a general brdf representation based on tensor decomposition
DESCRIPTION
A General BRDF Representation Based on Tensor Decomposition. Ahmet Bilgili 1 , Aydın Öztürk 2 and Murat Kurt 1 1 International Computer Institute, Ege University, TURKEY 2 Department of Computer Engineering, Yasar University, TURKEY. Our Goal. - PowerPoint PPT PresentationTRANSCRIPT
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
3
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
5
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
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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
9
)),,,(( 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.
10
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
11
• 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
12
Eurographics 2012, Cagliari, Italy
Importance Sampling – Isotropic Importance Sampling – Isotropic
13
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
14
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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