Eurographics 2012, Cagliari, Italy
A General BRDF
Representation Based on Tensor
Decomposition
Ahmet Bilgili1, Aydın Öztürk2 and Murat Kurt1
1
International Computer Institute, Ege University, TURKEY
2
Department of Computer Engineering, Yasar University, TURKEY
Eurographics 2012, Cagliari, Italy
Our Goal
• Given a set of precise reflectance
measurements from real surfaces is it possible
to represent these measurements compactly
and accurately?
• The proposed method should also lend itself to
developing an efficient and simple importance
sampling algorithm.
isotropic
anisotropic
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Eurographics 2012, Cagliari, Italy
Previous Work – Analytical Models
Analytical BRDF
Models
[CT81]
[EBJ*06]
Emprical
BRDF Models
Phong [Pho75]
Blinn-Phong [Bli77]
Ward [Ward92]
Lafortune et al. [LFTG97]
Ward-Duer [Due05]
Physically
based BRDF
Models
Torrance-Sparrow [TS67]
Cook-Torrance [CT81]
He et al. [HTSG91]
Oren-Nayar [ON94]
Anisotropic
BRDF Models
Kajiya [Kaj85]
Poulin-Fournier [PF90]
Ward [War92]
Lafortune et al. [LFTG97]
Ashikhmin-Shirley [AS00]
Ward-Duer [Due05]
Edwards et al. [EBJ*06]
Lineer BRDF
Models
Westin et al. [WAT92]
Koenderink et al. [KvDS96]
Schröder and Sweldens [SS95]
Lalonde and Fournier [LF97]
Stark et al. [SAS05]
Öztürk et al. [OKBG08]
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Eurographics 2012, Cagliari, Italy
Previous Work – Data-Driven Models
Data-Driven BRDF
Models
[MPBM03]
Measurement
based BRDF
Models
[LRR04]
p
q
Matusik et al. [MPBM03]
Romerio et al. [RVZ08]
Factorization
based BRDF
Models
Kautz and McCool [KM99]
McCool et al. [MAA01]
Lawrence et al. [LRR04]
f r  p(ωo )  q(ω h )  p(ωi )
[MAA01]
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Eurographics 2012, Cagliari, Italy
Previous Work – Importance
Sampling
400
samples/pixel
Importance
Sampling
[EBJ*06]
Analytical
BRDF Models
Phong [Pho75]
Blinn-Phong [Bli77]
Ward [War92]
Lafortune [LFTG97]
Ashikhmin-Shirley [AS00]
Ward-Duer [Due05]
Edwards et al. [EBJ∗06]
400
samples/pixel
[LRR04]
Factorization
based BRDF
Models
Lawrence et al. [LRR04]
General BRDF
Sampling
Methods
Lawrence et al. [LRR05]
Montes et al. [MUGL08]
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Eurographics 2012, Cagliari, Italy
Previous Work – Tensor Factorization
Computer
Graphics
[SZC∗07]
[VT04]
Data
Compression
BRDF Data
Representation
[WWS*05]
Sun et al. [SZC∗07]
Vasilescu and Terzopulos [VT04]
Wang et al. [WWS*05]
BTF Data
Representation
Original
[VT04]
[WWS*05]
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Eurographics 2012, Cagliari, Italy
Key Idea
1D Vector
IXP
X
K
J
1D Vector
JXQ
Y
g
I
T
A Scalar
PXQxR
Tucker
Z
3D Tensor Data
IXJXK
Project 3D Tensor data into products of
1D functions and a core tensor:
1D Vector
KXR
P = Q = R =1
T (i , j , k )  g X( i ) Y( j ) Z( k )
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Eurographics 2012, Cagliari, Italy
Our BRDF Representation
• Our BRDF model is based on halfway vector
representation.
• We used logarithmic transformation of
measured BRDF data (non-negativity).
• Our Tucker approximation for a 4D BRDF data:
log (  ( hi ,  hj ,  ok , ol ))  gf1 ( hi ) f 2 ( hj ) f 3 ( ok ) f 4 (ol )
• To improve the accuracy of the approximation
we propose applying the Tucker factorization
recursively (error modeling approach).
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Eurographics 2012, Cagliari, Italy
Error Modeling Approach
B o  log (  ( hi ,  hj ,  ok , ol ))
e1  B o  B o
Tucker
Tucker
Bo
'
e 2  e1  e1
'
e1
'
Tucker
'
The final logBRDF
values:
B o  B o  e 1  e 2    e L 1
'
'
'
'
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Eurographics 2012, Cagliari, Italy
Importance Sampling
• If the BRDF data is properly normalized, it can
be viewed as sampled frequencies of a multivariate probability distribution [ÖKB10].
ph  h , h , o , o  
  h , h , o , o sin  h
K
Normalizing
coefficient of
 h ,h ,o ,o 
• Then standard statistical methods can be used to
generate incident vectors for a given outgoing
direction.
ph  h , h ,  o , o 
ph  h , h |  o , o  
K'
Normalizing
coefficient of
ph  h , h ,o , o 
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Eurographics 2012, Cagliari, Italy
Importance Sampling
• 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.
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Eurographics 2012, Cagliari, Italy
Importance Sampling- Tucker Factors
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Eurographics 2012, Cagliari, Italy
Importance Sampling – Isotropic
ph h ,h | o ,o   ph h | o 


j


Ph  h |  o   ph  h | o  h
j
i
i 1
i  2(o  h )h  o
h  21 h  P (2 | o )
1
h


pi i | o 
ph  h , h | o , o 
4(i  o )
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Eurographics 2012, Cagliari, Italy
Importance Sampling – Anisotropic
ph h ,h | o , o   ph h , h 
 
j

Nh

Ph  h   ph  h ,   h h
j
k
i 1 k 1


i
h


i
ph  hm , hj
m 1
k
j
p

,

 h h h
Ph  hi |  hj   N
h


h  P (1 )
1
h


pi i | o 
ph  h , h | o , o 
k 1
h  Ph1 (2 | h )
4(i  o )
i  2(o  h )h  o
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Eurographics 2012, Cagliari, Italy
Results- Isotropic & Anisotropic
46.369
41.349
37.878
38.886
32.073
blue-fabric, blue-metallic-paint, nickel,
yellow-matte-plastic, grease-covered-steel
36.637
33.123
red-velvet,
yellow-satin
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Eurographics 2012, Cagliari, Italy
Results- Comparison on Isotropic
Materials
• 100 isotropic
materials from
MIT MERL
database.
• 6 well-known
BRDF models
are used in
comparison.
• Our proposed
model gives the
highest PSNR
values in 66
cases and
performing well
for the
remaining 34
materials.
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Eurographics 2012, Cagliari, Italy
Results- Alum-bronze
Reference Image
Ashikhmin-Shirley, 34.370
Cook-Torrance, 30.862
Edwards et al., 27.982
Lawrence et al., 32.629
Ward, 25.475
Ward-Duer, 26.146
Our model, 37.866
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Eurographics 2012, Cagliari, Italy
Results- Alum-bronze-Difference
Images
Reference Image
Ashikhmin-Shirley, 34.370
Cook-Torrance, 30.862
Edwards et al., 27.982
Lawrence et al., 32.629
Ward, 25.475
Ward-Duer, 26.146
Our model, 37.866
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Eurographics 2012, Cagliari, Italy
Results- Nylon
Reference Image
Ashikhmin-Shirley, 30.720
Cook-Torrance, 30.934
Edwards et al., 30.830
Lawrence et al., 23.720
Ward, 29.802
Ward-Duer, 30.105
Our model, 38.025
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Eurographics 2012, Cagliari, Italy
Results- Nylon-Difference Images
Reference Image
Ashikhmin-Shirley, 30.720
Cook-Torrance, 30.934
Edwards et al., 30.830
Lawrence et al., 23.720
Ward, 29.802
Ward-Duer, 30.105
Our model, 38.025
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Eurographics 2012, Cagliari, Italy
Results- Silver-metallic-paint
Reference Image
Ashikhmin-Shirley, 29.282
Cook-Torrance, 28.901
Edwards et al., 32.361
Lawrence et al., 33.190
Ward, 25.373
Ward-Duer, 28.910
Our model, 40.191
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Eurographics 2012, Cagliari, Italy
Results- Silver-metallic-paintDifference Images
Reference Image
Ashikhmin-Shirley, 29.282
Cook-Torrance, 28.901
Edwards et al., 32.361
Lawrence et al., 33.190
Ward, 25.373
Ward-Duer, 28.910
Our model, 40.191
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Eurographics 2012, Cagliari, Italy
Results- Comparison on Princeton
Scene
Reference Image
Ashikhmin-Shirley, 33.656
Cook-Torrance, 30.240
Edwards et al., 25.604
Lawrence et al., 33.403
Ward, 22.916
Ward-Duer, 31.126
Our model, 35.274
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Eurographics 2012, Cagliari, Italy
Results- Importance Sampling
BMP
Nickel
YMP
AshikhminShirley
0.5697
0.9432
0.7328
Edwards et al.
0.5330
1.8501
Lawrence et al.
0.4099
6.4845
Material
BMP
Nickel
YMP
AshikhminShirley
1.029
0.9903
0.9361
0.8134
Edwards et al.
0.8851
0.9672
0.9111
1.3159
Lawrence et al.
1.0158
1.2752
1.0759
Constant Environment
Material
Grace Environment
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Eurographics 2012, Cagliari, Italy
Results- Importance Sampling Comparison
on Princeton Scene
Ashikhmin-Shirley
Edwards et al.
sampling,
sampling,
256 samples/pixel, 256 samples/pixel,
Time: 1067.392 sec Time: 1109.015 sec
Lawrence et al.
Our factored
sampling,
sampling,
256 samples/pixel, 256 samples/pixel,
Time: 1161.327 sec Time: 1261.461 sec
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Eurographics 2012, Cagliari, Italy
Results- Comparison on Rendering
Times & Storage Needs
BRDF Model
BMP
Nickel
YMP
Measured
33.4 MB
33.4 MB
33.4 MB
Lawrence et al.
139.0 KB
96.5 KB
331.9 KB
Our factored model
76.7 KB
76.7 KB
73.2 KB
Storage Needs
BRDF Model
BMP
Nickel
YMP
Measured
1802.83
1894.33
1830.87
Cook-Torrance
1647.43
1759.23
1770.70
Lawrence et al.
1854.53
1795.97
1831.27
Ward
1465.93
1563.70
1591.10
Our factored model
2048.73
2122.40
2015.23
Rendering times (in seconds)
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Eurographics 2012, Cagliari, Italy
Conclusions
• Introduced a factored representation of the
BRDF that is general, accurate, compact and
amenable to importance 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 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 You
Thank You
http://ube.ege.edu.tr/~kurt/
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