On the scattering of light : various models and methods used in computer graphics and computer...

On the Scattering of light�

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Scattering

Volumetric scattering in CG

Tomoyuki Nishita, Yasuhiro Miyawaki, and Eihachiro Nakamae. 1987. A shading model for atmospheric scattering considering luminous intensity distribution of light sources. SIGGRAPH Comput. Graph. 21, 4 (August 1987), 303-310. DOI=10.1145/37402.37437 http://doi.acm.org/10.1145/37402.37437

Tomoyuki Nishita, Takao Sirai, Katsumi Tadamura, and Eihachiro Nakamae. 1993. Display of the earth taking into account atmospheric scattering. In Proceedings of the 20th annual conference on Computer graphics and interactive techniques (SIGGRAPH '93). ACM, New York, NY, USA, 175-182. DOI=10.1145/166117.166140 http://doi.acm.org/10.1145/166117.166140

Tomoyuki Nishita and Eihachiro Nakamae. 1994. Method of displaying optical effects within water using accumulation buffer. In Proceedings of the 21st annual conference on Computer graphics and interactive techniques (SIGGRAPH '94). ACM, New York, NY, USA, 373-379. DOI=10.1145/192161.192261 http://doi.acm.org/10.1145/192161.192261

Tomoyuki Nishita, Yoshinori Dobashi, and Eihachiro Nakamae. 1996. Display of clouds taking into account multiple anisotropic scattering and sky light. In Proceedings of the 23rd annual conference on Computer graphics and interactive techniques (SIGGRAPH '96). ACM, New York, NY, USA, 379-386. DOI=10.1145/237170.237277 http://doi.acm.org/10.1145/237170.237277

Yonghao Yue, Kei Iwasaki, Bing-Yu Chen, Yoshinori Dobashi, and Tomoyuki Nishita. 2010. Unbiased, adaptive stochastic sampling for rendering inhomogeneous participating media. ACM Trans. Graph. 29, 6, Article 177 (December 2010), 8 pages. In ACM SIGGRAPH Asia 2010 papers (SIGGRAPH ASIA '10). DOI=10.1145/1882261.1866199 http://doi.acm.org/10.1145/1882261.1866199

Volumetric scattering in CG

Outline

• Reflection – Diffuse reflection – Specular reflection – BRDF

• Transmission – Diffuse transmission – Specular transmission – BTDF

• Scattering – BSSRDF

• Dipole/Multipole • Plane-parallel

approximation • Empirical BSSRDF

• Why BSSDRF is bad?

• Scattering model – Participating medium

• Absorption • Emission • In-scattering • Out-scattering

– Rendering equations • Airlight approximation • Born series • Neumann series • M’s series

Reflection Diffuse reflection Specular reflection BRDF Transmission Diffuse transmission Specular transmission BTDF Scattering BSSRDF Dipole/Multipole Plane-parallel Empirical BSSRDF Why BSSDRF is bad? Scattering model Participating medium Absorption Emission In-scattering Out-scattering Rendering equations Airlight approximation Born series Neumann series M’s series

REFLECTION, TRANSMISSION, AND

SCATTERING OF LIGHT

Diffuse reflection 拡散反射

• Constant irradiance

– Independent to viewpoint

• Pure diffuse surface

– Lambertian

Specular reflection 鏡面反射

• Angle of reflection – is the angle of

incidence

• Highlights – More reflection near

to the angle of reflection

• Model – Phong

– Blinn

– Cook-Trrance

– Trrance-Sparrow Highlights

Mirror

Diffuse + Specular

Diffuse + Specular ?

𝜔 𝜔

• Bidirectional Reflectance Distribution Function – 双方向反射率分布

関数

• 6 parameters – Point 𝑥

– angles of incidence 𝜔 and reflection 𝜔

• If isotropic and homogeneous – 3 parameters

Wojciech Matusik, Hanspeter Pfister, Matthew Brand, and Leonard McMillan. 2003. Efficient isotropic BRDF measurement. In Proceedings of the 14th Eurographics workshop on Rendering (EGRW '03). Eurographics Association, Aire-la-Ville, Switzerland, Switzerland, 241-247.

Isotropic BRDF measurement

武田祐樹, 坂口嘉之, 田中弘美, 少数視点画像の反射光解析に基づくシルクライク織物の異方性反射レンダリング, 芸術科学会論文誌, Vol. 7, No. 4, pp.132-143, 2008 .

Anisotropic BRDF

Refraction 回折 Transmission 透過

• Transmitted light

– Translucent materials

– All non-metals

• Refraction

– Snell’s low

Specular Transmission 鏡面透過

• Angle of refraction

– depends on • the angle of

incidence

• the indexes of refraction of two materials

Diffuse transmission 拡散透過

• Constant irradiance

– Independent to viewpoint inside the material

Diffuse + Specular transmission

frosted glass

a light bulb

• Bidirectional Transmittance Distribution Function

– 双方向透過率分布関数

Reflection + Transmission

• Reflection

– BRDF

• Transmission

– BTDF

Subsurface scattering 表面下散乱

Subsurface scattering

• Scattering

– Inside medium

– Some come the surface

– Some absorbed

• Outgoing light

– From a point different from the incident point

Surface reflection

Henrik Wann Jensen, Stephen R. Marschner, Marc Levoy, and Pat Hanrahan. 2001. A practical model for subsurface light transport. In Proceedings of the 28th annual conference on Computer graphics and interactive techniques (SIGGRAPH '01). ACM, New York, NY, USA, 511-518.

Surface Subsurface

Surface reflection

Surface

Subsurface

Surface reflection

Surface reflection Surface Subsurface

Single scattering 単散乱

Double scattering 二重散乱

Multiple scattering 多重散乱

Multiple scattering only

Single scattering only

Single + Multiple scattering + Fresnel reflection

Single scattering + Multiple scattering + Fresnel reflection

Single scattering

Multiple scattering

Single + Multiple scattering

Craig Donner, Jason Lawrence, Ravi Ramamoorthi, Toshiya Hachisuka, Henrik Wann Jensen, and Shree Nayar. 2009. An empirical BSSRDF model. ACM Trans. Graph. 28, 3, Article 30 (July 2009), 10 pages. In ACM SIGGRAPH 2009 papers (SIGGRAPH '09). DOI=10.1145/1531326.1531336

Single scattering +

Multiple scattering

Single scattering

BSSRDF

• Bidirectional Scattering Surface Reflectance Distribution Function – 双方向散乱面（深層散

乱）反射率分布関数

• 8 parameters – Points of incidence 𝑥

and outgoing 𝑥 , angles of incidence 𝜔 and outgoing 𝜔

• If isotropic and homogeneous – 5 parameters

• Sub-Surface Scattering Irradiance Distribution – 表面下放射照度分布

• Outgoing light – Diffuse transmission

– Independent to viewpoint

• 3 parameters – Isotropic and

homogeneous

– Point of outgoing 𝑥 , angle of incidence 𝜔

高村幸平, 真鍋知久, 玉木徹, 金田和文 : 「表面下散乱シミュレーションと放射照度分布特性を考慮した表示モデル」, 電子情報通信学会技術報告パターン認識・メディア理解研究会 PRMU2009-77, Vol.109, No.249, pp.37-42, 広島大学, 広島(2009 10).

SUBSURFACE SCATTERING MODELS

Dipole model

• Homogeneous and semi-infinite medium

• Single scattering + multiple scattering (isometric and diffusion)

• Sum of two terms – Each represents a

virtual point light source

Multiple scattering with Dipole model

Single scattering

Multipole model

• Finite thickness

– Infinite sum of Dipoles

– A multipole model

• Finite-thick layers

– The use of some Multipole models

Craig Donner and Henrik Wann Jensen. 2005. Light diffusion in multi-layered translucent materials. ACM Trans. Graph. 24, 3 (July 2005), 1032-1039. In ACM SIGGRAPH 2005 Papers (SIGGRAPH '05). DOI=10.1145/1073204.1073308

Backlight Sidelight

Multipole approximation of human skin layers

Dipole approximation

Multipole approximation

Finite-thick medium (paper)

Plane-parallel approximation

• Dipole – Plane, semi-infinite,

homogeneous

• Multipole – Finite homogeneous

layers

• Plane-parallel – Layered

homogeneous

– Depends on the depth 𝑧 from the surface

Single layer, semi-infinite

Finite-thick layers

Layered homogeneous

Analytic solution to plane-parallel

新谷幹夫, 白石路雄, 土橋宜典, 岩崎慶, 西田友是, "固有解を用いた表面下散乱の高速表示", Visual Computing / グラフィクスと CAD 合同シンポジウム 2008, 2008. http://nis-lab.is.s.u-tokyo.ac.jp/nis/abs_cgi.html#ipsj09-2

Integro-differential equation

Integro-ordinary differential equation

Discretization

Eigensolution Eigenfunction

Eigenvalue

Anisotropic Plane-parallel model

Nelson Max, Suguru Saito, Kazuya Watanabe, Masayuki Nakajima, "Rendering Grass Blowing in The Wind with Global Illumination", Pacific Graphics 2009, 2009. Nelson Max, Kazuya Watanabe, Suguru Saito, Masayuki Nakajima, "Plane-Parallel Radiance Transport for Rendering Grass Blowing in The Wind", Visual Computing / グラフィクスと CAD 合同シンポジウム 2009, 2009.

新谷幹夫，白石路雄, 土橋宜典，岩崎慶，西田友是, "異方性Plane-Parallel散乱モデルによる毛髪状物体の高速レンダリング", Visual Computing / グラフィクスと CAD 合同シンポジウム 2010, 2010.

Empirical BSSRDF model

• Semi-infinite homogeneous plane medium

• Simulating 5D parameters of BSSRDF

– 6 parameters

– 0.85 million patterns

• Function fitting

– Each incident / outgoing angles / points

– 36GB data

Single scattering ＋ Dipole model

Single scattering ＋ Empirical BSSRDF

Monte Carlo Method

Monte Carlo simulation

Single scattering

+ Empirical BSSRDF

BSSRDF, A WRONG MODEL

𝜔 𝜔

• 6 parameters – Point 𝑥

– angles of incidence 𝜔 and reflection 𝜔

BRDF, OK!

𝜔 𝜔

• BRDF

– Works for a point on a curved surface

– Angles relative to the normal of the point

BSSRDF

• 8 parameters

– Points of incidence 𝑥 and outgoing 𝑥 , angles of incidence 𝜔 and outgoing 𝜔

BSSRDF ?

• 8 parameters

– Points of incidence 𝑥 and , angles of incidence 𝜔 and outgoing 𝜔

NG : curved surface Reflection Diffuse reflection Specular reflection BRDF Transmission Diffuse transmission Specular transmission BTDF Scattering BSSRDF Dipole/Multipole Plane-parallel Empirical BSSRDF Why BSSDRF is bad? Scattering model Participating medium Absorption Emission In-scattering Out-scattering Rendering equations Airlight approximation Born series Neumann series M’s series

BSSRDF ?

• 8 parameters

– Points of incidence 𝑥 and , angles of incidence 𝜔 and outgoing 𝜔

NG : curved surface

NG : transmission

Single scattering vs BSSRDF

Single scattering only Multiple scattering only with dipole model

Transmitted light

No light transmission

Blurred by BSSRDF

Curvature dependent BRDF

Hiroyuki Kubo, Yoshinori Dobashi, and Shigeo Morishima. 2010. Curvature-dependent reflectance function for rendering translucent materials. In ACM SIGGRAPH 2010 Talks (SIGGRAPH '10). ACM, New York, NY, USA, , Article 46 , 1 pages. DOI=10.1145/1837026.1837086 http://doi.acm.org/10.1145/1837026.1837086 久保尋之, 土橋宜典, 森島繁生, "半透明物体の高速描画に向けた曲率に依存する反射関数の近似式", Visual Computing / グラフィクスと CAD 合同シンポジウム 2010, 2010.

• Light is reflected

– Even when 𝜃 > 180°

𝜃 < 180°

𝜃 = 180°

𝜃 > 180°

SCATTERING, A CORRECT WAY

BSSRDF

• Subsurface scattering

Translucent material 半透明物質

Scattering model

• Volumetric scattering

Participating medium 関与媒質

Scattering model

• Volumetric scattering

Participating medium 関与媒質

Scattering model

Forward scattering

Backward scattering

Phase function

Scattering model

Phase function

Rendering by ray marching

the line of sight

Numerical integration along

Light in participating medium

Attenuation 減衰

Absorption 吸収

Out-scattering

In-scattering

Multiple scattering

Emission 発光

Usually ignored

Absorption

𝐿 𝐿

(Macroscopic) Beer-Lambert Law

モル吸光係数 Molar absorption coefficient

モル濃度 molarity

path length (thickness) 光路長，厚さ

光路長補正項

透明溶液 𝛽 = 1 混濁溶液 𝛽 ≠ 1

田村守, 「生体分光学の基本原理とその医学応用」，非侵襲・可視化技術ハンドブック, 小川誠二, 上野照剛監修, NTS, pp.253-266, 2007.

log𝐿𝐿

= 𝜀𝐶𝛽𝑑

Absorption

𝐿 𝐿

光路長補正項

Light pulse

log𝐿𝐿

= 𝜀𝐶𝛽𝑑

Absorption

𝐿 𝐿

光路長補正項

Light pulse

log𝐿𝐿

= 𝜀𝐶𝛽𝑑

100 ps = 3 cm by the speed of light

Absorption

𝐿 𝐿

吸収係数 Absorption coefficient Absorption cross section

path length (thickness)

log𝐿𝐿

= 𝜎 𝑑

Absorption

𝑑𝐿 = − 𝜎 𝐿 𝑑𝑠

𝐿 𝐿 + 𝑑𝐿

( scopic) Beer-Lambert Law

Infinitesimal path length

𝑑𝑠 Reflection Diffuse reflection Specular reflection BRDF Transmission Diffuse transmission Specular transmission BTDF Scattering BSSRDF Dipole/Multipole Plane-parallel Empirical BSSRDF Why BSSDRF is bad? Scattering model Participating medium Absorption Emission In-scattering Out-scattering Rendering equations Airlight approximation Born series Neumann series M’s series

Absorption

𝑑𝑠

log𝐿𝐿 = 𝜎 𝑑 𝐿 = 𝐿 𝑒

Integration from 0 to 𝑑

Exponential attenuation

Absorption

𝑑𝑠

𝐿 = 𝐿 𝑒 ∫

When 𝜎 is

not constant

optical depth

Out-scattering

(Microscopic) Beer-Lambert Law

散乱係数 Scattering coefficient Scattering cross section

𝑑𝑠

Attenuation = absorption + out-scattering

𝑑𝑠

𝐿 = 𝐿 𝑒

𝑑𝐿 = − 𝜎 𝐿 𝑑𝑠 Absorption

Out-scattering

𝑑𝐿 = −(𝜎 + 𝜎 ) 𝐿 𝑑𝑠 Attenuation

𝑑𝐿 = −𝜎 𝐿 𝑑𝑠

消散係数・消滅係数 Extinction coefficient

Emission

𝑑𝑠

𝑑𝐿 = 𝜎 𝐿 𝑑𝑠 Emission

Absorption coefficient (Emission of absorbed energy)

Emitted light Usually ignored

In-scattering

𝑑𝑠

𝑑𝐿(𝑥, 𝜔) = 𝜎 𝑝 𝑥, 𝜔, 𝜔 𝐿 𝑥, 𝜔 𝑑𝜔 𝑑𝑠

𝜔′

Phase function (from 𝜔′ to 𝜔 at 𝑥) (usually ignore 𝑥)

Incident light (from 𝜔′ at 𝑥)

Integrating all incoming lights over the sphere

(𝜔 ⋅ 𝛻)𝐿(𝑥, 𝜔) or

Rendering equation

𝑑𝑠

𝑑𝐿 𝑥, 𝜔 = −𝜎 𝑥 𝐿 𝑥, 𝜔 𝑑𝑠 + 𝜎 𝑥 𝑝 𝑥, 𝜔, 𝜔 𝐿 𝑥, 𝜔 𝑑𝜔 𝑑𝑠

𝜔′

Light transport equation Volume rendering equation (differential form)

In-scattering term Attenuation term

Rendering equation

𝑑𝑠

(𝜔 ⋅ 𝛻) 𝐿 𝑥, 𝜔 = −𝜎 𝑥 𝐿 𝑥, 𝜔 + 𝜎 𝑥 𝑝 𝑥, 𝜔, 𝜔 𝐿 𝑥, 𝜔 𝑑𝜔

𝜔′

Light transport equation Volume rendering equation (differential form)

In-scattering term Attenuation term

Rendering equation

𝐿 𝑥′, 𝜔 𝐿 𝑥, 𝜔

𝐿 𝑥, 𝜔 = 𝑒 ( , )𝐿 𝑥 + 𝑠𝜔,𝜔

+ 𝑒 ( , )𝜎 𝑥′ 𝑝 𝑥′, 𝜔, 𝜔 𝐿 𝑥′, 𝜔 𝑑𝜔 𝑑𝑥′

𝜔′

Light transport equation Volume rendering equation ( form)

In-scattering term

Attenuation term

𝜏(𝑥, 𝑥′) = 𝜎 𝑠 𝑑𝑠 Optical depth

Attenuation of scattered light

𝑥′ 𝑥

𝐿 𝑥 + 𝑠𝜔,𝜔

𝑥 + 𝑠𝜔

Airlight approximation to in-scattering

Schechner, Y.Y.; Karpel, N.; , "Clear underwater vision," Computer Vision and Pattern Recognition, 2004. CVPR 2004. Proceedings of the 2004 IEEE Computer Society Conference on , vol.1, no., pp. I-536- I-543 Vol.1, 27 June-2 July 2004 doi: 10.1109/CVPR.2004.1315078 URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1315078&isnumber=29133

Nayar, S.K.; Narasimhan, S.G.; , "Vision in bad weather," Computer Vision, 1999. The Proceedings of the Seventh IEEE International Conference on , vol.2, no., pp.820-827 vol.2, 1999 doi: 10.1109/ICCV.1999.790306 URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=790306&isnumber=17141

Vision in bad weather Clear underwater vision

𝐿 𝑥, 𝜔 = 𝑒 ( , )𝐿 𝑥 + 𝑠𝜔,𝜔 + 1 − 𝑒 , 𝐿

Airlight 天空光

Opacity 不透明度

Transparency 透明度

Born series

𝐿 𝑥, 𝜔 = 𝑒 ( , )𝐿 𝑥 + 𝑠𝜔,𝜔

+ 𝑒 ( , )𝜎 𝑥′ 𝑝 𝑥′, 𝜔, 𝜔 𝐿 𝑥′, 𝜔 𝑑𝜔 𝑑𝑥′

𝐿 𝑥, 𝜔 = 𝑒 ( , )𝐿 𝑥 + 𝑠𝜔,𝜔

+ 𝑒 ( , )𝜎 𝑥′ 𝑝 𝑥′, 𝜔, 𝜔 𝐿 𝑥′, 𝜔 𝑑𝜔 𝑑𝑥′

first-order Born approximation (single scattering)

second-order Born approximation (double scattering)

zero-order Born approximation (no scattering)

𝐿 , 𝐿 , 𝐿 , … are called Born series.

Born approximation

𝐿 = 𝐿 − 𝐿

Born approximation of scattering term

𝐿 = 𝐿 log𝐿𝐿

Rytov approximation of scattering term

Weak Scattering Acoustic Wave Field Analysis Using Backward Propagation Rytov Transform Akira Yamada Jpn. J. Appl. Phys. 36 (1997) 3203 http://jjap.jsap.jp/link?JJAP/36/3203

Neumann series

𝐿 𝑥, 𝜔 = 𝑒 ( , )𝐿 𝑥 + 𝑠𝜔,𝜔

+ 𝑒 ( , )𝜎 𝑥′ 𝑝 𝑥′, 𝜔, 𝜔 𝐿 𝑥′, 𝜔 𝑑𝜔 𝑑𝑥′

𝐿 = 𝑒 ( , )𝐿 𝑥 + 𝑠𝜔, 𝜔

𝑇 = 𝑑𝑥 𝑒 ( , )𝜎 𝑥′ 𝑑𝜔 𝑝 𝑥′, 𝜔, 𝜔

𝐿 = 𝐿 + 𝑇𝐿

　 = 1 − 𝑇 𝐿 = 𝐿 + 𝑇𝐿 + 𝑇 𝐿 + 𝑇 𝐿 +⋯ = 𝑇 𝐿

Incident light

Linear operator of scattering

Neumann series

Similar to interreflection cancellation operator (Seitz, ICCV2005)

Neumann series

𝐿 𝑥, 𝜔 = 𝑒 ( , )𝐿 𝑥 + 𝑠𝜔,𝜔

+ 𝑒 ( , )𝜎 𝑥′ 𝑝 𝑥′, 𝜔, 𝜔 𝐿 𝑥′, 𝜔 𝑑𝜔 𝑑𝑥′

𝐿 = 𝑒 ( , )𝐿 𝑥 + 𝑠𝜔, 𝜔

𝑇 = 𝑑𝑥 𝑒 ( , )𝜎 𝑥′ 𝑑𝜔 𝑝 𝑥′, 𝜔, 𝜔

𝐿 = 𝐿 + 𝑇𝐿

　 = 1 − 𝑇 𝐿 = 𝐿 + 𝑇𝐿 + 𝑇 𝐿 + 𝑇 𝐿 +⋯ = 𝑇 𝐿

Incident light

Linear operator of scattering

Neumann series

Mukaigawa series

𝐿 = 𝑇 𝐿

Neumann series of volume rendering equation

Linear operator (integral)

Continuous light

distribution

Mukaigawa series of light transport (CVPR2010, MIRU2010)

Light transport matrix

Discretized light field

Single, double, triple, … scattering

向川康博，ラメシュラスカル，八木康史， "散乱媒体中のライトトランスポートの解析'', MIRU2010, pp. 665-672, 2010. Y.Mukaigawa, Y.Yagi, R.Raskar, "Analysis of Light Transport in Scattering Media", Proc. CVPR2010, 2010.

Decomposition into each scattering

Outline

• Reflection – Diffuse reflection – Specular reflection – BRDF

• Transmission – Diffuse transmission – Specular transmission – BTDF

• Scattering – BSSRDF

• Dipole/Multipole • Plane-parallel

approximation • Empirical BSSRDF

• Why BSSDRF is bad?

• Scattering model – Participating medium

• Absorption • Emission • In-scattering • Out-scattering

– Rendering equations • Airlight approximation • Born approximation • Neumann series • M’s series

On the Scattering of light�

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