PG 2011Pacific Graphics 2011
The
19th
Pac
ific
Conf
eren
ce o
n Co
mpu
ter G
raph
ics a
nd A
pplic
ation
s (P
acifi
c Gr
aphi
cs 2
011)
will
be
held
on
Sept
embe
r 21
to 2
3, 2
011
in K
aohs
iung
, 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
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
Pacific Graphics 2012
Spherical HarmonicsWaveletsSRBFs
Pacific Graphics 2011
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 )=∫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 ′
Pacific Graphics 2012
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 )=∫Ω𝐺 ( 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
Pacific Graphics 2012
Pacific Graphics 2011
15 | Kaohsiung, Taiwan
Approximating Multiple Scattering
Accurate Translucent Material Rendering under Spherical Gaussian Lights
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
∞
𝑄 (𝑠 )𝑅 (𝑑 )𝑑𝑠)𝑑𝑖 ′
Pacific Graphics 2012
Pacific Graphics 2011
16 | Kaohsiung, Taiwan
• Diffusion Function
Approximating Multiple Scattering
Accurate Translucent Material Rendering under Spherical Gaussian Lights
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)
Pacific Graphics 2012
Pacific Graphics 2011
17 | Kaohsiung, Taiwan
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
Pacific Graphics 2012
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
Pacific Graphics 2012
Pacific Graphics 2011
20 | Kaohsiung, Taiwan
• Diffusion Function
Approximating Multiple Scattering
Accurate Translucent Material Rendering under Spherical Gaussian Lights
𝜙 (𝑑 )= 14𝜋 𝐷 ⋅ 𝑒
− √𝜎 𝑎/𝐷𝑑
𝑑
• Approximate withsum of Gaussians𝜙 (𝑑 )≈∑
𝑘𝑎𝑘𝑔 (𝑑 ;0 ,𝜆𝑘 )
𝑑
𝜙(𝑑)
Pacific Graphics 2012
Pacific Graphics 2011
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!
Pacific Graphics 2012
Pacific Graphics 2011
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
Pacific Graphics 2012
where
• Variable:
• Parameters:
Pacific Graphics 2011
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
Pacific Graphics 2012
Pacific Graphics 2011
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!
Pacific Graphics 2012
Pacific Graphics 2011
25 | Kaohsiung, Taiwan
• inner integral
Accurate Translucent Material Rendering under Spherical Gaussian Lights
𝐿𝐸=2 𝜆𝑘∫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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Pacific Graphics 2011
26 | Kaohsiung, Taiwan
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 ′)∫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 ′
Pacific Graphics 2012
𝑥𝑜
𝑜
𝑠 ′
scattering point
Pacific Graphics 2011
27 | Kaohsiung, Taiwan
• Single Scattering
Approximating Single Scattering
Accurate Translucent Material Rendering under Spherical Gaussian Lights
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 ′
Pacific Graphics 2012
𝑥𝑜Use SG center to approximate!
Pacific Graphics 2011
28 | Kaohsiung, Taiwan
• Single Scattering
Approximating Single Scattering
Accurate Translucent Material Rendering under Spherical Gaussian Lights
𝐽 (𝑠 ′ )=𝐸⋅𝑉∫Ω𝑝 (𝑖′ ,𝑜 ′)𝐺 (𝑖′ ; 𝑖 𝑗′ ,𝜆 𝑗
′ )𝑑𝑖 ′
• For Eddington phase function
Pacific Graphics 2012
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!
Pacific Graphics 2011
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