Eurographics 2012, Cagliari, Italy

A General BRDF A General BRDF RepresentationRepresentation Based on TensorBased on Tensor

DecompositionDecomposition

Ahmet Bilgili1, Aydın Öztürk2 and Murat Kurt1

1 International Computer Institute, Ege University, TURKEY2 Department of Computer Engineering, Yasar University, TURKEY

Eurographics 2012, Cagliari, Italy

• Given a set of precise reflectance measurements from real surfaces is it possible to represent

these measurements compactlycompactly and accuratelyaccurately?

• The proposed method should also lend itself to developing an efficientefficient and simple simple importance sampling algorithm.

Our GoalOur Goal

2

isotropic anisotropic

Eurographics 2012, Cagliari, Italy

Previous Work – Analytical ModelsPrevious Work – Analytical Models

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Phong [Pho75]Phong [Pho75]Blinn-Phong [Bli77]Blinn-Phong [Bli77]Ward [Ward92]Ward [Ward92]Lafortune et al. [LFTG97]Lafortune et al. [LFTG97]Ward-Duer [Due05]Ward-Duer [Due05]

Torrance-Sparrow [TS67]Torrance-Sparrow [TS67]Cook-Torrance [CT81]Cook-Torrance [CT81]He et al. [HTSG91]He et al. [HTSG91]Oren-Nayar [ON94]Oren-Nayar [ON94]

Analytical BRDF Analytical BRDF ModelsModels

Emprical Emprical BRDF ModelsBRDF Models

Physically Physically based BRDF based BRDF

ModelsModels

Anisotropic Anisotropic BRDF ModelsBRDF Models

Lineer BRDF Lineer BRDF ModelsModels

Kajiya [Kaj85]Kajiya [Kaj85]Poulin-Fournier [PF90] Poulin-Fournier [PF90] Ward [War92]Ward [War92]Lafortune et al. [LFTG97]Lafortune et al. [LFTG97]Ashikhmin-Shirley [AS00]Ashikhmin-Shirley [AS00]Ward-Duer [Due05]Ward-Duer [Due05]Edwards et al. [EBJ*06]Edwards et al. [EBJ*06]

Westin et al. [WAT92] Westin et al. [WAT92] Koenderink et al. [KvDS96]Koenderink et al. [KvDS96]SchrSchrööder and Sweldens [SS95]der and Sweldens [SS95]Lalonde and FournierLalonde and Fournier [LF97][LF97]Stark et al. [SAS05]Stark et al. [SAS05] ÖÖztztüürk et al. [OKBG08]rk et al. [OKBG08]

[EBJ*06][EBJ*06][CT81][CT81]

Eurographics 2012, Cagliari, Italy

Previous Work – Data-Driven ModelsPrevious Work – Data-Driven Models

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Matusik et al. [MPBM03]Matusik et al. [MPBM03]Romerio et al. [RVZ08]Romerio et al. [RVZ08]

Kautz and McCool [KM99]Kautz and McCool [KM99]McCool et al. [MAA01]McCool et al. [MAA01]Lawrence et al. Lawrence et al. [LRR04][LRR04]

Data-Driven BRDF Data-Driven BRDF ModelsModels

Measurement Measurement based BRDF based BRDF

ModelsModels

Factorization Factorization based BRDF based BRDF

ModelsModels

p q

)p(ω)q(ω)p(ωf ihor

[MPBM03][MPBM03]

[MAA01][MAA01]

[LRR04][LRR04]

Eurographics 2012, Cagliari, Italy

Previous Work – Importance Previous Work – Importance SamplingSampling

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Phong [Pho75]Phong [Pho75]Blinn-Phong [Bli77] Blinn-Phong [Bli77] Ward [War92]Ward [War92]Lafortune [LFTG97] Lafortune [LFTG97] Ashikhmin-Shirley [AS00]Ashikhmin-Shirley [AS00]Ward-Duer [Due05] Ward-Duer [Due05] Edwards et al. [EBJ 06]∗Edwards et al. [EBJ 06]∗

Lawrence et al. Lawrence et al. [LRR04][LRR04]

Importance Importance SamplingSampling

Analytical Analytical BRDF ModelsBRDF Models

Factorization Factorization based BRDF based BRDF

ModelsModels

General BRDF General BRDF Sampling Sampling MethodsMethods

Lawrence et al. Lawrence et al. [LRR0[LRR055]]Montes et al. [MUGL08]Montes et al. [MUGL08]

[LRR04][LRR04][EBJ*06][EBJ*06]

400 400 samples/pixelsamples/pixel

400 400 samples/pixelsamples/pixel

Eurographics 2012, Cagliari, Italy

Previous Work – Tensor FactorizationPrevious Work – Tensor Factorization

6

Vasilescu and Terzopulos [VT04]Vasilescu and Terzopulos [VT04]Wang et al. [WWS*05]Wang et al. [WWS*05]

Computer Computer GraphicsGraphics

Data Data CompressionCompression

BRDF Data BRDF Data RepresentationRepresentation

Sun et al. [SZC 07]∗Sun et al. [SZC 07]∗

[VT04[VT04]

[WWS*05][WWS*05]

[VT04[VT04] [WWS*05][WWS*05]OriginalOriginal

BTF Data BTF Data RepresentationRepresentation

[SZC 07]∗[SZC 07]∗

Eurographics 2012, Cagliari, Italy

Key IdeaKey Idea

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3D Tensor Data3D Tensor DataI I XX J J XX K K

II

JJKK

TT

1D Vector1D VectorI I XX P P

1D Vector1D VectorJ J XX Q Q

A ScalarA ScalarP P XX Q x R Q x R

1D Vector1D VectorK K XX R R

P = Q = R =1P = Q = R =1

XX

ZZ

gg

YY

Project Project 33D D Tensor dataTensor data into products of into products of 11D functionD functionss and and a core tensor: a core tensor:

)Z()Y()X(),,(T kjigkji )Z()Y()X(),,(T kjigkji

TuckerTucker

Eurographics 2012, Cagliari, Italy

Our BRDF RepresentationOur BRDF Representation• Our BRDF model is based on halfway vector representation. • We used logarithmic transformation of measured BRDF data (non-negativitynon-negativity).• Our Tucker approximation for a 4D BRDF data:

• To improve the accuracy of the approximation we propose applying the Tucker factorization recursively (error modeling approacherror modeling approach).

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)()()()()),,,(( 4321 olokhjhiolokhjhi fffgflog )()()()()),,,(( 4321 olokhjhiolokhjhi fffgflog

Eurographics 2012, Cagliari, Italy

Error Modeling ApproachError Modeling Approach

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)),,,(( olokhjhilog oB )),,,(( olokhjhilog oB

TuckerTucker

'oB'oB

'oo1 BBe 'oo1 BBe

TuckerTucker

'1e'1e

'112 eee '112 eee

TuckerTucker

'1L

'2

'1

'oo eeeBB

'1L

'2

'1

'oo eeeBB

The final logBRDF The final logBRDF values:values:

Eurographics 2012, Cagliari, Italy

Importance SamplingImportance Sampling• If the BRDF data is properly normalized, it can be viewed as sampled frequencies of a multi-variate

probability distribution [ÖKB10].

• Then standard statistical methods can be used to generate incident vectors for a given outgoing direction.

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K

p hoohhoohhh

sin,,,,,,

Normalizing Normalizing coefficient ofcoefficient of

'

,,,,|,

K

pp oohhh

oohhh

Normalizing Normalizing coefficient of coefficient of

oohhhp ,,,

oohh ,,,

Eurographics 2012, Cagliari, Italy

Importance SamplingImportance Sampling

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• We experimentally analyzed Tucker factors of both isotropic and anisotropic measured BRDF data set [MPBM03, NDM05].

• Based on the empirical properties explained, the Tucker factorization can be used to reduce the 4D sampling problem into a 2D case.

Eurographics 2012, Cagliari, Italy

Importance Sampling- Tucker FactorsImportance Sampling- Tucker Factors

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Eurographics 2012, Cagliari, Italy

Importance Sampling – Isotropic Importance Sampling – Isotropic

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ohhoohhh pp |,|,

ohhoi )(2

h

j

io

iho

jh hh

pP 1

|| 12 h )|( 21

ohh P

)(4

,|,|

oi

oohhhoii

pp

Eurographics 2012, Cagliari, Italy

Importance Sampling – Anisotropic Importance Sampling – Anisotropic

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hh

j

i

N

k

ih

kh

jh

h

hhpP

1 1

, )( 11 hh P

)|( 21

hhh P

hhhoohhh pp ,,|,

i

mN

k

jh

khh

jh

mhhji

hhhh

p

pP

1

1

,

,|

ohhoi )(2

)(4

,|,|

oi

oohhhoii

pp

Eurographics 2012, Cagliari, Italy

Results- Isotropic & AnisotropicResults- Isotropic & Anisotropic

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46.36946.369 41.34941.349 37.87837.878 38.88638.886 32.07332.073 36.63736.637 33.12333.123

blue-fabricblue-fabric, , blue-metallic-paintblue-metallic-paint, nickel, , nickel, yellow-matte-plasticyellow-matte-plastic, , grease-covered-steelgrease-covered-steel

red-velvetred-velvet, , yellow-yellow-satinsatin

Eurographics 2012, Cagliari, Italy

Results- Comparison on Isotropic Results- Comparison on Isotropic MaterialsMaterials

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• 100100 isotropic materials from MIT MERL database.

• 66 well-known BRDF models are used in comparison.

• Our proposed model gives the highesthighest PSNR values in 6666 cases and performing well well for the remaining 3434 materials.

Eurographics 2012, Cagliari, Italy

Results- Alum-bronzeResults- Alum-bronze

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Reference ImageReference Image Ashikhmin-Shirley, 34.370Ashikhmin-Shirley, 34.370 Cook-Torrance, 30.862Cook-Torrance, 30.862 Edwards et al., 27.982Edwards et al., 27.982

Lawrence et al., 32.629Lawrence et al., 32.629 Ward, 25.475Ward, 25.475 Ward-Duer, 26.146Ward-Duer, 26.146 Our model, Our model, 37.86637.866

Eurographics 2012, Cagliari, Italy

Results- Alum-bronze-Difference Results- Alum-bronze-Difference ImagesImages

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Reference ImageReference Image Ashikhmin-Shirley, 34.370Ashikhmin-Shirley, 34.370 Cook-Torrance, 30.862Cook-Torrance, 30.862 Edwards et al., 27.982Edwards et al., 27.982

Lawrence et al., 32.629Lawrence et al., 32.629 Ward, 25.475Ward, 25.475 Ward-Duer, 26.146Ward-Duer, 26.146 Our model, Our model, 37.86637.866

Eurographics 2012, Cagliari, Italy

Results- NylonResults- Nylon

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Reference ImageReference Image Ashikhmin-Shirley, 30.720Ashikhmin-Shirley, 30.720 Cook-Torrance, 30.934Cook-Torrance, 30.934 Edwards et al., 30.830Edwards et al., 30.830

Lawrence et al., 23.720Lawrence et al., 23.720 Ward, 29.802Ward, 29.802 Ward-Duer, 30.105Ward-Duer, 30.105 Our model, Our model, 38.02538.025

Eurographics 2012, Cagliari, Italy

Results- Nylon-Difference ImagesResults- Nylon-Difference Images

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Reference ImageReference Image Ashikhmin-Shirley, 30.720Ashikhmin-Shirley, 30.720 Cook-Torrance, 30.934Cook-Torrance, 30.934 Edwards et al., 30.830Edwards et al., 30.830

Lawrence et al., 23.720Lawrence et al., 23.720 Ward, 29.802Ward, 29.802 Ward-Duer, 30.105Ward-Duer, 30.105 Our model, Our model, 38.02538.025

Eurographics 2012, Cagliari, Italy

Results- Silver-metallic-paintResults- Silver-metallic-paint

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Reference ImageReference Image Ashikhmin-Shirley, 29.282Ashikhmin-Shirley, 29.282 Cook-Torrance, 28.901Cook-Torrance, 28.901 Edwards et al., 32.361Edwards et al., 32.361

Lawrence et al., 33.190Lawrence et al., 33.190 Ward, 25.373Ward, 25.373 Ward-Duer, 28.910Ward-Duer, 28.910 Our model, Our model, 40.19140.191

Eurographics 2012, Cagliari, Italy

Results- Silver-metallic-paint-Results- Silver-metallic-paint-Difference ImagesDifference Images

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Reference ImageReference Image Ashikhmin-Shirley, 29.282Ashikhmin-Shirley, 29.282 Cook-Torrance, 28.901Cook-Torrance, 28.901 Edwards et al., 32.361Edwards et al., 32.361

Lawrence et al., 33.190Lawrence et al., 33.190 Ward, 25.373Ward, 25.373 Ward-Duer, 28.910Ward-Duer, 28.910 Our model, Our model, 40.19140.191

Eurographics 2012, Cagliari, Italy

Results- Comparison on Princeton Results- Comparison on Princeton SceneScene

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Reference ImageReference Image Ashikhmin-Shirley, 33.656Ashikhmin-Shirley, 33.656 Cook-Torrance, 30.240Cook-Torrance, 30.240 Edwards et al., 25.604Edwards et al., 25.604

Lawrence et al., 33.403Lawrence et al., 33.403 Ward, 22.916Ward, 22.916 Ward-Duer, 31.126Ward-Duer, 31.126 Our model, Our model, 35.27435.274

Eurographics 2012, Cagliari, Italy

Results- Importance SamplingResults- Importance Sampling

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Material BMP Nickel YMP

Ashikhmin-Shirley 0.56970.5697 0.94320.9432 0.73280.7328

Edwards et al. 0.53300.5330 1.85011.8501 0.81340.8134

Lawrence et al. 0.40990.4099 6.48456.4845 1.31591.3159

Material BMP Nickel YMP

Ashikhmin-Shirley 1.0291.029 0.99030.9903 0.93610.9361

Edwards et al. 0.88510.8851 0.96720.9672 0.91110.9111

Lawrence et al. 1.01581.0158 1.27521.2752 1.07591.0759

Constant EnvironmentConstant Environment Grace EnvironmentGrace Environment

Eurographics 2012, Cagliari, Italy

Results- Importance Sampling Comparison Results- Importance Sampling Comparison on Princeton Sceneon Princeton Scene

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Ashikhmin-Shirley Ashikhmin-Shirley sampling, sampling,

256 samples/pixel,256 samples/pixel,Time: 1067.392 secTime: 1067.392 sec

Edwards et al. Edwards et al. sampling,sampling,

256 samples/pixel,256 samples/pixel,Time: 1109.015 secTime: 1109.015 sec

Lawrence et al. Lawrence et al. sampling,sampling,

256 samples/pixel,256 samples/pixel,Time: 1161.327 secTime: 1161.327 sec

Our factored Our factored sampling,sampling,

256 samples/pixel,256 samples/pixel,Time: 1261.461 secTime: 1261.461 sec

Eurographics 2012, Cagliari, Italy

Results- Comparison on Rendering Results- Comparison on Rendering Times & Storage NeedsTimes & Storage Needs

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BRDF Model BMP Nickel YMP

Measured 33.4 MB33.4 MB 33.4 MB33.4 MB 33.4 MB33.4 MB

Lawrence et al. 139.0 KB139.0 KB 96.5 KB96.5 KB 331.9 KB331.9 KB

Our factored model 76.7 KB76.7 KB 76.7 KB76.7 KB 73.2 KB73.2 KB

Storage NeedsStorage Needs

Rendering times (in seconds)Rendering times (in seconds)

BRDF Model BMP Nickel YMP

Measured 1802.831802.83 1894.331894.33 1830.871830.87

Cook-Torrance 1647.431647.43 1759.231759.23 1770.701770.70

Lawrence et al. 1854.531854.53 1795.971795.97 1831.271831.27

Ward 1465.931465.93 1563.701563.70 1591.101591.10

Our factored model 2048.732048.73 2122.402122.40 2015.232015.23

Eurographics 2012, Cagliari, Italy

ConclusionConclusionss

• Introduced a factored representation of the BRDF that is generalgeneral, accurateaccurate, compact compact and amenable to importance samplingimportance sampling:– Correct parameterization of incoming direction.– Decomposition into small set of one-dimensional factored

forms.– Importance sampling with numerical inversion.

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Eurographics 2012, Cagliari, Italy

Future WorksFuture Works

• Factored forms for– Higher dimensional data: SvBRDFs, BTF, BSSRDF..

• Implementation of our factored BRDF representation in real-time global illumination algorithms.

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Eurographics 2012, Cagliari, Italy

Thank YouThank You

Thank YouThank Youhttp://ube.ege.edu.tr/~kurt/http://ube.ege.edu.tr/~kurt/

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