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jinchao
2025-02-23 20:51:53 +08:00
parent 2bf0000c22
commit 791c17ebdf
21 changed files with 1946 additions and 1827 deletions
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docs/note/graphics/illumination_model/microfacets/microfacet_3.svg
+129 -119
View File
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@@ -25,9 +25,11 @@ G_1=\dfrac{x}{y}=\dfrac{2\sin\gamma\sin\alpha}{\sin(\beta+\pi/2)}
$$
其中
$$
\sin\alpha=\vec{n}\cdot\vec{h} \\
\sin(\beta+\pi/2)=\cos\beta=\vec{v}\cdot\vec{h} \\
\sin\gamma=\vec{n}\cdot\vec{l}
\begin{aligned}
\sin\alpha&=\vec{n}\cdot\vec{h} \\
\sin(\beta+\pi/2)&=\cos\beta=\vec{v}\cdot\vec{h} \\
\sin\gamma&=\vec{n}\cdot\vec{l}
\end{aligned}
$$
所以可以求得
$$
@@ -39,16 +41,21 @@ G_2=\dfrac{2(\vec{n}\cdot\vec{h})(\vec{n}\cdot\vec{v})}{\vec{v}\cdot\vec{h}}
$$
所以光线受几何衰减因子的影响共有三种可能性，完全不受影响，照射光被阻挡，反射光被阻挡，最终的表达式为
$$
G(\vec{l},\vec{v},\vec{h})=\min\{1,G_1,G_2\}=\min\left\{1,\dfrac{2(\vec{n}\cdot\vec{h})(\vec{n}\cdot\vec{l})}{\vec{v}\cdot\vec{h}}
\begin{aligned}
G(\vec{l},\vec{v},\vec{h})&=\min\{1,G_1,G_2\}\\
&=\min\left\{1,\dfrac{2(\vec{n}\cdot\vec{h})(\vec{n}\cdot\vec{l})}{\vec{v}\cdot\vec{h}}
,\dfrac{2(\vec{n}\cdot\vec{h})(\vec{n}\cdot\vec{v})}{\vec{v}\cdot\vec{h}}
\right\}
\end{aligned}
$$
### 7.3 Schlick-Beckmann
在Unreal Engine中，使用了这种模型，在这种模型中
$$
G(\vec{l},\vec{v},\vec{h})=G_1(\vec{l})G_1(\vec{v}) \\
G_1(\vec{v})=\dfrac{\vec{n}\cdot\vec{v}}{(\vec{n}\cdot\vec{v})(1-k)+k}
\begin{aligned}
G(\vec{l},\vec{v},\vec{h})&=G_1(\vec{l})G_1(\vec{v}) \\
G_1(\vec{v})&=\dfrac{\vec{n}\cdot\vec{v}}{(\vec{n}\cdot\vec{v})(1-k)+k}
\end{aligned}
$$
在原始的Schlick算法中，定义
$$
@@ -60,11 +67,17 @@ k=\frac{\alpha}{2}
$$
根据所计算的光源的不同，$\alpha$的定义也不同，当使用间接光源(IBL)时
$$
\alpha=roughness^2, k=\dfrac{roughness^2}{2}
\begin{aligned}
\alpha&=roughness^2\\
k&=\dfrac{roughness^2}{2}
\end{aligned}
$$
使用直接光源时，
$$
\alpha=\left(\frac{roughness+1}{2}\right)^2, k=\dfrac{(roughness+1)^2}{8}
\begin{aligned}
\alpha&=\left(\frac{roughness+1}{2}\right)^2\\
k&=\dfrac{(roughness+1)^2}{8}
\end{aligned}
$$
## 8. 菲尼尔反射方程