简单总结一些关于PBR和IBL的知识点，顺便贴一下自己的实现仓库软光栅实现IBL
一些定义
一个光子携带的能量
Q = h c λ Q = \frac{hc}{\lambda}
Q = λ h c
h h h c c c λ \lambda λ
单位时间内光穿过某个区域的能量
Φ = lim Δ t → 0 Δ Q Δ t = d Q d t \Phi = \lim^{}_{\Delta t \to 0}{\frac{\Delta Q}{\Delta t}} = \frac{dQ}{dt}
Φ = Δ t → 0 lim Δ t Δ Q = d t d Q
单位时间通过单位面积的能量
E ( p ) = lim Δ A → 0 Δ Φ ( p ) Δ A = d Φ ( p ) d A ⊥ = d Φ ( p ) d A cos θ E(p) =\lim^{}_{\Delta A \to 0}{\frac{\Delta \Phi(p)}{\Delta A}}
= \frac{d\Phi(p)}{dA^\bot}= \frac{d\Phi(p)}{dA \cos\theta} E ( p ) = Δ A → 0 lim Δ A Δ Φ ( p ) = d A ⊥ d Φ ( p ) = d A cos θ d Φ ( p )
单位时间内通过单位立体角的能量
I = lim Δ ω → 0 Δ Φ Δ ω = d Φ d ω I = \lim^{}_{\Delta \omega \to 0}\frac{\Delta \Phi}{\Delta \omega}
=\frac{d\Phi}{d\omega} I = Δ ω → 0 lim Δ ω Δ Φ = d ω d Φ
单位时间内通过单位立体角单位面积的能量
L = lim Δ ω → 0 Δ E ω ( p ) Δ ω = d E ω ( p ) d ω = d Φ d ω d A ⊥ = d Φ d ω d A cos θ L = \lim^{}_{\Delta \omega \to 0}\frac{\Delta E_{\omega}(p)}{\Delta \omega}
= \frac{dE_{\omega}(p)}{d\omega}
= \frac{d\Phi}{d\omega dA^\bot}
= \frac{d\Phi}{d\omega dA \cos\theta}
L = Δ ω → 0 lim Δ ω Δ E ω ( p ) = d ω d E ω ( p ) = d ω d A ⊥ d Φ = d ω d A cos θ d Φ
L o ( p , ω o ) = ∫ Ω f r ( p , ω i , ω o ) L i ( p , ω i ) n ⋅ ω i d ω i L_{o}(p,\omega_{o}) = \int^{}_{\Omega}f_{r}(p,\omega_{i},\omega_{o})L_{i}(p,\omega_{i})
n \cdot \omega_{i}d\omega_{i}
L o ( p , ω o ) = ∫ Ω f r ( p , ω i , ω o ) L i ( p , ω i ) n ⋅ ω i d ω i
我们之前常说的漫反射和镜面反射现象其实是光线通过微表面模型反射后的一种统计学现象。
BRDF定义为 单位面积上的 出射辐射度 与 入射辐照度 的比值：
f r ( p , ω i , ω o ) = d L o ( p , ω o ) d E i ( p , ω i ) f_{r}(p,\omega_{i},\omega_{o}) = \frac{dL_{o}(p,\omega_{o})}{dE_{i}(p,\omega_{i})}
f r ( p , ω i , ω o ) = d E i ( p , ω i ) d L o ( p , ω o )
目前一般实时使用的BRDF模型为Cook-Torrance BRDF模型，其分为漫反射和镜面反射两个部分
f r = k d f l a m b e r t + k s f c o o k − t o r r a n c e f_{r} = k_{d}f_{lambert} + k_{s}f_{cook-torrance}
f r = k d f l a m b e r t + k s f c o o k − t o r r a n c e
光线在微表面发生碰撞时会产生折射和反射现象，其中折射是光线进入物体内部，两者的能量比满足菲涅尔方程，光线在物体内部折射时能量可能被吸收，同时路径也可能会改，也可能因为被介质散射（考虑体介质参与），最后又从物体表面射出，如果重新射出的表面点离原来的入射点很近，那么可以简单将其考虑为一种漫反射现象，一般是朝各个方向均匀射出，如果距离比较远，比如说超过一个像素范围，那么如果要达到好的渲染效果，需要进一步考虑BSSRDF反射模型。
上式中k d kd k d k s ks k s k s + k d = 1 ks + kd = 1 k s + k d = 1 f d f_{d} f d
f l a m b e r t = c π f_{lambert} = \frac{c}{\pi}
f l a m b e r t = π c
c c c π \pi π
∫ 0 2 π ∫ 0 π 2 cos θ sin θ d θ d ϕ = π \int^{2\pi}_{0}\int^{\frac{\pi}{2}}_{0}\cos\theta\sin\theta d\theta d\phi = \pi
∫ 0 2 π ∫ 0 2 π cos θ sin θ d θ d ϕ = π
而镜面项的形式为：
f c o o k − t o r r a n c e = D F G 4 ( ω o ⋅ n ) ( ω i ⋅ n ) f_{cook-torrance} = \frac{DFG}{4(\omega_{o}\cdot n)(\omega_{i}\cdot n)}
f c o o k − t o r r a n c e = 4 ( ω o ⋅ n ) ( ω i ⋅ n ) D F G
这个公式也可以通过推导得到：
假设 θ h \theta_{h} θ h ω i \omega_{i} ω i ω o \omega_{o} ω o ω h \omega_{h} ω h
假设 θ \theta θ ω h \omega_{h} ω h n n n
假设 n n n a a a
假设 θ i \theta_{i} θ i θ o \theta_{o} θ o
类似于d ω = sin θ d θ d ϕ d\omega = \sin \theta d\theta d\phi d ω = sin θ d θ d ϕ
d A ( ω h ) = D ( ω h ) d ω h d A dA(\omega_{h}) = D(\omega_{h})d\omega_{h}dA
d A ( ω h ) = D ( ω h ) d ω h d A
D就是法线分布函数
这个点的能量通量为：
d Φ h = d E h d A ( ω h ) d\Phi_{h} = dE_{h}dA(\omega_{h})
d Φ h = d E h d A ( ω h )
出射的辐射度定义为：
L ( ω o ) = d Φ o d ω o d A ( ω o ) L(\omega_{o}) = \frac{d\Phi_{o}}{d\omega_{o}dA(\omega_{o})}
L ( ω o ) = d ω o d A ( ω o ) d Φ o
根据菲涅尔定理有：
d Φ o = F r ( ω o ) d Φ h d\Phi_{o} = F_{r}(\omega_{o})d\Phi_{h}
d Φ o = F r ( ω o ) d Φ h
代入替换一下 d Φ o d\Phi_{o} d Φ o L ( ω o ) L(\omega_{o}) L ( ω o )
L ( ω o ) = F r ( ω o ) d Φ h d ω o d A ( ω o ) L(\omega_{o}) = \frac{F_{r}(\omega_{o})d\Phi_{h}}{d\omega_{o}dA(\omega_{o})}
L ( ω o ) = d ω o d A ( ω o ) F r ( ω o ) d Φ h
再替换当中的Φ h \Phi_{h} Φ h
L ( ω o ) = F r ( ω o ) d E h d A ( ω h ) d ω o d A ( ω o ) L(\omega_{o}) = \frac{F_{r}(\omega_{o})dE_{h}dA(\omega_{h})}{d\omega_{o}dA(\omega_{o})}
L ( ω o ) = d ω o d A ( ω o ) F r ( ω o ) d E h d A ( ω h )
在这里需要注意的是d A ( ω h ) dA(\omega_{h}) d A ( ω h ) d A ( ω o ) dA(\omega_{o}) d A ( ω o )
d A ( ω o ) = cos θ o d A dA(\omega_{o}) = \cos \theta_{o} dA
d A ( ω o ) = cos θ o d A
另外d E h dE_{h} d E h
d E h = L i d ω i cos θ h dE_{h} = L_{i}d\omega_{i} \cos \theta_{h}
d E h = L i d ω i cos θ h
然后再替换一下得到：
L ( ω o ) = F r L i d ω i cos θ h D ( ω h ) d ω h d A d ω o cos θ o d A = F r L i D ( ω h ) cos θ h d ω h d ω i cos θ o d ω o L(\omega_{o}) = \frac{F_{r}L_{i}d\omega_{i} \cos \theta_{h}D(\omega_{h})d\omega_{h}dA}
{d\omega_{o}\cos \theta_{o} dA}
=\frac{F_{r}L_{i} D(\omega_{h})\cos \theta_{h}d\omega_{h}d\omega_{i}}
{\cos \theta_{o}d\omega_{o}}
L ( ω o ) = d ω o cos θ o d A F r L i d ω i cos θ h D ( ω h ) d ω h d A = cos θ o d ω o F r L i D ( ω h ) cos θ h d ω h d ω i
然后w o w_{o} w o w h w_{h} w h θ o ′ \theta^{'}_{o} θ o ′ θ o = 2 ⋅ θ h \theta_{o} = 2 \cdot \theta_{h} θ o = 2 ⋅ θ h d ω h d\omega_{h} d ω h
d ω h d ω o = sin θ h d θ h d ϕ h sin ω o d θ o d ϕ o = sin θ h d θ h d ϕ h sin ( 2 θ h ) d ( 2 θ h ) d ϕ o = 1 4 cos θ h \frac{d\omega_{h}}{d\omega_{o}} = \frac{\sin \theta_{h} d\theta_{h} d\phi_{h}}
{\sin \omega_{o}d\theta_{o} d\phi_{o}} = \frac
{\sin \theta_{h} d\theta_{h} d\phi_{h}}{\sin (2\theta_{h}) d (2\theta_{h}) d\phi_{o}}
=\frac{1}{4 \cos \theta_{h}}
d ω o d ω h = sin ω o d θ o d ϕ o sin θ h d θ h d ϕ h = sin ( 2 θ h ) d ( 2 θ h ) d ϕ o sin θ h d θ h d ϕ h = 4 cos θ h 1
所以代入后，最后化简为：
L ( ω o ) = F r ( ω o ) L i ( ω i ) D ( ω h ) d ω i 4 cos θ o = L i ( ω i ) ⋅ F r ( ω o ) D ( ω h ) 4 cos θ o ⋅ d ω i L(\omega_{o}) = \frac{F_r(\omega_{o})L_{i}(\omega_{i})D(\omega_{h})d\omega_{i}}
{4 \cos \theta_{o}}
=L_{i}(\omega_{i}) \cdot \frac{F_r(\omega_{o})D(\omega_{h})}
{4 \cos \theta_{o}} \cdot d\omega_{i}
L ( ω o ) = 4 cos θ o F r ( ω o ) L i ( ω i ) D ( ω h ) d ω i = L i ( ω i ) ⋅ 4 cos θ o F r ( ω o ) D ( ω h ) ⋅ d ω i
那么BRDF的定义为：
L ( ω o ) = F r ( ω o ) G ( ω o , ω i ) D ( ω h ) 4 cos θ o cos θ i L(\omega_{o}) = \frac{F_r(\omega_{o})G(\omega_{o},\omega_{i})D(\omega_{h})}
{4 \cos \theta_{o} \cos \theta_{i}}
L ( ω o ) = 4 cos θ o cos θ i F r ( ω o ) G ( ω o , ω i ) D ( ω h )
一般来说，F r F_{r} F r ω o , ω h , F 0 \omega_{o},\omega_{h},F_{0} ω o , ω h , F 0 D D D ω h , n , α \omega_{h},n,\alpha ω h , n , α G G G n , ω i , ω o , k n,\omega_{i},\omega_{o},k n , ω i , ω o , k k k k α \alpha α
而 G G G G G G
G ( n , ω o , ω i , α ) = G S c h l i c k G G X ( n , ω i , k ( α ) ) ⋅ G S c h l i c k G G X ( n , ω o , k ( α ) ) G(n,\omega_{o},\omega_{i},\alpha) = G_{SchlickGGX}(n,\omega_{i},k(\alpha)) \cdot
G_{SchlickGGX}(n,\omega_{o},k(\alpha))
G ( n , ω o , ω i , α ) = G S c h l i c k G G X ( n , ω i , k ( α ) ) ⋅ G S c h l i c k G G X ( n , ω o , k ( α ) )
其中：
G S c h l i c k G G X ( n , v , k ) = n ⋅ v ( n ⋅ v ) ( 1 − k ) + k G_{SchlickGGX}(n,v,k) = \frac{n \cdot v}{(n \cdot v)(1 - k) + k}
G S c h l i c k G G X ( n , v , k ) = ( n ⋅ v ) ( 1 − k ) + k n ⋅ v
k d i r e c t = ( α + 1 ) 2 8 k_{direct} = \frac{(\alpha + 1)^2}{8}
k d i r e c t = 8 ( α + 1 ) 2
k I B L = α 2 2 k_{IBL} = \frac{\alpha^2}{2}
k I B L = 2 α 2
常用的法线分布函数D有Trowbrideg-ReitzGGX，定义为：
N D F G G X T R ( n , h , α ) = α 2 π ( ( n ⋅ h ) 2 ( α 2 − 1 ) + 1 ) 2 NDF_{GGXTR}(n,h,\alpha) = \frac{\alpha^2}
{\pi((n \cdot h)^2(\alpha^2-1)+1)^2}
N D F G G X T R ( n , h , α ) = π ( ( n ⋅ h ) 2 ( α 2 − 1 ) + 1 ) 2 α 2
关于电介质的折射系数n的测量，其实是根据可见光在电介质和真空中的相对速度求得的
r ∥ = η t cos θ i − η i cos θ t η t cos θ i + η i cos θ t r_{\parallel} = \frac
{\eta_{t} \cos \theta_{i} - \eta_{i} \cos \theta_{t}}
{\eta_{t} \cos \theta_{i} + \eta_{i} \cos \theta_{t}}
r ∥ = η t cos θ i + η i cos θ t η t cos θ i − η i cos θ t
r ⊥ = η i cos θ i − η t cos θ t η i cos θ i + η t cos θ t r_{\bot} = \frac
{\eta_{i} \cos \theta_{i} - \eta_{t} \cos \theta_{t}}
{\eta_{i} \cos \theta_{i} + \eta_{t} \cos \theta_{t}}
r ⊥ = η i cos θ i + η t cos θ t η i cos θ i − η t cos θ t
F r = 1 2 ( r ∥ 2 + r ⊥ 2 ) F_{r} = \frac{1}{2}(r^2_{\parallel} + r^2_{\bot})
F r = 2 1 ( r ∥ 2 + r ⊥ 2 )
导体的 折射指数 定义为复数，形式为 η ‾ = η + i k \overline\eta = \eta + ik η = η + i k
菲涅尔方程比较复杂，求解代价比较大，一般使用Fresnel-Schlick近似法求解：
F S c h l i c k ( h , v , F 0 ) = F 0 + ( 1 − F 0 ) ( 1 − ( h ⋅ v ) ) 5 F_{Schlick}(h,v,F_{0}) = F_{0} + (1 - F_{0})(1-(h \cdot v))^5
F S c h l i c k ( h , v , F 0 ) = F 0 + ( 1 − F 0 ) ( 1 − ( h ⋅ v ) ) 5
F 0 F_{0} F 0 F 0 F0 F 0
L o ( p , ω o ) = ∫ Ω ( k d c π + k s D F G 4 ( ω o ⋅ n ) ( ω i ⋅ n ) ) L i ( p , ω i ) n ⋅ ω i d ω i L_{o}(p,\omega_{o})=\int_{\Omega}
(k_{d}\frac{c}{\pi}+k_{s}\frac{DFG}{4(\omega_{o}\cdot n)(\omega_{i}\cdot n)})
L_{i}(p,\omega_{i})n \cdot \omega_{i}d\omega_{i}
L o ( p , ω o ) = ∫ Ω ( k d π c + k s 4 ( ω o ⋅ n ) ( ω i ⋅ n ) D F G ) L i ( p , ω i ) n ⋅ ω i d ω i
L d ( ω o ) = c π ∫ Ω k d L i ( ω i ) n ⋅ ω i d ω i L_{d}(\omega_{o}) = \frac{c}{\pi}\int_{\Omega}k_{d}L_{i}(\omega_{i})n\cdot \omega_{i}d\omega_{i}
L d ( ω o ) = π c ∫ Ω k d L i ( ω i ) n ⋅ ω i d ω i
因为 k d k_{d} k d
k d = ( 1 − F ( n ⋅ ω i ) ) ( 1 − m e t a l n e s s ) ≈ ( 1 − F r o u g h n e s s ( n ⋅ ω o ) ) ( 1 − m e t a l n e s s ) = k d ∗ k_{d} = (1 - F(n\cdot \omega_{i}))(1 - metalness) \\
\approx (1-F_{roughness}(n\cdot \omega_{o}))(1-metalness)
=k^{*}_{d}
k d = ( 1 − F ( n ⋅ ω i ) ) ( 1 − m e t a l n e s s ) ≈ ( 1 − F r o u g h n e s s ( n ⋅ ω o ) ) ( 1 − m e t a l n e s s ) = k d ∗
其中的 F r o u g h n e s s ( n ⋅ ω o ) F_{roughness}(n\cdot \omega_{o}) F r o u g h n e s s ( n ⋅ ω o )
F r o u g h n e s s ( n ⋅ ω o ) = F 0 + ( m a x ( 1 − r o u g h n e s s , F 0 ) − F 0 ) ( 1 − n ⋅ ω o ) 5 F_{roughness}(n\cdot \omega_{o}) = F_{0} + (max(1-roughness,F_{0}) - F_{0})(1-n\cdot \omega_{o})^5
F r o u g h n e s s ( n ⋅ ω o ) = F 0 + ( m a x ( 1 − r o u g h n e s s , F 0 ) − F 0 ) ( 1 − n ⋅ ω o ) 5
采用蒙特卡洛方法进行预计算，均匀采样每一个方向的L i L_{i} L i
∫ a b f ( x ) d x = b − a N ∑ i = 1 N f ( X i ) \int^{b}_{a}f(x)dx = \frac{b-a}{N} \sum^{N}_{i=1}f(X_{i})
∫ a b f ( x ) d x = N b − a i = 1 ∑ N f ( X i )
转换为：
∫ Ω 1 π L i ( p , ω i ) d ω i ≈ π n 1 ⋅ n 2 ∑ m = 0 n 1 ∑ n = 0 n 2 L i ( p , ϕ m , θ n ) cos ( θ n ) s i n ( θ n ) \int_{\Omega}\frac{1}{\pi}L_{i}(p,\omega_{i})d\omega_{i} \approx
\frac{\pi}{n1 \cdot n2}\sum^{n1}_{m=0} \sum^{n2}_{n=0}
L_{i}(p,\phi_{m},\theta_{n})\cos(\theta_{n})sin(\theta_{n})
∫ Ω π 1 L i ( p , ω i ) d ω i ≈ n 1 ⋅ n 2 π m = 0 ∑ n 1 n = 0 ∑ n 2 L i ( p , ϕ m , θ n ) cos ( θ n ) s i n ( θ n )
也可以用余弦为权重进行半球采样 p ( ω i ) = cos θ π p(\omega_i) = \frac{\cos \theta}{\pi} p ( ω i ) = π cos θ
1 π ∫ Ω L i ( ω i ) n ⋅ ω i d ω i ≈ 1 π 1 N ∑ i N L ( ω i ) ( n ⋅ ω i ) n ⋅ ω i π = 1 N ∑ i N L ( ω i ) \frac{1}{\pi}\int_{\Omega}L_{i}(\omega_{i})n\cdot \omega_{i}d\omega_{i} \approx
\frac{1}{\pi}\frac{1}{N}\sum^{N}_{i}\frac{L(\omega_{i})(n\cdot \omega_{i})}
{\frac{n\cdot \omega_{i}}{\pi}}
=\frac{1}{N}\sum^{N}_{i}L(\omega_{i})
π 1 ∫ Ω L i ( ω i ) n ⋅ ω i d ω i ≈ π 1 N 1 i ∑ N π n ⋅ ω i L ( ω i ) ( n ⋅ ω i ) = N 1 i ∑ N L ( ω i )
使用了split sum近似，将L i ( ω i ) L_{i}(\omega_{i}) L i ( ω i ) f s ( ω i , ω o ) n ⋅ ω i f_{s}(\omega_{i},\omega_{o})n\cdot\omega_{i} f s ( ω i , ω o ) n ⋅ ω i
L s ( ω o ) = ∫ Ω f s ( ω i , ω o ) L i ( ω i ) n ⋅ ω i d ω i = L c ( ω o ) ∫ Ω f s ( ω i , ω o ) n ⋅ ω i d ω i L c ( ω o ) = ∫ Ω f s ( ω i , ω o ) L i ( ω i ) n ⋅ ω i d ω i ∫ Ω f s ( ω i , ω o ) n ⋅ ω i d ω i L_{s}(\omega_{o})=\int_{\Omega}f_{s}(\omega_{i},\omega_{o})L_{i}(\omega_{i})n\cdot \omega_{i}d\omega_{i}
=L_{c}(\omega_{o})\int_{\Omega}f_{s}(\omega_{i},\omega_{o})n\cdot \omega_{i}d\omega_{i}\\
L_{c}({\omega_{o}})=\frac
{\int_{\Omega}f_{s}(\omega_{i},\omega_{o})L_{i}(\omega_{i})n\cdot \omega_{i}d\omega_{i}}
{\int_{\Omega}f_{s}(\omega_{i},\omega_{o})n\cdot \omega_{i}d\omega_{i}}
L s ( ω o ) = ∫ Ω f s ( ω i , ω o ) L i ( ω i ) n ⋅ ω i d ω i = L c ( ω o ) ∫ Ω f s ( ω i , ω o ) n ⋅ ω i d ω i L c ( ω o ) = ∫ Ω f s ( ω i , ω o ) n ⋅ ω i d ω i ∫ Ω f s ( ω i , ω o ) L i ( ω i ) n ⋅ ω i d ω i
之后的L c ( ω o ) L_{c}(\omega_{o}) L c ( ω o ) F r F_{r} F r G G G
L c ( ω o ) ≈ ∑ i N L i ( ω i ) n ⋅ ω i ∑ i N n ⋅ ω i L_{c}(\omega_{o}) \approx \frac{\sum^{N}_{i}L_{i}(\omega_{i})n\cdot \omega_{i}}
{\sum^{N}_{i}n\cdot \omega_{i}}
L c ( ω o ) ≈ ∑ i N n ⋅ ω i ∑ i N L i ( ω i ) n ⋅ ω i
在实际计算过程中，如果只是从原始的环境贴图采样，计算出来的预过滤贴图在明亮区域周围会出现点状图案
prefiltered_color += texture (environmentMap,L).rgb * NdotL; total_weight += NdotL;
可以将一个采样点实际代表的立体角大小和pdf结合考虑，估算应该从哪个lod采样，pdf越大，说明这个方向采样的概率越大，应该采样更高分辨率的贴图，pdf越小，应该采样更低分辨率的贴图，出现点状图案是因为采样没有很好的收敛，采样次数不够多，方差还较大，如果pdf小的时候采样高分辨率的贴图就会有很大概率引入较大的方差。
float D = DistributionGGX(N,H,roughness);float NdotH = max (dot (N, H), 0. f);float HdotV = max (dot (H, V), 0. f);float pdf = D * NdotH / (4. f * HdotV) + 0.0001 f;float resolution = PrefilterMapSize;float sa_texel = 4. f * PI / (6. f * resolution *resolution);float sa_sample = 1. f / (float (SampleCount) * pdf + 0.0001 f);float mip_level = roughness == 0. f ? 0. f : 0.5 f * log2 (sa_sample / sa_texel);prefiltered_color += textureLod (environmentMap, L, mip_level).rgb * NdotL; total_weight += NdotL;
原来的积分式子移掉L c ( ω o ) L_{c}(\omega_{o}) L c ( ω o )
∫ Ω f s ( ω i , ω o ) n ⋅ ω i d ω i = ∫ Ω f s ( ω i , ω o ) F ( ω o , h ) F ( ω o , h ) n ⋅ ω i d ω i = ∫ Ω f s ( ω i , ω o ) F ( ω o , h ) ( F 0 + ( 1 − F 0 ) ( 1 − ω o ⋅ h ) 5 ) n ⋅ ω i d ω i F 0 ∫ Ω f s ( ω i , ω o ) F ( ω o , h ) ( 1 − ( 1 − ω o ⋅ h ) 5 ) n ⋅ ω i d ω i + ∫ Ω f s ( ω i , ω o ) F ( ω o , h ) ( 1 − ω o ⋅ h ) 5 n ⋅ ω i d ω i = F 0 ∗ s c a l e + b i a s \int_{\Omega}f_{s}(\omega_{i},\omega_{o})n\cdot\omega_{i}d\omega_{i}\\
=\int_{\Omega}f_{s}(\omega_{i},\omega_{o})\frac{F(\omega_{o},h)}{F(\omega_{o},h)}n\cdot\omega_{i}d\omega_{i}\\
=\int_{\Omega}\frac{f_{s}(\omega_{i},\omega_{o})}{F(\omega_{o},h)}(F_{0}+(1-F_{0})(1-\omega_{o}\cdot h)^5)n\cdot \omega_{i}d\omega_{i}\\
F_{0}\int_{\Omega}\frac{f_{s}(\omega_{i},\omega_{o})}{F(\omega_{o},h)}(1-(1-\omega_{o}\cdot h)^5)n\cdot\omega_{i}d\omega_{i}
+\int_{\Omega}\frac{f_{s}(\omega_{i},\omega_{o})}{F(\omega_{o},h)}(1-\omega_{o}\cdot h)^5n\cdot\omega_{i}d\omega_{i}\\
=F_{0}*scale+bias
∫ Ω f s ( ω i , ω o ) n ⋅ ω i d ω i = ∫ Ω f s ( ω i , ω o ) F ( ω o , h ) F ( ω o , h ) n ⋅ ω i d ω i = ∫ Ω F ( ω o , h ) f s ( ω i , ω o ) ( F 0 + ( 1 − F 0 ) ( 1 − ω o ⋅ h ) 5 ) n ⋅ ω i d ω i F 0 ∫ Ω F ( ω o , h ) f s ( ω i , ω o ) ( 1 − ( 1 − ω o ⋅ h ) 5 ) n ⋅ ω i d ω i + ∫ Ω F ( ω o , h ) f s ( ω i , ω o ) ( 1 − ω o ⋅ h ) 5 n ⋅ ω i d ω i = F 0 ∗ s c a l e + b i a s
这个积分的未知量维度为9维，包括ω o , n , F 0 ( 由 m e t a l n e s s 和 a l b e d o 决 定 ) , 粗 糙 度 α \omega_{o},n,F_{0}(由metalness和albedo决定),粗糙度\alpha ω o , n , F 0 ( 由 m e t a l n e s s 和 a l b e d o 决 定 ) , 粗 糙 度 α ϕ \phi ϕ cos θ \cos\theta cos θ ω o 与 n 的 夹 角 c o s 值 \omega_{o}与n的夹角cos值 ω o 与 n 的 夹 角 c o s 值 ( ω o ⋅ n ) 和 α (\omega_{o}\cdot{n})和\alpha ( ω o ⋅ n ) 和 α
实际离散计算积分的时候，需要对s c a l e scale s c a l e ω h \omega_{h} ω h ω i \omega_{i} ω i
∫ Ω D ( ω h ) n ⋅ ω h d ω h d A = d A ⇒ ∫ Ω D ( ω h ) n ⋅ ω h d ω h = 1 (1) \int_{\Omega}D(\omega_{h})n\cdot\omega_{h}d\omega_{h}dA = dA \Rightarrow
\int_{\Omega}D(\omega_{h})n\cdot\omega_{h}d\omega_{h} = 1 \tag 1
∫ Ω D ( ω h ) n ⋅ ω h d ω h d A = d A ⇒ ∫ Ω D ( ω h ) n ⋅ ω h d ω h = 1 ( 1 )
d ω h d ω i = 1 4 ( ω h ⋅ ω i ) (2) \frac{d\omega_{h}}{d\omega_{i}} = \frac{1}{4(\omega_{h}\cdot\omega_{i})} \tag 2
d ω i d ω h = 4 ( ω h ⋅ ω i ) 1 ( 2 )
把(2)带入(1)得到：
p d f ( ω i ) = ω h ⋅ ω n 4 ( ω h ⋅ ω i ) D ( ω h ) pdf(\omega_{i}) = \frac{\omega_{h}\cdot\omega_{n}}{4(\omega_{h}\cdot\omega_{i})}D(\omega_{h})
p d f ( ω i ) = 4 ( ω h ⋅ ω i ) ω h ⋅ ω n D ( ω h )
因此对s c a l e scale s c a l e
s c a l e ≈ 1 N ∑ i N D ( ω h ( k ) ) G ( ω o , ω i k ) 4 ( ω o ⋅ n ) ( ω i ( k ) ⋅ n ) ( 1 − ( 1 − ω o ⋅ ω h ( k ) ) 5 ) ( ω i ( k ) ⋅ n ) p ( ω i ( k ) ) = 1 N ∑ i N D ( ω h ( k ) ) G ( ω o , ω i k ) 4 ( ω o ⋅ n ) ( ω i ( k ) ⋅ n ) ( 1 − ( 1 − ω o ⋅ ω h ( k ) ) 5 ) ( ω i ( k ) ⋅ n ) D ( ω h ( k ) ) ( ω h ( k ) ⋅ n ) 4 ( ω o ⋅ ω h ( k ) ) = 1 N ∑ i N G ( ω o , ω i k ) ( ω o ⋅ ω h ( k ) ) ( 1 − ( 1 − ω o ⋅ ω h ( k ) ) 5 ) ( ω h ( k ) ⋅ n ) ( ω o ⋅ n ) scale \approx \frac{1}{N}\sum^{N}_{i}
\frac
{
\frac{
D(\omega^{(k)}_{h})G(\omega_{o},\omega^{k}_{i})}{
4(\omega_{o}\cdot n)(\omega^{(k)}_{i}\cdot n)
}
(1-(1-\omega_{o}\cdot\omega^{(k)}_{h})^5)(\omega^{(k)}_{i}\cdot n)
}
{p(\omega^{(k)}_{i})}\\
=\frac{1}{N}\sum^{N}_{i}
\frac
{
\frac{
D(\omega^{(k)}_{h})G(\omega_{o},\omega^{k}_{i})}{
4(\omega_{o}\cdot n)(\omega^{(k)}_{i}\cdot n)
}
(1-(1-\omega_{o}\cdot\omega^{(k)}_{h})^5)(\omega^{(k)}_{i}\cdot n)
}
{\frac
{D(\omega^{(k)}_{h})(\omega^{(k)}_{h}\cdot n)}
{4(\omega_{o}\cdot\omega^{(k)}_{h})}}\\
=\frac{1}{N}\sum^{N}_{i}
\frac
{
G(\omega_{o},\omega^{k}_{i})(\omega_{o}\cdot\omega^{(k)}_{h})
(1-(1-\omega_{o}\cdot\omega^{(k)}_{h})^5)
}
{(\omega^{(k)}_{h}\cdot n)(\omega_{o}\cdot n)}
s c a l e ≈ N 1 i ∑ N p ( ω i ( k ) ) 4 ( ω o ⋅ n ) ( ω i ( k ) ⋅ n ) D ( ω h ( k ) ) G ( ω o , ω i k ) ( 1 − ( 1 − ω o ⋅ ω h ( k ) ) 5 ) ( ω i ( k ) ⋅ n ) = N 1 i ∑ N 4 ( ω o ⋅ ω h ( k ) ) D ( ω h ( k ) ) ( ω h ( k ) ⋅ n ) 4 ( ω o ⋅ n ) ( ω i ( k ) ⋅ n ) D ( ω h ( k ) ) G ( ω o , ω i k ) ( 1 − ( 1 − ω o ⋅ ω h ( k ) ) 5 ) ( ω i ( k ) ⋅ n ) = N 1 i ∑ N ( ω h ( k ) ⋅ n ) ( ω o ⋅ n ) G ( ω o , ω i k ) ( ω o ⋅ ω h ( k ) ) ( 1 − ( 1 − ω o ⋅ ω h ( k ) ) 5 )
类似的b i a s bias b i a s
b i a s ≈ 1 N ∑ i N G ( ω o , ω i k ) ( ω o ⋅ ω h ( k ) ) ( 1 − ω o ⋅ ω h ( k ) ) 5 ( ω h ( k ) ⋅ n ) ( ω o ⋅ n ) bias \approx
\frac{1}{N}\sum^{N}_{i}
\frac
{
G(\omega_{o},\omega^{k}_{i})(\omega_{o}\cdot\omega^{(k)}_{h})
(1-\omega_{o}\cdot\omega^{(k)}_{h})^5
}
{(\omega^{(k)}_{h}\cdot n)(\omega_{o}\cdot n)}
b i a s ≈ N 1 i ∑ N ( ω h ( k ) ⋅ n ) ( ω o ⋅ n ) G ( ω o , ω i k ) ( ω o ⋅ ω h ( k ) ) ( 1 − ω o ⋅ ω h ( k ) ) 5
BRDF的LUT计算代码如下：
vec2 IntegrateBRDF(float NdotV, float roughness){ vec3 V; V.x = sqrt (1. f - NdotV * NdotV); V.y = 0. f; V.z = NdotV; float A = 0. f; float B = 0. f; vec3 N = vec3 (0. f,0. f,1. f); for (uint i = 0 u; i < SampleCount; i++){ vec2 Xi = Hammersley(i, SampleCount); vec3 H = ImportanceSampleGGX(Xi, N, roughness); vec3 L = normalize (2. f * dot (V, H) * H - V); float NdotL = max (L.z, 0. f); float NdotH = max (H.z, 0. f); float VdotH = max (dot (V, H), 0. f); if (NdotL > 0. f){ float G = GeometrySmith(N, V, L, roughness); float G_Vis = (G * VdotH) / (NdotH * NdotV); float Fc = pow (1. f - VdotH, 5. f); A += (1. f - Fc) * G_Vis; B += Fc * G_Vis; } } A /= float (SampleCount); B /= float (SampleCount); return vec2 (A, B); }
vec3 F0 = mix (vec3 (0.04 ),albedo,metalness);vec3 F = fresnelSchlickRoughness(wo,normal,F0,roughness);vec3 kS = F;vec3 kD = (vec3 (1 ) - kS) * (1 - metalness);vec3 irradiance = texture (irradianceMap,n).rgb;vec3 diffuse_color = kD * albedo * irradiance;vec3 prefilteredColor = textureLod (prefilterMap,wi,roughness*MAX_LOD).rgb;vec2 brdf = texture (brdfLUT,vec2 (max (dot (n,wo),0.0 ),roughness)).rg;vec3 specular_color = (F0 * brdf.x + brdf.y) * prefilteredColor;vec3 color = diffuse_color + specular_color;
