations:• Vertical scattering path

• Unable to handle area lights

(require sampling)



Related Works


• Translucent Rendering under Environment Lighting• Wang et al. 2005• Xu et al. 2007• ……

Based on pre-computation!







Introduction



Use SG lights

Account for oblique scattering path

No extra numerical integration or scene-dependent precomputation



• Main Contribution• An extended BSSRDF model• under Spherical Gaussian light• account for oblique scattering path• include multiple and single scattering

Overview





Approximating Multiple Scattering


𝐿𝐷 (𝑥𝑜 ,𝑜 )=∫Ω𝐺 ( 𝑖; 𝑖 𝑗 ,𝜆 𝑗 )(𝐹∫0

∞

𝑄 (𝑠 ) 𝑅 (𝑑 )𝑑𝑠)𝑑𝑖

• Multiple Scattering

𝑥𝑜

𝐿𝐷 (𝑥𝑜 ,𝑜 )=∫Ω𝐺 ( 𝑖′ ; 𝑖 𝑗 ′ ,𝜆 𝑗 ′ )(𝐹∫0

∞

𝑄 (𝑠 )𝑅 (𝑑 )𝑑𝑠)𝑑𝑖 ′


𝑖 𝑗𝜆 𝑗

𝑖 𝑗 ′𝜆 𝑗 ′

𝑖𝑖𝑖

𝑖𝑖

𝑖 ′𝑖 ′ 𝑖 ′ 𝑖 ′

𝑖 ′

𝑜

𝑠 𝑑𝑥𝑖





𝐿𝐷 (𝑥𝑜 ,𝑜 )=∫Ω𝐺 ( 𝑖′ ; 𝑖 𝑗 ′ ,𝜆 𝑗 ′ )(𝐹∫0

∞


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







𝐿𝐷 (𝑥𝑜 ,𝑜 )=∫Ω𝐺 ( 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 Scattering𝐿𝐷 (𝑥𝑜 ,𝑜 )=∫

Ω𝐺 ( 𝑖′ ; 𝑖 𝑗 ′ ,𝜆 𝑗 ′ )(𝐹∫0

∞





• Diffusion Function



𝜙 (𝑑 )= 14𝜋 𝐷 ⋅ 𝑒

− √𝜎 𝑎/𝐷𝑑

𝑑

𝑑

𝜙(𝑑)𝑥𝑜𝑥𝑖 𝑟

𝑠 𝑑

𝑖 ′

𝑖2

𝑑=√𝑠2+𝑟2−2 𝑠𝑟 (𝑖′ ⋅𝑖2)




𝐿𝐷 (𝑥𝑜 ,𝑜 )=𝐹 𝐶𝜙𝐿𝜙+𝐹𝐷𝐶𝐸𝐿𝐸




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



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

𝐿𝜙=∫Ω∫

0

∞

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



¿∫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




𝐿𝜙=∫Ω∫

0

∞

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



¿∫0

∞

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

• Our key insight• Can be represented by spherical

functions?• YES




• Diffusion Function



𝜙 (𝑑 )= 14𝜋 𝐷 ⋅ 𝑒

− √𝜎 𝑎/𝐷𝑑

𝑑

• Approximate withsum of Gaussians𝜙 (𝑑 )≈∑

𝑘𝑎𝑘𝑔 (𝑑 ;0 ,𝜆𝑘 )

𝑑

𝜙(𝑑)






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!




𝐿𝜙≈∫0

∞

𝑄 (𝑠 )𝑒−𝜆𝑘 (𝑠−𝑟 )2𝑁 𝑑𝑠



• inner integral


where

• Variable:

• Parameters:



𝐿𝜙≈∫0

∞

𝑄 (𝑠 )𝑒−𝜆𝑘 (𝑠−𝑟 )2𝑁 𝑑𝑠



• inner integral

Pre-fit into a 2D table of and

Now has analytical solution!

𝑠

𝑁




• directional derivative


𝐿𝐸=∫Ω∫0

∞

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

• Flux Integral


Additional term!




• inner integral


𝐿𝐸=2 𝜆𝑘∫0

∞

𝑄 (𝑠 )𝑒−𝜆𝑘 (𝑠 −𝑟 )2 𝑀𝑑𝑠

• Flux Integral


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




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

• Single Scattering

Approximating Single Scattering


𝐿(1) (𝑥𝑜 ,𝑜 )=𝜎𝑠∫0

∞

𝐹 𝑡𝐸(𝑠 ′)∫Ω𝑝𝐺 (𝑖′ ; 𝑖 𝑗′ ,𝜆 𝑗

′ )𝐸 (𝑠)𝑉 (𝑖 ′)𝑑𝑖 ′ 𝑑𝑠 ′

Fresnel transmittance termAttenuation term

𝐽 (𝑠 ′)=∫Ω𝑝 (𝑖′ ,𝑜 ′)𝐺 (𝑖 ′ ; 𝑖 𝑗′ ,𝜆 𝑗

′ )𝐸 (𝑠)𝑉 (𝑖 ′)𝑑𝑖 ′


𝑥𝑜

𝑜

𝑠 ′

scattering point






𝐽 (𝑠 ′)=∫Ω𝑝 (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∫Ω𝑝 (𝑖′ ,𝑜 ′)𝐺 (𝑖′ ; 𝑖 𝑗′ ,𝜆 𝑗

′ )𝑑𝑖 ′


𝑥𝑜Use SG center to approximate!






𝐽 (𝑠 ′ )=𝐸⋅𝑉∫Ω𝑝 (𝑖′ ,𝑜 ′)𝐺 (𝑖′ ; 𝑖 𝑗′ ,𝜆 𝑗

′ )𝑑𝑖 ′

• For Eddington phase function


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!



Results





Conclusion


• An extended BSSRDF model• under Spherical Gaussian light• accounts for oblique scattering path• including multiple and single scattering




• Heterogeneous translucent materials• Participating media• ……

Future Works





Thank you!Any questions?

The End



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