はじめに
Microfacet Model for Refraction through Rough Surfaces についてまとめます. 論文自体はimportance samplingについでですが, microfacetの要約が簡潔にまとまっていると思います.
Microfacet Model for Refraction through Rough Surfaces
Microfacet Theory
BSDF (Bidirectional Scattering Distribution Function)
BSDFとは, 入射方向i \boldsymbol{i} i o \boldsymbol{o} o f s ( i , o , n ) f_{s}(i, o, n) f s ( i , o , n ) n \boldsymbol{n} n f s = f r + f t f_{s} = f_{r} + f_{t} f s = f r + f t
定義
放射束 (radiant flux) (W W W
放射照度 (irradiance) (W / m 2 W/m^{2} W / m 2
放射輝度 (radiance) (W / s r / m 2 W/sr/m^{2} W / s r / m 2
Microfacet Models
マイクロサーフェスの細かな凹凸は, 完全な反射（透過）特性を持った微小平面の集合として表現する. マイクロサーフェスの特徴は２つの統計量を用いて, マクロサーフェスとして簡略化される. ２つの統計量は, マイクロファセット（の法線）の分布D D D G G G D D D G G G
Microfacet Distribution Function D D D
マイクロファセットの法線分布D ( m ) D(m) D ( m ) m m m d ω m d\omega_{m} d ω m d A dA d A D ( m ) d ω m d A D(m)d\omega_{m}dA D ( m ) d ω m d A D D D D D D
0以上
0 ≤ D ( m ) ≤ ∞ 0 \leq D(m) \leq \infty 0 ≤ D ( m ) ≤ ∞
マイクロファセットの総面積は少なくとも, 対応するマクロファセットの面積に等しい
1 ≤ D ( c m ) d ω m 1 \leq D(cm)d\omega_{m} 1 ≤ D ( c m ) d ω m
マイクロファセットの符号付き投影面積は, 任意の方向v v v
( v ⋅ n ) = ∫ D ( m ) ( v ⋅ m ) d ω m (v \cdot n) = \int D(m)(v \cdot m)d\omega_{m} ( v ⋅ n ) = ∫ D ( m ) ( v ⋅ m ) d ω m v = n v=n v = n
1 = ∫ D ( m ) 1 = \int D(m) 1 = ∫ D ( m )
Shadowing Masking Function G
幾何減衰関数G ( i , o , m ) G(i,o,m) G ( i , o , m ) m m m G G G
0以上1以下
0 ≤ G ( i , o , m ) ≤ 1 0 \leq G(i,o,m) \leq 1 0 ≤ G ( i , o , m ) ≤ 1
入出射方向に対して対象
G ( i , o , m ) = G ( o , i , m ) G(i,o,m)=G(o,i,m) G ( i , o , m ) = G ( o , i , m )
表面側から裏面は見えない（その逆も同様）
G ( i , o , m ) = 0 i n ( i ⋅ m ) ( i ⋅ n ) ≤ o r ( o ⋅ m ) ( o ⋅ n ) ≤ 0 G(i,o,m)=0 in (i \cdot m)(i \cdot n) \leq or (o \cdot m)(o \cdot n) \leq 0 G ( i , o , m ) = 0 i n ( i ⋅ m ) ( i ⋅ n ) ≤ o r ( o ⋅ m ) ( o ⋅ n ) ≤ 0
マイクロサーフェスBSDFの積分
BSDFの積分は, マイクロサーフェスの可視部分からの放射を積分する. 放射の割合はマイクロサーフェスのBSDFf s m f_{s}^{m} f s m D D D G G G (1) f s ( i , o , n ) = ∫ ∣ i ⋅ m i ⋅ n ∣ f s m ( i , o , m ) ∣ o ⋅ m o ⋅ n ∣ G ( i , o , m ) D ( m ) d w m f_{s}(i,o,n) = \int |\frac{i \cdot m}{i \cdot n}|f_{s}^{m}(i,o,m)|\frac{o \cdot m}{o \cdot n}|G(i,o,m)D(m)dw_{m} \tag{1} f s ( i , o , n ) = ∫ ∣ i ⋅ n i ⋅ m ∣ f s m ( i , o , m ) ∣ o ⋅ n o ⋅ m ∣ G ( i , o , m ) D ( m ) d w m ( 1 ) f s m f_{s}^{m} f s m F F F
マイクロサーフェススペキュラー BSDF
多くのマイクロファセットモデルが理想のスペキュラー反射を仮定する. このとき, マイクロサーフェスは極小の鏡の集まりのように振る舞い, この極小の鏡のことをマイクロファセットと呼ぶ.i i i s s s ρ \rho ρ ρ \rho ρ s s s i i i m m m (2) f s m ( i , o , m ) = ρ δ ω o ( s , o ) ∣ o ⋅ m ∣ f_{s}^{m}(i,o,m) = \rho\frac{\delta_{\omega_{o}}(s,o)}{|o \cdot m|}\tag{2} f s m ( i , o , m ) = ρ ∣ o ⋅ m ∣ δ ω o ( s , o ) ( 2 ) δ ω o \delta_{\omega_{o}} δ ω o s = 0 s=0 s = 0 ∞ \infin ∞ i i i o o o h ( i , o ) h(i,o) h ( i , o ) ( 2 ) (2) ( 2 ) h h h (3) f s m ( i , o , m ) = ρ ( i , m ) δ ω m ( h ( i , o ) , m ) ∣ o ⋅ m ∣ ∣ ∣ ∂ ω h ∂ ω o ∣ ∣ f_{s}^{m}(i,o,m)=\rho(i,m)\frac{\delta_{\omega_{m}}(h(i,o),m)}{|o \cdot m|}||\frac{\partial \omega_{h}}{\partial \omega_{o}}|| \tag{3} f s m ( i , o , m ) = ρ ( i , m ) ∣ o ⋅ m ∣ δ ω m ( h ( i , o ) , m ) ∣ ∣ ∂ ω o ∂ ω h ∣ ∣ ( 3 ) ∣ ∣ ∂ ω h ∂ ω o ∣ ∣ ||\frac{\partial \omega_{h}}{\partial \omega_{o}}|| ∣ ∣ ∂ ω o ∂ ω h ∣ ∣ ω h \omega_{h} ω h ω o \omega_{o} ω o
理想的な反射f r m f_{r}^{m} f r m
理想的な反射を考えたとき, ハーフベクトルh r h_{r} h r h r = h r ( i , o ) = h r ⃗ ∣ ∣ h r ⃗ ∣ ∣ h_{r}=h_{r}(i,o)=\frac{\vec{h_{r}}}{||\vec{h_{r}}||} h r = h r ( i , o ) = ∣ ∣ h r ∣ ∣ h r h r ⃗ = s i g n ( i ⋅ n ) ( i + o ) \vec{h_{r}}=sign(i \cdot n)(i+o) h r = s i g n ( i ⋅ n ) ( i + o ) ∣ ∣ ∂ ω h r ∂ ω o ∣ ∣ = ∣ o ⋅ h r ∣ ∣ ∣ h r ⃗ ∣ ∣ 2 = 1 4 ∣ o ⋅ h r ∣ ||\frac{\partial \omega_{h_{r}}}{\partial \omega_{o}}|| = \frac{|o \cdot h_{r}|}{||\vec{h_{r}}||^{2}}=\frac{1}{4|o \cdot h_{r}|} ∣ ∣ ∂ ω o ∂ ω h r ∣ ∣ = ∣ ∣ h r ∣ ∣ 2 ∣ o ⋅ h r ∣ = 4 ∣ o ⋅ h r ∣ 1
ρ \rho ρ F F F ( 3 ) (3) ( 3 ) (4) f r m ( i , o , m ) = F ( i , m ) δ ω m ( h r , m ) 4 ( i ⋅ h r ) 2 f_{r}^{m}(i,o,m)=F(i,m)\frac{\delta_{\omega_{m}}(h_{r},m)}{4(i \cdot h_{r})^{2}} \tag{4} f r m ( i , o , m ) = F ( i , m ) 4 ( i ⋅ h r ) 2 δ ω m ( h r , m ) ( 4 ) f r m f_{r}^{m} f r m ∣ i ⋅ h r ∣ |i \cdot h_{r}| ∣ i ⋅ h r ∣
理想的な透過f t m f_{t}^{m} f t m
スネルの法則の屈折率をη i \eta_{i} η i η o \eta_{o} η o
(5) h t = h t ( i , o ) = h t ⃗ ∣ ∣ h t ⃗ ∣ ∣ w h e r e h t ⃗ = − ( η i i + η o o ) h_{t}=h_{t}(i,o)=\frac{\vec{h_{t}}}{\vec{||h_{t}}||} \ where \ \vec{h_{t}} = -(\eta_{i}i + \eta_{o}o) \tag{5} h t = h t ( i , o ) = ∣ ∣ h t ∣ ∣ h t w h e r e h t = − ( η i i + η o o ) ( 5 )
η i ≠ η o \eta_{i} \neq \eta_{o} η i ̸ = η o
(6) ∣ ∣ ∂ ω h t ∂ ω o ∣ ∣ = η o 2 ∣ o ⋅ h t ∣ ∣ ∣ h t ⃗ ∣ ∣ 2 = η o 2 ∣ o ⋅ h t ∣ ( η i ( i ⋅ h t ) + η o ( o ⋅ h t ) ) 2 ||\frac{\partial \omega_{h_{t}}}{\partial \omega_{o}}|| = \frac{\eta_{o}^{2}|o \cdot h_{t}|}{||\vec{h_{t}}||^{2}}=\frac{\eta_{o}^{2}|o \cdot h_{t}|}{(\eta_{i}(i \cdot h_{t}) + \eta_{o}(o \cdot h_{t}))^{2}} \tag{6} ∣ ∣ ∂ ω o ∂ ω h t ∣ ∣ = ∣ ∣ h t ∣ ∣ 2 η o 2 ∣ o ⋅ h t ∣ = ( η i ( i ⋅ h t ) + η o ( o ⋅ h t ) ) 2 η o 2 ∣ o ⋅ h t ∣ ( 6 )
光が吸収されないと仮定すると, ρ \rho ρ 1 − F 1-F 1 − F
(7) f t m ( i , o , m ) = ( 1 − F ( i , m ) ) δ ω m ( h t , m ) η o 2 ( η i ( i ⋅ h t ) + η o ( o ⋅ h t ) ) 2 f_{t}^{m}(i,o,m)=(1-F(i,m))\frac{\delta_{\omega_{m}}(h_{t},m)\eta_{o}^{2}}{(\eta_{i}(i \cdot h_{t}) + \eta_{o}(o \cdot h_{t}))^{2}} \tag{7} f t m ( i , o , m ) = ( 1 − F ( i , m ) ) ( η i ( i ⋅ h t ) + η o ( o ⋅ h t ) ) 2 δ ω m ( h t , m ) η o 2 ( 7 )
BTDFは相反性がなく, f t m ( i , o , m ) / η o 2 = f t m ( o , i , m ) / η i 2 f_{t}^{m}(i,o,m)/\eta_{o}^{2}=f_{t}^{m}(o,i,m)/\eta_{i}^{2} f t m ( i , o , m ) / η o 2 = f t m ( o , i , m ) / η i 2 1 .
1. 正反射方向からずれた方向にスペキュラーのピークがあることを, オフスペキュラーと呼ぶ.
粗い面のBSDF
マイクロサーフェスのBSDFと式( 1 ) (1) ( 1 ) f s f_{s} f s
f s ( i , o , m ) = f r ( i , o , m ) + f t ( i , o , m ) f_{s}(i,o,m) = f_{r}(i,o,m) + f_{t}(i,o,m) f s ( i , o , m ) = f r ( i , o , m ) + f t ( i , o , m )
反射の項は,
(8) f r ( i , o , n ) = F ( i , h r ) G ( i , o , h r ) D ( h r ) 4 ∣ i ⋅ n ∣ ∣ o ⋅ n ∣ f_{r}(i,o,n)=\frac{F(i,h_{r})G(i,o,h_{r})D(h_{r})}{4|i \cdot n||o \cdot n|} \tag{8} f r ( i , o , n ) = 4 ∣ i ⋅ n ∣ ∣ o ⋅ n ∣ F ( i , h r ) G ( i , o , h r ) D ( h r ) ( 8 )
分母のπ \pi π
(9) f t ( i , o , n ) = ∣ i ⋅ h t ∣ ∣ o ⋅ h t ∣ ∣ i ⋅ n ∣ ∣ o ⋅ n ∣ η o 2 ( 1 − F ( i , h t ) ) G ( i , o , h t ) D ( h t ) ( η i ( i ⋅ h t ) + η o ( o ⋅ h t ) ) 2 f_{t}(i,o,n)=\frac{|i \cdot h_{t}||o \cdot h_{t}|}{|i \cdot n||o \cdot n|}\frac{\eta_{o}^{2}(1-F(i,h_{t}))G(i,o,h_{t})D(h_{t})}{(\eta_{i}(i \cdot h_{t}) + \eta_{o}(o \cdot h_{t}))^{2}} \tag{9} f t ( i , o , n ) = ∣ i ⋅ n ∣ ∣ o ⋅ n ∣ ∣ i ⋅ h t ∣ ∣ o ⋅ h t ∣ ( η i ( i ⋅ h t ) + η o ( o ⋅ h t ) ) 2 η o 2 ( 1 − F ( i , h t ) ) G ( i , o , h t ) D ( h t ) ( 9 )
F, D, Gの選択
フレネル項は次のようになる.
(10) F ( i , m ) = 1 2 ( g − c ) 2 ( g + c ) 2 ( 1 + ( c ( g + c ) − 1 ) 2 ( c ( g − c ) + 1 ) 2 ) F(i,m)=\frac{1}{2}\frac{(g-c)^{2}}{(g+c)^{2}}(1+\frac{(c(g+c)-1)^{2}}{(c(g-c)+1)^{2}}) \tag{10} F ( i , m ) = 2 1 ( g + c ) 2 ( g − c ) 2 ( 1 + ( c ( g − c ) + 1 ) 2 ( c ( g + c ) − 1 ) 2 ) ( 1 0 ) w h e r e g = η t 2 η i 2 − 1 + c 2 a n d c = ∣ i ⋅ m ∣ where \ g=\sqrt{\frac{\eta_{t}^{2}}{\eta_{i}^{2}}-1+c^{2}} \ and \ c=|i \cdot m| w h e r e g = η i 2 η t 2 − 1 + c 2 a n d c = ∣ i ⋅ m ∣
フレネル項F F F CT82 , Sch94 ].
F ( θ ) = R 0 + ( 1 − R 0 ) ( 1 − c o s θ ) 5 w h e r e R 0 = ( η 1 − η 2 η 1 + η 2 ) 2 F(\theta)=R_{0}+(1-R_{0})(1-cos\theta)^5 \ where \ R_{0}=(\frac{\eta_{1}-\eta_{2}}{\eta_{1}+\eta_{2}})^2 F ( θ ) = R 0 + ( 1 − R 0 ) ( 1 − c o s θ ) 5 w h e r e R 0 = ( η 1 + η 2 η 1 − η 2 ) 2
マイクロファセット法線分布関数D D D G G G D D D Smi67 ]. SmithのG G G G 1 G_{1} G 1
G ( i , o , m ) ≈ G 1 ( i , m ) G 1 ( o , m ) G(i,o,m) \approx G_{1}(i,m)G_{1}(o,m) G ( i , o , m ) ≈ G 1 ( i , m ) G 1 ( o , m ) G 1 G_{1} G 1 D D D Smi67 , Bro80 , BBS02 ], と付録A).
分布関数と関連する関数
法線分布関数Beckmann, GGXについて, Smithの幾何減衰関数G 1 G_{1} G 1 [ 0 , 1 ) [0,1) [ 0 , 1 ) ξ 1 \xi_{1} ξ 1 ξ 2 \xi_{2} ξ 2 m m m
p m ( m ) = D ( m ) ∣ m ⋅ n ∣ p_{m}(m)=D(m)|m \cdot n| p m ( m ) = D ( m ) ∣ m ⋅ n ∣
以降, θ m \theta_{m} θ m m m m n n n θ v \theta_{v} θ v v v v n n n χ + ( a ) \chi^{+}(a) χ + ( a ) D D D m ⋅ n ≤ 0 m \cdot n \leq 0 m ⋅ n ≤ 0 D ( m ) = 0 D(m)=0 D ( m ) = 0
Beckmann Distribution
Beckmann分布は幅のパラメータα b \alpha_{b} α b
(11) D ( m ) = χ + ( m ⋅ n ) π α b 2 c o s 4 θ m e − t a n 2 θ m α b 2 D(m)=\frac{\chi^{+}(m \cdot n)}{\pi\alpha_{b}^{2}cos^{4}\theta_{m}}e^{\frac{-tan^{2}\theta_{m}}{\alpha_{b}^{2}}} \tag{11} D ( m ) = π α b 2 c o s 4 θ m χ + ( m ⋅ n ) e α b 2 − t a n 2 θ m ( 1 1 ) (12) G 1 ( v , m ) = χ + ( v ⋅ m v ⋅ n ) 2 1 + e r f ( a ) + 1 a π e − a 2 G_{1}(v,m)=\chi^{+}(\frac{v \cdot m}{v \cdot n})\frac{2}{1+erf(a)+\frac{1}{a\sqrt{\pi}}e^{-a^{2}}} \tag{12} G 1 ( v , m ) = χ + ( v ⋅ n v ⋅ m ) 1 + e r f ( a ) + a π 1 e − a 2 2 ( 1 2 ) w i t h a = ( α b t a n θ v ) − 1 with \ a=(\alpha_{b}tan\theta_{v})^{-1} w i t h a = ( α b t a n θ v ) − 1
誤差関数e r f ( x ) = 2 π ∫ 0 x e − x 2 d x erf(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-x^{2}}dx e r f ( x ) = π 2 ∫ 0 x e − x 2 d x Sch94 ]は上記Smithの幾何減衰関数の近似を示している.
(13) G 1 ( v , m ) ≈ χ + ( v ⋅ m v ⋅ n ) = { 3.535 a + 2.181 a 2 1 + 2.276 a + 2.577 a 2 i f a < 1.6 1 o t h e r w i s e G_{1}(v,m) \approx \chi^{+}(\frac{v \cdot m}{v \cdot n})=\begin{cases}\frac{3.535a + 2.181a^{2}}{1+2.276a+2.577a^{2}} & if \ a<1.6\\1 & otherwise\end{cases} \tag{13} G 1 ( v , m ) ≈ χ + ( v ⋅ n v ⋅ m ) = { 1 + 2 . 2 7 6 a + 2 . 5 7 7 a 2 3 . 5 3 5 a + 2 . 1 8 1 a 2 1 i f a < 1 . 6 o t h e r w i s e ( 1 3 )
分布関数のサンプルD ( m ) ∣ m ⋅ n ∣ D(m)|m \cdot n| D ( m ) ∣ m ⋅ n ∣ θ m = a r c t a n − α b 2 l o g ( 1 − ξ 1 ) ( 14 ) ϕ m = 2 π ξ 2 ( 15 ) \begin{array}{ll}
\theta_{m}=arctan\sqrt{-\alpha_{b}^{2}log(1-\xi_{1})} & (14)\\
\phi_{m}=2\pi\xi_{2} & (15)
\end{array} θ m = a r c t a n − α b 2 l o g ( 1 − ξ 1 ) ϕ m = 2 π ξ 2 ( 1 4 ) ( 1 5 )
GGX Distribution
GGX分布は幅のパラメータα g \alpha_{g} α g
(16) D ( m ) = α g 2 χ + ( m ⋅ n ) π c o s 4 θ m ( α g 2 + t a n 2 θ m ) 2 D(m)=\frac{\alpha_{g}^{2}\chi^{+}(m \cdot n)}{\pi cos^{4}\theta_{m}(\alpha_{g}^{2} + tan^{2}\theta_{m})^{2}} \tag{16} D ( m ) = π c o s 4 θ m ( α g 2 + t a n 2 θ m ) 2 α g 2 χ + ( m ⋅ n ) ( 1 6 )
式変形すると,(17) D ( m ) = α g 2 χ + ( m ⋅ n ) π ( ( α g 2 − 1 ) c o s 2 θ + 1 ) 2 D(m)=\frac{\alpha_{g}^{2}\chi^{+}(m \cdot n)}{\pi ((\alpha_{g}^{2}-1)cos^{2}\theta + 1)^{2}} \tag{17} D ( m ) = π ( ( α g 2 − 1 ) c o s 2 θ + 1 ) 2 α g 2 χ + ( m ⋅ n ) ( 1 7 )
(18) G 1 ( v , m ) = χ + ( v ⋅ m v ⋅ n ) 2 1 + 1 + α g 2 t a n 2 θ v G_{1}(v,m)=\chi^{+}(\frac{v \cdot m}{v \cdot n})\frac{2}{1+\sqrt{1+\alpha_{g}^{2}tan^{2}\theta_{v}}} \tag{18} G 1 ( v , m ) = χ + ( v ⋅ n v ⋅ m ) 1 + 1 + α g 2 t a n 2 θ v 2 ( 1 8 )
分布関数のサンプルD ( m ) ∣ m ⋅ n ∣ D(m)|m \cdot n| D ( m ) ∣ m ⋅ n ∣ θ m = a r c t a n ( α g ξ 1 1 − ξ 1 ) ( 19 ) θ m = a r c c o s 1 − ξ 1 ξ 1 ( α g 2 − 1 ) + 1 ( 20 ) ϕ m = 2 π ξ 2 ( 21 ) \begin{array}{ll}
\theta_{m}=arctan(\frac{\alpha_{g}\sqrt{\xi_{1}}}{\sqrt{1-\xi_{1}}}) & (19)\\
\theta_{m}= arccos\sqrt{\frac{1-\xi_{1}}{\xi_{1}(\alpha_{g}^{2}-1)+1}}& (20)\\
\phi_{m}=2\pi\xi_{2} & (21)
\end{array} θ m = a r c t a n ( 1 − ξ 1 α g ξ 1 ) θ m = a r c c o s ξ 1 ( α g 2 − 1 ) + 1 1 − ξ 1 ϕ m = 2 π ξ 2 ( 1 9 ) ( 2 0 ) ( 2 1 )
重み
サンプリングの重み係数は,(22) w e i g h t ( o ) = f s ( i , o , n ) ∣ o ⋅ n ∣ p o ( o ) weight(o) = \frac{f_{s}(i,o,n)|o \cdot n|}{p_{o}(o)} \tag{22} w e i g h t ( o ) = p o ( o ) f s ( i , o , n ) ∣ o ⋅ n ∣ ( 2 2 ) (23) p o ( o ) = p m ( m ) ∣ ∣ ∂ ω h ∂ ω o ∣ ∣ p_{o}(o)=p_{m}(m)||\frac{\partial\omega_{h}}{\partial\omega_{o}}|| \tag{23} p o ( o ) = p m ( m ) ∣ ∣ ∂ ω o ∂ ω h ∣ ∣ ( 2 3 )
反射方向o r o_{r} o r o r = 2 ∣ i ⋅ m ∣ m − i o_{r}=2|i \cdot m|m-i o r = 2 ∣ i ⋅ m ∣ m − i o t o_{t} o t o t = ( η c − s i g n ( i ⋅ n ) 1 + η ( c 2 − 1 ) m − η i w i t h c = ( i ⋅ m ) a n d η = η i / η t \begin{array}{l}
o_{t}=(\eta c-sign(i \cdot n)\sqrt{1+\eta(c^{2}-1)}m-\eta i\\
with \ c=(i \cdot m) \ and \ \eta=\eta_{i}/\eta_{t}
\end{array} o t = ( η c − s i g n ( i ⋅ n ) 1 + η ( c 2 − 1 ) m − η i w i t h c = ( i ⋅ m ) a n d η = η i / η t
反射と透過どちらの場合も重み係数は次のようになる.
(24) w e i g h t ( o ) = f s ( i , o , n ) ∣ o ⋅ n ∣ p o ( o ) = ∣ i ⋅ m ∣ G ( i , o , m ) ∣ i ⋅ n ∣ ∣ m ⋅ n ∣ weight(o) = \frac{f_{s}(i,o,n)|o \cdot n|}{p_{o}(o)} = \frac{|i \cdot m|G(i,o,m)}{|i \cdot n||m \cdot n|} \tag{24} w e i g h t ( o ) = p o ( o ) f s ( i , o , n ) ∣ o ⋅ n ∣ = ∣ i ⋅ n ∣ ∣ m ⋅ n ∣ ∣ i ⋅ m ∣ G ( i , o , m ) ( 2 4 )
この重みは, パラメータによってはグレイジングアングル付近で, 過大な値になることがある.
参照
[CT82] COOK R. L., TORRANCE K. E.: A reflectance model for computer graphics. ACM Transactions on Graphics 1, 1 (Jan. 1982), 7–24.[Sch94] SCHLICK C.: An inexpensive BRDF model for physically-based rendering. Computer Graphics Forum 13, 3 (1994), 233–246.[Smi67] SMITH B. G.: Geometrical shadowing of a random rough surface. IEEE Trans. on Antennas and Propagation (1967), 668–671.[Bro80] BROWN G. S.: Shadowing by non-Gaussian random surfaces. IEEE Trans. on Antennas and Propagation 28 (Nov. 1980), 788–790.[BBS02] BOURLIER C., BERGINC G., SAILLARD J.: One- and two-dimensional shadowing functions for any height and slope stationary uncorrelated surface in the monostatic and bistatic configurations. IEEE Trans. on Antennas and Propagation 50 (Mar. 2002), 312–324.https://agraphicsguy.wordpress.com/2015/11/01/sampling-microfacet-brdf/ https://www.tobias-franke.eu/log/2014/03/30/notes_on_importance_sampling.html
付録A: Smithの幾何学減衰関数G 1 G_{1} G 1
付録B: 法線分布関数D D D
逆関数法
ある確率分布P ( x ) P(x) P ( x ) U U U P ( U ≤ u ) = u ( 0 ≤ u ≤ 1 ) P(U \le u) = u \ (0 \le u \le 1) P ( U ≤ u ) = u ( 0 ≤ u ≤ 1 ) P ( x ) P(x) P ( x ) F ( x ) F(x) F ( x ) U ≤ u ⇔ F − 1 ( U ) ≤ F − 1 ( u ) U \le u \Leftrightarrow F^{-1}(U) \le F^{-1}(u) U ≤ u ⇔ F − 1 ( U ) ≤ F − 1 ( u ) P ( F − 1 ( U ) ≤ F − 1 ( u ) ) = u P(F^{-1}(U) \le F^{-1}(u)) = u P ( F − 1 ( U ) ≤ F − 1 ( u ) ) = u F − 1 ( u ) = x F^{-1}(u)=x F − 1 ( u ) = x P ( F − 1 ( U ) ≤ x ) = F ( x ) P(F^{-1}(U) \le x) = F(x) P ( F − 1 ( U ) ≤ x ) = F ( x ) F − 1 ( U ) F^{-1}(U) F − 1 ( U ) F ( x ) F(x) F ( x ) P ( x ) P(x) P ( x )
GGX
分布関数を半球面で積分したいので, 立体角方向にc o s θ cos \theta c o s θ s i n θ sin \theta s i n θ (25) D ( m ) = α g 2 χ + ( m ⋅ n ) π ( ( α g 2 − 1 ) c o s 2 θ + 1 ) 2 D(m)=\frac{\alpha_{g}^{2}\chi^{+}(m \cdot n)}{\pi ((\alpha_{g}^{2}-1)cos^{2}\theta + 1)^{2}} \tag{25} D ( m ) = π ( ( α g 2 − 1 ) c o s 2 θ + 1 ) 2 α g 2 χ + ( m ⋅ n ) ( 2 5 ) p h ( θ , ϕ ) = α g 2 c o s θ s i n θ π ( ( α g 2 − 1 ) c o s 2 θ + 1 ) 2 p_{h}(\theta, \phi) =\frac{\alpha_{g}^{2}cos \theta sin \theta}{\pi ((\alpha_{g}^{2}-1)cos^{2}\theta + 1)^{2}} p h ( θ , ϕ ) = π ( ( α g 2 − 1 ) c o s 2 θ + 1 ) 2 α g 2 c o s θ s i n θ ∫ 0 2 π ∫ 0 2 π p h ( θ , ϕ ) d θ d ϕ = ∫ 0 2 π 2 π p h ( θ ) d θ \int_{0}^{2 \pi} \int_{0}^{2 \pi}p_{h}(\theta, \phi)d\theta d\phi = \int_{0}^{2 \pi} 2 \pi p_{h}(\theta)d\theta ∫ 0 2 π ∫ 0 2 π p h ( θ , ϕ ) d θ d ϕ = ∫ 0 2 π 2 π p h ( θ ) d θ
累積分布関数は次のようになる.P h ( θ ) = ∫ 0 θ 2 α g 2 s i n ( t ) c o s ( t ) ( ( α g 2 − 1 ) c o s 2 ( t ) + 1 ) 2 d t = ∫ θ 0 α g 2 ( ( α g 2 − 1 ) c o s 2 ( t ) + 1 ) 2 d ( c o s 2 t ) = α g 2 α g 2 − 1 ∫ 0 θ d ( 1 ( α g 2 − 1 ) c o s 2 ( t ) + 1 ) = α g 2 α g 2 − 1 ( 1 ( α g 2 − 1 ) c o s 2 θ + 1 − 1 α g 2 ) = α g 2 ( α g 2 − 1 ) 2 c o s 2 θ + ( α g 2 − 1 ) − 1 α g 2 − 1 \begin{array}{lll}
P_{h}(\theta)&=&\int_{0}^{\theta} \frac{2 \alpha_{g}^{2} sin(t) cos(t)}{((\alpha_{g}^{2}-1)cos^{2}(t) + 1)^{2}} dt \\
&=&\int_{\theta}^{0} \frac{\alpha_{g}^{2}}{((\alpha_{g}^{2}-1)cos^{2}(t) + 1)^{2}} d(cos^{2}t) \\
&=&\frac{\alpha_{g}^{2}}{\alpha_{g}^2-1}\int_{0}^{\theta} d(\frac{1}{(\alpha_{g}^{2}-1)cos^{2}(t) + 1}) \\
&=&\frac{\alpha_{g}^{2}}{\alpha_{g}^2-1}(\frac{1}{(\alpha_{g}^{2}-1)cos^{2}\theta + 1} - \frac{1}{\alpha_{g}^{2}}) \\
&=&\frac{\alpha_{g}^{2}}{(\alpha_{g}^{2}-1)^{2}cos^{2}\theta + (\alpha_{g}^{2}-1)} - \frac{1}{\alpha_{g}^{2}-1}
\end{array} P h ( θ ) = = = = = ∫ 0 θ ( ( α g 2 − 1 ) c o s 2 ( t ) + 1 ) 2 2 α g 2 s i n ( t ) c o s ( t ) d t ∫ θ 0 ( ( α g 2 − 1 ) c o s 2 ( t ) + 1 ) 2 α g 2 d ( c o s 2 t ) α g 2 − 1 α g 2 ∫ 0 θ d ( ( α g 2 − 1 ) c o s 2 ( t ) + 1 1 ) α g 2 − 1 α g 2 ( ( α g 2 − 1 ) c o s 2 θ + 1 1 − α g 2 1 ) ( α g 2 − 1 ) 2 c o s 2 θ + ( α g 2 − 1 ) α g 2 − α g 2 − 1 1
逆関数P h − 1 P_{h}^{-1} P h − 1 P h − 1 = α g 2 ( α g 2 − 1 ) 2 c o s 2 θ + ( α g 2 − 1 ) − 1 α g 2 − 1 \begin{array}{lll}
P_{h}^{-1}&=&\frac{\alpha_{g}^{2}}{(\alpha_{g}^{2}-1)^{2}cos^{2}\theta + (\alpha_{g}^{2}-1)} - \frac{1}{\alpha_{g}^{2}-1}
\end{array} P h − 1 = ( α g 2 − 1 ) 2 c o s 2 θ + ( α g 2 − 1 ) α g 2 − α g 2 − 1 1
一様分布にしたがう乱数をξ \xi ξ
ξ = α g 2 ( α g 2 − 1 ) 2 c o s 2 θ + ( α g 2 − 1 ) − 1 α g 2 − 1 \xi = \frac{\alpha_{g}^{2}}{(\alpha_{g}^{2}-1)^{2}cos^{2}\theta + (\alpha_{g}^{2}-1)} - \frac{1}{\alpha_{g}^{2}-1} ξ = ( α g 2 − 1 ) 2 c o s 2 θ + ( α g 2 − 1 ) α g 2 − α g 2 − 1 1 θ \theta θ θ = a r c c o s 1 − ξ ξ ( α g 2 − 1 ) + 1 o r θ = a r c t a n ( α g ξ 1 − ξ ) \theta = arccos\sqrt{\frac{1-\xi}{\xi(\alpha_{g}^{2}-1)+1}} \ or \ \theta = arctan(\alpha_{g}\sqrt{\frac{\xi}{1-\xi}}) θ = a r c c o s ξ ( α g 2 − 1 ) + 1 1 − ξ o r θ = a r c t a n ( α g 1 − ξ ξ )
Importance Sampling Microfacet-Based BSDFs using the Distribution of Visible Normals
GGX
GGX分布は幅のパラメータα x , α y \alpha_{x}, \alpha_{y} α x , α y
(16) D ( m ) = χ + ( m ⋅ n ) π α x α y c o s 4 θ m ( 1 + t a n 2 θ m ( c o s 2 ϕ m α x 2 + s i n 2 ϕ m α y 2 ) ) 2 D(m)=\frac{\chi^{+}(m \cdot n)}{\pi \alpha_{x}\alpha_{y}cos^{4}\theta_{m}(1 + tan^{2}\theta_{m}(\frac{cos^{2} \phi_{m}}{\alpha_{x}^{2}} + \frac{sin^{2}\phi_{m}}{\alpha_{y}^{2}}))^{2}} \tag{16} D ( m ) = π α x α y c o s 4 θ m ( 1 + t a n 2 θ m ( α x 2 c o s 2 ϕ m + α y 2 s i n 2 ϕ m ) ) 2 χ + ( m ⋅ n ) ( 1 6 )
α = c o s 2 ϕ v α x 2 + s i n 2 ϕ v α y 2 \alpha = \sqrt{cos^{2} \phi_{v} \alpha_{x}^2 + sin^{2} \phi_{v} \alpha_{y}^{2}} α = c o s 2 ϕ v α x 2 + s i n 2 ϕ v α y 2 (18) G 1 ( v , m ) = χ + ( v ⋅ m v ⋅ n ) 2 1 + 1 + α 2 t a n 2 θ v G_{1}(v,m)=\chi^{+}(\frac{v \cdot m}{v \cdot n})\frac{2}{1+\sqrt{1+\alpha^{2}tan^{2}\theta_{v}}} \tag{18} G 1 ( v , m ) = χ + ( v ⋅ n v ⋅ m ) 1 + 1 + α 2 t a n 2 θ v 2 ( 1 8 )
Sampling the GGX Distribution of Visible Normals
まとめ
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