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Supplementary Document:Real-time Rendering of Layered Materials with
Anisotropic Normal DistributionsTomoya YamaguchiWaseda University
Tatsuya YatagawaThe University of [email protected]
Yusuke TokuyoshiSQUARE ENIX CO., LTD. (now at Intel Corporation)
Shigeo MorishimaWaseda [email protected]
A ADDITIONAL FORMULASA.1 Derivation of Jacobian MatricesTo derive Jacobian matrices, we partly followed the derivation byStam [2001]. Different from his derivation, we derived an approxi-mate solution for the Jacobian matrices over the region near to theorigin of P, while Stam derived the exact solution only at the origin.Without loss of generality, we can assume incident direction ωias (θi , 0). Let ωr and ωt be directions for reflection and refraction,respectively. We denote the directions ωi , ωr , ωt , and h as follows:
ωi = (sinθi , 0, cosθi ),ωr = (xr ,yr , zr ),
ωt = (xt ,yt , zt ),
h = (xh ,yh , zh ).
Let η be a relative refractive index between two interfaces, we canwrite ωr and ωt as follows:
ωr = 2(ωi · h)h − ωi ,
ηωt =
(ωi · h −
√(ωi · h)2 + η2 − 1
)h − ωi .
Using these equations, we can obtain the projected 2D coordinates(xr ,yr ) and (xt ,yt ) of ωr and ωt :{
xr = 2Axh − sinθiyr = 2Ayhηxt =
(A −
√A2 + η2 − 1
)xh − sinθi
ηyt =(A −
√A2 + η2 − 1
)yh
where A = xh sinθi + cosθi√1 − x2h − y2h .
Therefore, for reflection, the Jacobian matrix is obtained as in themain body of the paper. For refraction, the Jacobian matrix is calcu-lated as follows:
Jt =
[ ∂xt∂xh
∂xt∂yh
∂yt∂xh
∂yt∂yh
],
η∂xt∂xh= A −
√A2 + η2 − 1 + xh
∂A
∂xh
(1 −
2A√A2 + η2 − 1
)η∂xt∂yh= xh
∂A
∂yh
(1 −
2A√A2 + η2 − 1
)η∂yt∂xh= yh
∂A
∂xh
(1 −
2A√A2 + η2 − 1
)η∂yt∂yh= A −
√A2 + η2 − 1 + yh
∂A
∂yh
(1 −
2A√A2 + η2 − 1
)
where
∂A∂xh= sinθi − xh cos θi√
1−x 2h−y
2h
,
∂A∂yh= −
yh cos θi√1−x 2
h−y2h
Aswewrote in the main body of the paper, we assume xh ,yh , and θiare small enough that we can ignore the second- and higher-orderterms of xh , yh , and sinθi . Then, we can approximate Jt as follows:
Jt ≈1η
[cosθi −
√cos2 θi + η2 − 1 00 cosθi −
√cos2 θi + η2 − 1
]
=1η
[cosθi − cosθt 0
0 cosθi − cosθt
]Thus, the Jacobian matrix for refraction is also diagonal and itsdiagonal entries are the same.
A.2 Adding-Doubling for Two-layer MaterialsFor two-layer materials, Belcour [2018] provided the result of theadding-doubling method in Section 5 of his paper. To extend theirformulas using our result for anisotropic distribution is easy. Byreplacing the scalar variances σ {T ,R }
i j with covariance matrices
Σ{T ,R }i j . The series of interactions that are possible in two-layer
materials are only R and TR+T . The atomic operators for R aregiven by
eR = r12,
µR = −µi ,
ΣR = r12ΣR12 .
, , Yamaguchi et al.
For TR+T , the atomic operators are obtained as follows:
eTR+T =
t12r23t121 − r23r12
,
µTR+T = −µi ,
ΣTR+T =
t12r23t121 − r23r12
[ΣT12 + ΣT21 + K21
(ΣR23 +
r23r211 − r23r21
ΣR21
)].
In these formulas, r jk and tjk denote reflection and transmissioncoefficients between j-th and k-th interfaces, and Kjk is a trans-mission scaling factor which scales the roughness parameters. Asexplained in the main body of the paper, Σ{R,T }
12 can be obtained asfollows:
Σ{R,T }
12 =[tx ty
]⊤ [σ{R,T }
12,x 00 σ
{R,T }
12,y
] [tx ty
],
σR12, {x,y } = h(α {x,y }
), σT12, {x,y } = h
(s × α {x,y }
).
REFERENCESL. Belcour. 2018. Efficient rendering of layeredmaterials using an atomic decomposition
with statistical operators. ACM Trans. Graph. 37, 4, Article 73 (2018), 15 pages.https://doi.org/10.1145/3197517.3201289
J. Stam. 2001. An illumination model for a skin layer bounded by rough surfaces.In Eurographics Workshop on Rendering. 39–52. https://doi.org/10.2312/EGWR/EGWR01/039-052
(Appendix B starts from the next page)
Supplementary Document: Real-time Rendering of Layered Materials with Anisotropic Normal Distributions , ,
B ADDITIONAL RESULTSB.1 Results for varying roughness parameters
Reference Ours Error
= 0.01, = 0.01top:
RMSE: 9.16E-5
= 0
.01, =
0.0
1bo
ttom
:
RMSE: 9.16E-5
= 0
.05, =
0.0
1bo
ttom
:
RMSE: 9.16E-5
= 0
.1, =
0.0
1bo
ttom
:
RMSE: 9.16E-5
= 0
.2, =
0.0
1bo
ttom
:
RMSE: 9.16E-5
= 0
.5, =
0.0
1bo
ttom
:
RMSE: 7.21E-5
RMSE: 7.32E-5
RMSE: 7.01E-5
RMSE: 9.16E-5
RMSE: 7.24E-5
Figure 1: Rendering results with varying roughness parameters on the bottom layer ranging from 0.01 to 0.5. The roughnessparameters of the top layer are fixed at(αx ,αy ) = (0.01, 0.01)
.
, , Yamaguchi et al.
= 0.05, = 0.01top:
Reference Ours Error
RMSE: 9.16E-5
= 0
.01, =
0.0
1bo
ttom
:
RMSE: 9.16E-5
= 0
.05, =
0.0
1bo
ttom
:
RMSE: 9.16E-5
= 0
.1, =
0.0
1bo
ttom
:
RMSE: 9.16E-5
= 0
.2, =
0.0
1bo
ttom
:
RMSE: 9.16E-5
= 0
.5, =
0.0
1bo
ttom
:
RMSE: 4.45E-5
RMSE: 4.60E-5
RMSE: 5.15E-5
RMSE: 5.58E-5
RMSE: 6.03E-5
Figure 2: Rendering results with varying roughness parameters on the bottom layer ranging from 0.01 to 0.5. The roughnessparameters of the top layer are fixed at (αx ,αy ) = (0.05, 0.01)
.
B.2 Results for varying rotation of local coordinate system
Supplementary Document: Real-time Rendering of Layered Materials with Anisotropic Normal Distributions , ,
= 0.1, = 0.01top:
Reference Ours Error
RMSE: 9.16E-5
= 0
.01, =
0.0
1bo
ttom
:
RMSE: 9.16E-5
= 0
.05, =
0.0
1bo
ttom
:
RMSE: 9.16E-5
= 0
.1, =
0.0
1bo
ttom
:
RMSE: 9.16E-5
= 0
.2, =
0.0
1bo
ttom
:
RMSE: 9.16E-5
= 0
.5, =
0.0
1bo
ttom
:
RMSE: 6.77E-5
RMSE: 6.84E-5
RMSE: 6.45E-5
RMSE: 6.41E-5
RMSE: 6.02E-5
Figure 3: Rendering results with varying roughness parameters on the bottom layer ranging from 0.01 to 0.5. The roughnessparameters of the top layer are fixed at (αx ,αy ) = (0.1, 0.01)
.
, , Yamaguchi et al.
Reference Ours Error
= 0.2, = 0.01top:
RMSE: 9.16E-5
= 0
.01, =
0.0
1bo
ttom
:
RMSE: 9.16E-5
= 0
.05, =
0.0
1bo
ttom
:
RMSE: 9.16E-5
= 0
.1, =
0.0
1bo
ttom
:
RMSE: 9.16E-5
= 0
.2, =
0.0
1bo
ttom
:
RMSE: 9.16E-5
= 0
.5, =
0.0
1bo
ttom
:
RMSE: 9.27E-5
RMSE: 8.84E-5
RMSE: 8.77E-5
RMSE: 7.98E-5
RMSE: 6.77E-5
Figure 4: Rendering results with varying roughness parameters on the bottom layer ranging from 0.01 to 0.5. The roughnessparameters of the top layer are fixed at (αx ,αy ) = (0.2, 0.01)
.
Supplementary Document: Real-time Rendering of Layered Materials with Anisotropic Normal Distributions , ,
Reference Ours Error
= 0.5, = 0.01top:
RMSE: 9.16E-5
= 0
.01, =
0.0
1bo
ttom
:
RMSE: 9.16E-5
= 0
.05, =
0.0
1bo
ttom
:
RMSE: 9.16E-5
= 0
.1, =
0.0
1bo
ttom
:
RMSE: 9.16E-5
= 0
.2, =
0.0
1bo
ttom
:
RMSE: 9.16E-5
= 0
.5, =
0.0
1bo
ttom
:
RMSE: 1.38E-4
RMSE: 1.28E-4
RMSE: 1.19E-4
RMSE: 1.09E-4
RMSE: 1.07E-4
Figure 5: Rendering results with varying roughness parameters on the bottom layer ranging from 0.01 to 0.5. The roughnessparameters of the top layer are fixed at (αx ,αy ) = (0.5, 0.01)
.
, , Yamaguchi et al.
top: θ = 0
Reference Ours Error
RMSE: 9.16E-5
RMSE: 9.16E-5
bott
om: θ
= 0
bott
om: θ
= 3
0
RMSE: 9.11E-5
RMSE: 8.95E-5
bott
om: θ
= -9
0
RMSE: 9.16E-5RMSE: 8.42E-5
RMSE: 9.16E-5
bott
om: θ
= -6
0
RMSE: 8.22E-5
RMSE: 9.16E-5
bott
om: θ
= -3
0
RMSE: 8.60E-5
RMSE: 9.16E-5
bott
om: θ
= 6
0
RMSE: 9.01E-5
RMSE: 9.16E-5
bott
om: θ
= 9
0
RMSE: 8.67E-5
Figure 6: Rendering results for rotated local coordinate systems for the bottom layer, while the local coordinate system of thetop layer is fixed.
Supplementary Document: Real-time Rendering of Layered Materials with Anisotropic Normal Distributions , ,
Reference Ours Error
RMSE: 9.16E-5
RMSE: 9.16E-5
bottom: θ = 0
top
: θ
= 0
top
: θ
= 30
RMSE: 9.16E-5
top
: θ
= -9
0
RMSE: 7.73E-5
RMSE: 9.16E-5
top
: θ
= -6
0
RMSE: 8.40E-5
RMSE: 9.16E-5
top
: θ
= -3
0
RMSE: 9.82E-5
RMSE: 9.27E-5
RMSE: 9.38E-5
RMSE: 9.16E-5
top
: θ
= 60
RMSE: 8.51E-5
RMSE: 9.16E-5
top
: θ
= 90
RMSE: 7.82E-5
Figure 7: Rendering results for rotated local coordinate systems for the top layer, while the local coordinate system of thebottom layer is fixed.
