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|
||||
# 微平面理论(一)
|
||||
----
|
||||
## 1. 概述
|
||||
微表面理论（Microfacet Theory）认为我们看到的表面上的一点是由很多朝向各异且光学平的微小表面组成。当光线从$\vec{l}$方向照射到这点，而我们在$\vec{v}$方向观察时，由于光学平面只会将光线$\vec{l}$反射到关于法线对称的$\vec{v}$方向，而$\vec{l}$和$\vec{v}$已经确定，所以只有法线朝向正好是$\vec{l}$和$\vec{v}$的半角向量$\vec{h}$的微表面才会将光线发射到$\vec{v}$方向，从而被我们看见。
|
||||
|
||||
|
||||
|
||||
## 2. 法线分布函数
|
||||
### 2.1 定义
|
||||
法线分布函数（Normal Distribution Function，简写为NDF）用来描述微表面法线的概率分布，可以这样理解：向NDF输入一个朝向h，NDF会返回法线是$\vec{h}$的微表面数占微表面总数的比例，需要注意的是$D(h)$是描述微表面**数量**的概率分布函数
|
||||
|
||||
### 2.2 微表面的面积
|
||||
设一个小平面，面积为$dA$，此小平面由更小的微表面组成，其中法线为$\vec{h}$的微表面的概率为$D(h)$，这些微表面的面积总和为$D(h)dA$，考虑到法线范围，定义微表面的法线为$\vec{h}$，范围在立体角$d\omega_h$之内的所有微表面的表面积为
|
||||
$$
|
||||
dA(\vec{h})=D(\vec{h})d\omega_h dA
|
||||
$$
|
||||
|
||||
## 3. 几何衰减因子
|
||||
|
||||
### 3.1 定义
|
||||
由于微表面之间的互相遮挡(Shadowing)，并不是所有微表面能够接收到光线，还有一部分微表面反射的光线会被阻挡(Masking)，还有一部分光线会在微表面之间互相反射.
|
||||
Shadowing和Masking效应用几何衰减因子(Geometrical Attenuation Factor)来表示，用$G(l,v)$表示
|
||||
|
||||
|
||||
|
||||
## 4. 菲涅尔方程
|
||||
### 4.1 定义
|
||||
光学平面会将一部分光线反射，一部分光线折射，其中反射光线的比例用菲尼尔方程(Fresnel Equations)计算，用$F(\vec{\omega}_i,\vec{\omega}_h)$表示，因为菲尼尔函数一般只和光线和法线之间的夹角相关，有时候也会用$F(\vec{\omega}_i\cdot\vec{\omega}_h)$表示
|
||||
|
||||
|
||||
## 5.Cook-Torrance模型
|
||||
|
||||
### 5.1 定义
|
||||
Cook-Torrance模型用来描述反射平面的的BRDF模型中的高光(Specular)部分，公式为
|
||||
$$
|
||||
f(\vec{\omega}_i,\vec{\omega}_o)=\frac{F(\vec{\omega}_i,\vec{\omega}_h)G(\vec{\omega}_i,\vec{\omega}_o)D(\vec{\omega}_h)}{4(\vec{n}\cdot \vec{\omega}_i)(\vec{n}\cdot\vec{\omega}_o)}
|
||||
$$
|
||||
|
||||
### 5.2 推导过程
|
||||
设一个小平面，面积为$dA$，法线方向$\vec{n}$，入射光方向$\vec{\omega}_i$，微分变量为$d\omega_i$，观察方向$\vec{\omega}_o$，微分变量为$d\omega_o$
|
||||
|
||||
|
||||
|
||||
根据上面的推导，这个小平面上所有法线为$\vec{\omega_h}$的微平面的面积为
|
||||
$$
|
||||
dA(\vec{\omega_h})=D(\vec{\omega_h})d\omega_h dA
|
||||
$$
|
||||
由于其他微平面接受的入射光通量不会反射到$\vec{\omega}_o$方向上，所以不用考虑，所以在$\vec{\omega}_h$方向上有效入射光通量为
|
||||
$$\begin{split}
|
||||
d\Phi_h&=L_i(\vec{\omega}_i)\cos(\theta_h)dA(\vec{\omega_h})d\omega_i \\
|
||||
&=L_i(\vec{\omega}_i)\cos(\theta_h)D(\vec{\omega_h})d\omega_h dA
|
||||
d\omega_i
|
||||
\end{split}$$
|
||||
出射光通量为
|
||||
$$
|
||||
d\Phi_o=dL_o(\vec{\omega}_o)\cos(\theta_o)dAd\omega_o
|
||||
$$
|
||||
根据菲尼尔公式，出射光通量和入射光通量的比例可以用如下公式表示
|
||||
$$
|
||||
d\Phi_o = F(\vec{\omega}_i, \vec{\omega}_h)d\Phi_h
|
||||
$$
|
||||
联立以上公式可得
|
||||
$$\begin{split}
|
||||
dL_o(\vec{\omega}_o)&=\dfrac{d\Phi_o}{\cos(\theta_o)dAd\omega_o}\\
|
||||
&=\dfrac{F(\vec{\omega}_i, \vec{\omega}_h)d\Phi_h}{\cos(\theta_o)dAd\omega_o} \\
|
||||
&=\dfrac{F(\vec{\omega}_i, \vec{\omega}_h)L_i(\vec{\omega}_i)\cos(\theta_h)D(\vec{\omega_h})d\omega_hd\omega_i
|
||||
}{\cos(\theta_o)d\omega_o}
|
||||
\end{split}$$
|
||||
根据BRDF的定义
|
||||
$$\begin{split}
|
||||
f(\vec{\omega}_i,\vec{\omega}_o)&=\dfrac{dL_o(\vec{\omega}_o)}{L_i(\vec{\omega}_i)\cos(\theta_i)d\omega_i} \\
|
||||
&=\dfrac{F(\vec{\omega}_i, \vec{\omega}_h)D(\vec{\omega}_h)\cos(\theta_h)}{\cos(\theta_o)\cos(\theta_i)}\cdot\dfrac{d\omega_h}{d\omega_o}
|
||||
\end{split}$$
|
||||
|
||||
下面考察$d\omega_o$和$d\omega_h$的关系，设小平面上的一个微平面，接受的入射光线方向为$\vec{\omega}_i$, 法线为$\vec{\omega}_h$，出射光线方向为$\vec{\omega}_o$，和单位圆的交点分别是$I,Q,R$
|
||||
|
||||
|
||||
|
||||
立体角$d\omega_h$和立体角$d\omega_o$之间的约束关系在于，保持$\vec{\omega}_i$不变，让法线$\vec{\omega}_o$在立体角$d\omega_o$范围之内移动，则$\vec{\omega}_h$需要保证是$\vec{\omega}_i$和$d\omega_o$的半角法线，它的活动范围$d\omega_h$比较小，具体可以计算一下
|
||||
$$
|
||||
\dfrac{d\omega_h}{d\omega_o}=\dfrac{dA_h}{dA_r}
|
||||
$$
|
||||
连接$IR$相交法线$\vec{\omega}_h$于交点$P$，由于$IR=2IP$，所以
|
||||
$$
|
||||
\dfrac{dA_r}{dA''}=4
|
||||
$$
|
||||
由于$OP=\cos(\theta_h)$，所以
|
||||
$$
|
||||
\dfrac{dA'}{dA''}=\dfrac{1}{\cos^2(\theta_h)}
|
||||
$$
|
||||
由于$dA_h=dA'\cos(\theta_h)$，联立以上公式，可以得到
|
||||
$$
|
||||
\dfrac{d\omega_h}{d\omega_o}=\dfrac{1}{4\cos(\theta_h)}
|
||||
$$
|
||||
带入上面的BRDF公式，可以得到
|
||||
$$
|
||||
f(\vec{\omega}_i,\vec{\omega}_o)=\dfrac{F(\vec{\omega}_i, \vec{\omega}_h)D(\vec{\omega}_h)}{4\cos(\theta_o)\cos(\theta_i)}
|
||||
$$
|
||||
将几何衰减因子G考虑在内，可以得到
|
||||
$$
|
||||
f(\vec{\omega}_i,\vec{\omega}_o)=\dfrac{F(\vec{\omega}_i, \vec{\omega}_h)D(\vec{\omega}_h)G(\vec{\omega}_i, \vec{\omega}_o)}{4\cos(\theta_o)\cos(\theta_i)}
|
||||
$$
|
||||
@@ -0,0 +1,95 @@
|
||||
# 微平面理论(二)
|
||||
-----
|
||||
## 6. 法线分布函数详解
|
||||
### 6.1 归一化约束
|
||||
所有的法线分布函数都应该符合以下约束
|
||||
$$
|
||||
(\vec{v}\cdot\vec{n})=\int_{\Omega}D(m)(\vec{v}\cdot\vec{m})d\omega_m
|
||||
$$
|
||||
可以理解为，这些为平面在某个指定方向上的投影面积，等于整体宏平面的投影面积
|
||||
|
||||
|
||||
|
||||
当$v=n$时，有
|
||||
$$
|
||||
1=\int_{\Omega}D(\vec{m})(\vec{n}\cdot\vec{m})d\omega_m
|
||||
$$
|
||||
|
||||
### 6.2 Phong
|
||||
Phong模型认为，反射强度正比于$(\vec{n}\cdot\vec{m})^\alpha$，其中$\alpha$表示材质的“光滑程度”，当$\alpha=0$，材质是绝对的漫反射材质，当$\alpha\rightarrow\infty$，材质成为绝对光滑的镜面，设
|
||||
$$
|
||||
D_{phong}(\vec{m})=R_p(\vec{n}\cdot\vec{m})^\alpha
|
||||
$$
|
||||
由上面法线分布函数的归一性可知
|
||||
$$
|
||||
1=\int_{\Omega}R_p(\vec{n}\cdot\vec{m})^{\alpha+1}d\omega_m
|
||||
$$
|
||||
由于$d\omega_m=\sin(\theta)d\phi d\theta$
|
||||
|
||||
$$
|
||||
1=\int_0^{2\pi}\int_0^{\pi/2}R_p\cos^{\alpha+1}(\theta)\sin(\theta)d\theta d\phi=\dfrac{2\pi}{2+\alpha}R_p
|
||||
$$
|
||||
可以得到最终D函数为
|
||||
$$
|
||||
D_{phong}(\vec{m})=\dfrac{2+\alpha}{2\pi}(\vec{n}\cdot\vec{m})^\alpha
|
||||
$$
|
||||
一般引擎用粗糙度（取值范围[0,1])表示材质的光滑程度，Phong模型中的$\alpha$常用如下公式计算
|
||||
$$
|
||||
\alpha_{phong}=\dfrac{2}{roughness^4}-2
|
||||
$$
|
||||
用Mathmatica绘制函数图
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
### 6.3 Beckmann
|
||||
Beckmann的法线分布函数为
|
||||
$$
|
||||
\displaystyle{D_{beckmann}(\vec{m})=R_{beckmann}\dfrac{1}{\alpha^2(\vec{n}\cdot\vec{m})^4}e^{-\left(\dfrac{1-(\vec{n}\cdot\vec{m})^2}{(\vec{n}\cdot\vec{m})^2\alpha^2}\right)}}
|
||||
$$
|
||||
经过积分计算$R_{beckmann}=1/\pi$，所以
|
||||
$$
|
||||
\displaystyle{D_{beckmann}(\vec{m})=\dfrac{1}{\pi\alpha^2(\vec{n}\cdot\vec{m})^4}e^{-\left(\dfrac{1-(\vec{n}\cdot\vec{m})^2}{(\vec{n}\cdot\vec{m})^2\alpha^2}\right)}}
|
||||
$$
|
||||
Beckmann法线分布函数中，$\alpha$同样代表了材质的粗糙度，趋向于0时表示绝对光滑，取值却大约粗糙，但当取值超过1时，材质会变得“超级粗糙”（诡异），$\alpha$用如下公式获取
|
||||
$$
|
||||
\alpha_{beckmann}=roughness^2
|
||||
$$
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
当$\alpha$取值在0.5以下时，Beckmann模型和Phong模型的结果非常相似，这两个参数可以有如下对应关系
|
||||
$$
|
||||
\alpha_{phong}=\dfrac{2}{\alpha_{beckmann}^2}-2
|
||||
$$
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
### 6.4 Trowbridge-Reitz(GGX)
|
||||
这个模型又被称为GGX模型，是UnrealEngine采用的法线分布函数
|
||||
$$
|
||||
D_{GGX}(\vec{m})=R_{GGX}\dfrac{1}{((\vec{n}\cdot\vec{m})^2(\alpha^2-1)+1)^2}
|
||||
$$
|
||||
经过积分$R_{GGX}=\alpha^2/\pi$，所以
|
||||
$$
|
||||
D_{GGX}(\vec{m})=\dfrac{\alpha^2}{\pi((\vec{n}\cdot\vec{m})^2(\alpha^2-1)+1)^2}
|
||||
$$
|
||||
在Mathmatica中的模拟
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
可以看出，当$\alpha$趋向于0，表示材质越光滑，当$\alpha=1$时，GGX称为一条直线，也就是纯粹的漫反射材质，当$\alpha>1$，GGX模型的表现类似于Beckmann，表现出“超级粗糙”特性。
|
||||
和Phong模型比较，可以看出在粗糙度小于0.4时，ggx和phong模型比较类似，当粗糙度大于0.4时，ggx的表现更“平顺”一些，高光中心更集中，但长尾区域更长。
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
在Unreal实现中，定义了$\alpha=roughness^2$， roughness是用户输入的材质粗糙度，范围在[0,1]之间
|
||||
@@ -0,0 +1,95 @@
|
||||
# 微表面理论(三)
|
||||
-----
|
||||
## 7. 几何衰减因子
|
||||
### 7.1 定义
|
||||
一束光线照射到微平面上，在到达之前或者反射之后，有可能被其他微平面阻挡，这种阻挡造成族中出射光线的衰减，可以用几何衰减因子$G$描述这种衰减程度
|
||||
|
||||
|
||||
|
||||
### 7.2 Torrance-Sparrow
|
||||
以入射光线被阻挡为例，计算衰减因子$G$
|
||||
|
||||
|
||||
|
||||
根据上图可知
|
||||
$$
|
||||
\sin\alpha=\dfrac{1}{y}
|
||||
$$
|
||||
根据正弦定理
|
||||
$$
|
||||
\dfrac{x}{\sin\gamma}=\dfrac{2}{\sin(\beta+\pi/2)}
|
||||
$$
|
||||
联立以上两式，可以得到入射光线的有效比例为
|
||||
$$
|
||||
G_1=\dfrac{x}{y}=\dfrac{2\sin\gamma\sin\alpha}{\sin(\beta+\pi/2)}
|
||||
$$
|
||||
其中
|
||||
$$
|
||||
\sin\alpha=\vec{n}\cdot\vec{h} \\
|
||||
\sin(\beta+\pi/2)=\cos\beta=\vec{v}\cdot\vec{h} \\
|
||||
\sin\gamma=\vec{n}\cdot\vec{l}
|
||||
$$
|
||||
所以可以求得
|
||||
$$
|
||||
G_1=\dfrac{2(\vec{n}\cdot\vec{h})(\vec{n}\cdot\vec{l})}{\vec{v}\cdot\vec{h}}
|
||||
$$
|
||||
同理，反转$\vec{v}$和$\vec{l}$，可以求得反射光线被微表面阻挡的情况
|
||||
$$
|
||||
G_2=\dfrac{2(\vec{n}\cdot\vec{h})(\vec{n}\cdot\vec{v})}{\vec{v}\cdot\vec{h}}
|
||||
$$
|
||||
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
|
||||
$$
|
||||
G(\vec{l},\vec{v},\vec{h})=\min\{1,G_1,G_2\}=\min\left\{1,\dfrac{2(\vec{n}\cdot\vec{h})(\vec{n}\cdot\vec{l})}{\vec{v}\cdot\vec{h}}
|
||||
,\dfrac{2(\vec{n}\cdot\vec{h})(\vec{n}\cdot\vec{v})}{\vec{v}\cdot\vec{h}}
|
||||
\right\}
|
||||
$$
|
||||
|
||||
### 7.3 Schlick-Beckmann
|
||||
在Unreal Engine中，使用了这种模型，在这种模型中
|
||||
$$
|
||||
G(\vec{l},\vec{v},\vec{h})=G_1(\vec{l})G_1(\vec{v}) \\
|
||||
G_1(\vec{v})=\dfrac{\vec{n}\cdot\vec{v}}{(\vec{n}\cdot\vec{v})(1-k)+k}
|
||||
$$
|
||||
在原始的Schlick算法中，定义
|
||||
$$
|
||||
k=\alpha\sqrt{\frac{2}{\pi}}
|
||||
$$
|
||||
在Unreal实现中，调整为
|
||||
$$
|
||||
k=\frac{\alpha}{2}
|
||||
$$
|
||||
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
|
||||
$$
|
||||
\alpha=roughness^2, k=\dfrac{roughness^2}{2}
|
||||
$$
|
||||
使用直接光源时，
|
||||
$$
|
||||
\alpha=\left(\frac{roughness+1}{2}\right)^2, k=\dfrac{(roughness+1)^2}{8}
|
||||
$$
|
||||
|
||||
## 8. 菲尼尔反射方程
|
||||
### 8.1 定义
|
||||
当入射光线碰到一个反射平面时，菲尼尔反射方程会根据观察角度计算出反射光线所占的比例。当光线垂直入射时，材质反射的光线的比例称为基础反射率(Base Reflectivity)，一般用$F_0$表示，随着入射角的增大，反射的比例会增加，这种现象被称为菲涅尔现象。
|
||||
|
||||
### 8.2 Schlick
|
||||
菲涅尔方程是一个相当复杂的方程式，一般用Fresnel-Schlick近似:
|
||||
$$
|
||||
F_{schlick}(\vec{l},\vec{h},F_0)=F_0+(1-F_0)(1-(\vec{l}\cdot\vec{h}))^5
|
||||
$$
|
||||
在Mathmatica中模拟
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
在Unreal引擎中，使用了一个模拟运算以加快速度
|
||||
$$
|
||||
F_{schlick}(\vec{l},\vec{h},F_0)=F_0+(1-F_0)2^{(-5.55473(\vec{l}\cdot\vec{h})-6.98316)(\vec{l}\cdot\vec{h})}
|
||||
$$
|
||||
### 8.3 导体和绝缘体
|
||||
对于大多数绝缘体（非金属），$F_0$的范围在0.02和0.05之间，实际运算中一般取经验平均值$F_0=0.04$，并且绝缘体的$F_0$对不同波长的光的数值是相同的，可以视作一个灰度值。
|
||||
对于导体（金属），$F_0$的范围在0.5到1.0之间，并且对不同的波长的光的数值不同，所以在实际运算中，$F_0$的取值一般使用如下公式获得
|
||||
$$
|
||||
F_0=\text{mix}(0.04,Color,Metalness)
|
||||
$$
|
||||
其中$Metalness$表示金属度，0表示存粹的绝缘体(非金属)，1表示导体(金属)
|
||||
Reference in New Issue
Block a user