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docs/note/graphics/illumination_model/BRDF/BRDF.md
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# BRDF(双向反射分布函数)
----
## 1. 物体对光的反应
### 1.1 金属
![](./metal.png)
对于金属，折射进表面的光线的能量会立即被金属中的自由电子吸收，转换成电子的能量，不再可见
### 1.2 非金属
![](./nonmetal.png)
对于非金属（电介质或绝缘体），它们往往不是由单一成分构成，而可以认为其中包含了很多折射率不同的微粒，光线遇到这些粒子后发生反射折射，在物质内部不断传播，散射到不同方向，其中一部分会再次穿过表面被观察到,这种现象称为次表面散射(Subsurface Scattering)
#### 1.2.1 观察尺度
根据物质属性和观察尺度的不同，次表面散射会表现出不同的效果
![](./diffuse.png)
如果光线在物质中传播距离小于观察尺度（绿色区域，可以认为是一个像素区域），我们看到情况如下面的右上图，入射点、反射点、次表面散射的出射点看起来是同一个点。其中反射部分（图中浅棕色出射光）就是我们常说的高光（Specular Light），常聚集在一个方向周围，向这个方向观察该点会看到明显的高光，从其他方向观察该点时高光则比较微弱；次表面散射部分（图中蓝色出射光）是漫射光（Diffuse Light），光线被散射到各个方向。
![](./Subsurface.png)
如果光线在物质中的传播距离大于观察尺度，就需要使用次表面散射算法来建模
## 2. BRDF
### 2.1 定义
BRDF(bidirectional reflectance distribution function),是用来描述反光小平面的反射光和入射光的比例关系的函数，设一个不透明的、不发光的小平面，BRDF的定义为
$$
f(\vec{\omega_i},\vec{\omega_o})=\frac{dL_o(\vec{\omega_o})}{dE_i(\vec{\omega_i})}
$$
![](./BRDF.svg)
其中$f$就是BRDF函数
$\vec{\omega_i}$是入射光方向，
$\vec{\omega_o}$是观察方向
$L_o(\vec{\omega_o})$是$\vec{\omega_o}$方向反射光的亮度(Luminance)
$E_i(\vec{\omega_i})$是小平面接受的照度在$\vec{\omega_i}$方向上的分量, 由照度和亮度的定义可知，来自入射方向的入射光的辐射率（亮度）为
$$
L_i(\vec{\omega_i})=\frac{d\Phi}{d\omega_i dA^{Proj}}=\frac{d\Phi}{d\omega_i dA\cos\theta_i}=\frac{dE(\vec{\omega_i})}{d\omega_i\cos\theta_i}
$$
得到$dE_i(\vec{\omega_i})=L_i(\vec{\omega_i})\cos\theta_i d\omega_i$，所以
$$
f(\vec{\omega_i},\vec{\omega_o})=\frac{dL_o(\vec{\omega_o})}{L_i(\vec{\omega_i})\cos\theta_i d\omega_i}
$$
### 2.2 特性
$$\begin{split}
f(\vec{\omega_i},\vec{\omega_o})&\ge 0 \\
f(\vec{\omega_i},\vec{\omega_o})&=f(\vec{\omega_o},\vec{\omega_i}) \\
\forall \vec{\omega_i}, \int_{\Omega}f(\vec{\omega_i},\vec{\omega_o})\cos\theta_o d\omega_o&\le 1
\end{split}$$
### 2.3 **为什么BRDF方程是L和E的比值**
[参考](https://zhuanlan.zhihu.com/p/21376124)
#### 2.3.1 从测量方面来看
测量入射光的设备测量小平面的照度（E）最方便，不用关心方向，而测量出射光则是测量辐射率(L)最方便，用狭长的测光筒
#### 2.3.2 从数学角度来解释
一束光照到平面后，被平面反射到各个方向，其中一个出射方向的光通量只是整个反射光通量极小的一部分，当出射方向立体角趋于0时，
$$\lim_{\omega_o\rightarrow 0}\frac{dL_o}{L_i}=0$$
所以在实际计算中使用辐射率和辐射率比值是没有意义的。而如果分母改成表面上接收到的来自光源方向的微分辐照度，$dE=L_i(l)d\omega\cos\theta_i$，$dE$会小很多，所以比值$dL_o/dE_i$是有意义的
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#光照模型(Illumination Model)
----
##1. Phong模型
###1.1 基本定义
![](/graphics/illumination_model/Phong_Vectors.svg)
$$\vec{\mathrm{N}}$$表示表面的单位法线矢量
$$\vec{\mathrm{L}}$$是指向光源的单位矢量
$$\vec{\mathrm{R}}$$是反射光线的单位矢量
$$\vec{\mathrm{V}}$$是指向相机的单位矢量
###1.2 算法
https://en.wikipedia.org/wiki/Phong_reflection_model
![](/graphics/illumination_model/Phong_components_version_4.png)
$$
I_\mathrm{phong}=I_\mathrm{ambient}+\sum({I_\mathrm{diffuse}+I_\mathrm{specular}})
$$
1.2.1 **环境光（Ambient Reflection）**
$$
I_\mathrm{ambient}=L_aK_a
$$
其中$$L_a$$是环境光亮度，$$I_a$$是环境光反射系数$$0\le K_a\le 1$$
1.2.2 **漫反射（Diffuse Reflection）**
![](/graphics/illumination_model/Diffuse_Reflection.svg)
根据Lambert规则，材质的漫反射是均匀发射到周围的，和观察角度无关，只和光源的入射角度有关
$$
I_\mathrm{diffuse}=(\vec{L}\cdot\vec{N})L_dK_d
$$
其中$$L_d$$是光源的光亮度，$$K_d$$是漫反射系数$$0\le K_d\le 1$$
1.2.3 **高光反射(Specular Reflection)**
![](/graphics/illumination_model/Specular_Reflection.svg)
$$
I_\mathrm{specular}=(\vec{\mathrm{R}}\cdot\vec{\mathrm{V}})^nL_sK_s
$$
其中$$L_s$$是光源亮度，$$K_s$$是高光反射系数$$0\le K_s\le 1$$，$$n$$高光强度
1.2.3 **反射射线**
在计算高光反射时，需要计算反射射线$$\vec{\mathrm{R}}$$，
$$
\vec{\mathrm{R}}=2(\vec{\mathrm{L}}\cdot\vec{\mathrm{N}})\vec{\mathrm{N}}-\vec{\mathrm{L}}
$$
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#光照模型（二）
----
##2. Blinn-Phong模型
###2.1 基本定义
![](/graphics/illumination_model/Blinn-Phong_Vectors.svg)
和Phong模型很类似，只不过增加了半角向量$$\vec{\mathrm{H}}$$，也就是指向光源的向量$$\vec{\mathrm{L}}$$和指向相机的向量$$\vec{\mathrm{V}}$$的中间向量
$$
\vec{\mathrm{V}}=\frac{\vec{\mathrm{L}}+\vec{\mathrm{V}}}{\|\vec{\mathrm{L}}+\vec{\mathrm{V}}\|}
$$
###2.2 算法
https://en.wikipedia.org/wiki/Blinn%E2%80%93Phong_shading_model
$$
I_\mathrm{BlinnPhong}=I_\mathrm{ambient}+\sum({I_\mathrm{diffuse}+I_\mathrm{specular}})
$$
其中环境光和漫反射和Phong相同
$$
I_\mathrm{diffuse}=(\vec{L}\cdot\vec{N})L_dK_d \\
I_\mathrm{specular}=(\vec{\mathrm{R}}\cdot\vec{\mathrm{V}})^\alpha L_sK_s
$$
高光部分使用$$\vec{\mathrm{N}}\cdot\vec{\mathrm{H}}$$代替$$\vec{\mathrm{R}}\cdot\vec{\mathrm{V}}$$
$$
I_\mathrm{specular}=(\vec{\mathrm{N}}\cdot\vec{\mathrm{H}})^nL_sK_s
$$
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# 光照模型
* [光度学](./luminosity/luminosity.md)
* [双向反射分布函数\(BRDF\)](./BRDF/BRDF.md)
* [微平面理论(一)](./microfacets/microfacets_1.md)
* [微平面理论(二)](./microfacets/microfacets_2.md)
* [微平面理论(三)](./microfacets/microfacets_3.md)
* [光照方程](./render_function/render_function_1.md)
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# 光度学
----
## 1. 辐射度学
### 1.1 定义
辐射度学([radiometry](https://en.wikipedia.org/wiki/Radiometry))是一门研究电磁辐射能测量的科学与技术。它是用能量单位描述光辐射能的客观物理量，适用于整个电磁波段
### 1.2 基本单位
|名称|符号|单位|量纲|解释|
|:-|:-|:-|:-|:-|
|辐射能(Radiant energy)|$Q_e$|焦耳($J$)|$ML^2T^{−2}$|发射源所放出的能量|
|辐射通量(Radiant flux)|$\Phi_e=dQ_e/dt$|瓦特($W$)|$ML^2T^{−3}$|每单位时间的辐射能量，亦作“辐射功率”|
|辐射强度(Radiant intensity)|$I_e=d\Phi_e/d\omega$|瓦特每球面度<br>($W/sr$)|$ML^2T^{−3}$|给定传输方向上，由一个点光源发出的单位立体角内的辐射通量|
|辐射亮度(Radiance)|$\begin{split}L_e&=\frac{dI_e}{dA\cos\theta}\\&=\frac{d^2\Phi_e}{dAd\omega\cos\theta}\end{split}$|瓦特每球面度每平方米<br>($W/(sr·m^2$))|$MT^{−3}$|辐射表面在垂直其辐射传输方向上单位表面积单位立体角内发出的辐射通量|
|辐射出射度(Radiant exitance)|$\displaystyle M_e=\frac{d\Phi_e}{dA}$|瓦特每平方米<br>($W/m^2$)|$MT^{−3}$|离开光源表面单位面积内的辐射通量|
|辐照度(Irradiance)|$\displaystyle E_e=\frac{d\Phi_e}{dA}$|瓦特每平方米<br>($W/m^2$)|$MT^{−3}$|物体表面单位面积内接收到的辐射通量|
## 2. 光度学
### 2.1 定义
光度学([Photometry ](https://en.wikipedia.org/wiki/Photometry_%28optics%29))只限于可见光范围内（380nm-780nm），并以人眼的视觉特性为基础。
### 2.2 光度函数
各个波长的光给人眼带来的“明亮”的感觉并不一致，而且这种视觉反应因个体的差异而有所不同，因此光度学把不同波长的辐射功率用光度函数([Luminosity function](https://en.wikipedia.org/wiki/Luminosity_function))加权，这个函数经由国际照明委员会（CIE）确认后，现已成为所有色彩科学使用的标准函数。
#### 2.2.1 定义
光谱光效能
$$
K(\lambda)=\frac{\Phi_v(\lambda)}{\Phi_e(\lambda)}
$$
表示在某一波长上每一瓦辐射通量可以产生多少流明的光通量。
$\Phi_e(\lambda)$即为上文所说的辐射通量，是真实的亮度，单位是瓦特;
$\Phi_v(\lambda)$叫光通量，即人眼感觉到的假的亮度，单位是流明。
#### 2.2.2 数据
光谱光效度（视觉函数）
$$
V(\lambda) =\frac{K( \lambda) }{K_{\mathrm{max}}}
$$
就是归一化的光谱光效能
![](./LuminosityCurve.svg)
蓝色线是人眼的暗视觉视觉函数数据，红色线是人眼的明视觉函数数据
### 2.3 基本单位
#### 2.3.1 基本单位
|名称|符号|单位|量纲|解释|
|:-|:-|:-|:-|:-|
|光量(Luminous energy)|$Q_v$|流明·秒<br>$lm·s$|$JT$||
|光通量(Luminous flux)|$\Phi_v=dQ_v/dt$|流明<br>$lm=cd·sr$|$J$|单位时间内由光源所发出或由被照物所吸收的总光能|
|发光强度(Luminous intensity)|$I_v=d\Phi_v/d\omega$|坎德拉<br>$cd=lm/sr$|$J$|光源在给定方向上，每单位立体角内所发出的的光通量|
|亮度(Luminance)|$\begin{split}L_v&=\frac{dI_v}{dA\cos\theta}\\&=\frac{d^2\Phi_v}{dAd\omega\cos\theta}\end{split}$|坎德拉每平方米<br>$cd/m^2$|$JL^{-2}$|单位面积光源在给定方向上，在每单位立体角内所发出的总光通量。其单位有时也称尼特（nit）|
|光出射度(Luminous exitance)|$\displaystyle M_v=\frac{d\Phi_v}{dA}$|勒克斯<br>$lx= lm/m^2$|$JL^{-2}$|物体表面每单位面积所发出可见光的光通量，用于出射表面的光|
|照度(Illuminance)|$\displaystyle E_v=\frac{d\Phi_v}{dA}$|勒克斯<br>$lx=lm/m^2$|$JL^{-2}$|物体表面每单位面积所接收的可见光的光通量，用于出入射表面的光|
#### 2.3.2 坎德拉
坎德拉([Candela](https://en.wikipedia.org/wiki/Candela))是描述发光强度(Luminous intensity)的单位，是七大基本单位之一，给定一个频率为540.0154×1012 Hz的单色辐射光源（黄绿色可见光）与一个方向，且该辐射源在该方向的辐射强度为1/683瓦特每球面度，则该辐射源在该方向的发光强度为1坎德拉
#### 2.3.3 流明
流明([lumen](https://en.wikipedia.org/wiki/Lumen_%28unit%29))光通量（luminous flux）,如果一个光源发射1坎德拉的发光强度到1个立体角的范围里,则到那个立体角的总发射光通量就是1流明，或者说，一个各向同性的一坎德拉光源发射的总光通量为$4\pi$流明
### 2.4 单位解释
#### 2.4.1 光通量$\phi$
![](./flux.png)
单位时间内，光源发射的能量，对应辐射度学中的单位为焦耳
#### 2.4.2 发光强度$I$
![](./intensity.png)
单位时间内，光源在单位立体角内发射的能量
#### 2.4.3 照度$E$/出射度$M$
![](./illuminance.png)
和发光强度类似，光源通过单位面积发射的总光通量为出射度$M$，受光体表面单位面积接收到的总光通量为照度$E$
![](./Illuminance_device.png)
测量照度的设备
#### 2.4.4 照度的衰减
![](./illuminance2.png)
对于理想的方向光，照度不随距离而变化，但位于点光源来说，照度随着距离的增大而衰减，反比于距离的平方
$$
E=\frac{Id\omega}{dA}=\frac{I}{dA}\frac{dA}{r^2}=\frac{I}{r^2}
$$
#### 2.4.5 亮度$L$
![](./luminance.png)
对于一个受光体小平面$dA$，其法线为$\vec{\boldsymbol n}$，指向光源的方向为$\vec{\omega}$，夹角为$\theta=\cos^{-1}(\vec{\boldsymbol n}\cdot\vec{\omega})$，那么定义该小平面在$\vec{\omega}$方向上的亮度为
$$
L(\vec{\omega})=\frac{d^2{\Phi}}{d\omega dA\cos\theta}
$$
### 2.5 亮度(luminance)的特性
#### 2.5.1
在光源传播方向上，lumiance保持不变，与距离无关
![](./luminance_theorem.svg)
设一个发光小平面，面积为$dA_s$，一个受光小平面，面积为$dA_r$，距离为$r$，两个平面相对于传播方向的夹角分别是$\theta_s$和$\theta_r$，发光平面发出的光中的一部分光通量被接受平面收到
那么对于发光平面来说，分析它在光的传播方向上的luminance，首先立体角
$$
\omega_s=\frac{dA_r\cos\theta_r}{r^2}
$$
同理
$$
\omega_r=\frac{dA_s\cos\theta_s}{r^2}
$$
![](./luminance_theorem2.svg)
发光面在光传播方向上的光亮度为$L_s$，根据光亮度的定义，发光面在此传播方向上发射的光通量
$$
d^2\Phi=L_s dA_s d\omega_s \cos\theta_s
$$
受光面在传播方向上的光亮度$L_r$，根据能量守恒，此方向传播的光通量全部被受光面接受，所以
$$
d^2\Phi=L_r dA_r d\omega_r \cos\theta_r
$$
代入前面的立体角公式可得
$$
\displaystyle{ d^2\Phi=L_sdA_s\cos\theta_s\frac{dA_r\cos\theta_r}{r^2}=L_rdA_r\cos\theta_r\frac{dA_s\cos\theta_s}{r^2}}
$$
整理可得
$$
L_s=L_r
$$
#### 2.5.2
图像传感器、人眼或者照相机感受或测量的量直接正比于Luminance, 因此图形学中主要计算的量就是这个量
### 2.6 这几个量的关系
$$\begin{split}
E&=\int_{2\pi}{L_i(\vec\omega)\cos\theta d\omega} \\
M&=\int_{2\pi}{L_o(\vec\omega)\cos\theta d\omega} \\
I(\vec\omega)&=\int_{A}{L(\vec\omega)\cos\theta dA} \\
\Phi&=\int_{A}\int_{2\pi}{L(\vec\omega)\cos\theta d\omega dA}
\end{split}$$
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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。
![](./microfacet_1.svg)
## 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
![](./microfacet_2.svg)
## 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$d\omega_o$
![](./microfacet_3.svg)
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$I,Q,R$
![](./microfacet_4.svg)
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
$$
f(\vec{\omega}_i,\vec{\omega}_o)=\dfrac{F(\vec{\omega}_i, \vec{\omega}_h)D(\vec{\omega}_h)G(\vec{\omega}_i, \vec{\omega}_o)}{4\cos(\theta_o)\cos(\theta_i)}
$$
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# 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
![](./microfacet_5.svg)
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
![](./Mathmatica_Phong.gif)
![](./microfacet_phong.svg)
### 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
$$
\alpha_{beckmann}=roughness^2
$$
![](./Mathmatica_Beckmann.gif)
![](./microfacet_beckmann.svg)
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
$$
\alpha_{phong}=\dfrac{2}{\alpha_{beckmann}^2}-2
$$
![](./Mathmatica_bp.gif)
![](./microfacet_bp.svg)
### 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
![](./Mathmatica_ggx.gif)
![](./microfacet_ggx.svg)
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。
![](./Mathmatica_gp.gif)
![](./microfacet_gp.svg)
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
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# 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
![](./microfacet_6.svg)
### 7.2 Torrance-Sparrow
以入射光线被阻挡为例，计算衰减因子$G$
![](./microfacet_7.svg)
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
![](./Mathmatica_fresnel.gif)
![](./microfacet_8.svg)
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)
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# 渲染方程（一）
----
## 1. 定义
### 1.1 反射方程
计算反射表面反射到$\vec{\omega_o}$方向上的来自于上半球所有方向入射光的辐射率
![](../BRDF/reflectance_equation.jpg)
$$
L_o(\vec{\omega_o})=\int_{\Omega}f(\vec{\omega_i},\vec{\omega_o})L_i(\vec{\omega_i})\cos\theta_id\omega_i
$$
上式称为反射方程（Reflectance Equation），用来计算表面反射辐射率
### 1.2 渲染方程
将反射方程增加上自发光因素，可以得到渲染方程([Rendering equation](https://en.wikipedia.org/wiki/Rendering_equation))
$$
L_{o}(\vec{\omega_o}) = L_{e}(\vec{\omega_o})+\int_{\Omega}f(\vec{\omega_i},\vec{\omega_o})L_i(\vec{\omega_i})\cos\theta_id\omega_i
$$
完整形式的渲染方程，需要考虑到反射平面所处的位置、时间，光的波长这些因素，具体形式为
$$
L_o(\mathbf{x},\vec{\omega_o},\lambda,t)=L_e(\mathbf{x},\vec{\omega_o},\lambda,t)+\int_{\Omega}f(\mathbf{x},\vec{\omega_i},\vec{\omega_o},\lambda,t)L_i(\mathbf{x},\vec{\omega_i},\lambda,t)(\vec{\omega_i}\cdot\vec{\mathbf {n}}) d\omega_i
$$
## 2. 反射方程分解
根据前面的微表面理论分析，入射光能量在材质表面分解为两部分，高光反射部分，比例为$k_s=F$，折射到材质内部的部分，比例为$1-F$，这部分能量一部分在材质内部转换为其他能量消失掉（对于金属材质为全部），一部分重新从表面发射出去，称为漫反射，所以漫反射的比例为$k_s=(1-F)(1-M)$，其中$M$表示金属度，范围从0到1，所以可以将反射方程中的BRDF分解为两部分
$$
f(\vec{\omega_i},\vec{\omega_o})=k_df_d(\vec{\omega_i},\vec{\omega_o})+k_sf_s(\vec{\omega_i},\vec{\omega_o})
$$
## 3. Lambert漫反射模型
[参考](https://zhuanlan.zhihu.com/p/21489591)
Lambert模型认为，材质的漫反射是均匀发射到所有方向的，所以$f_d$是一个常数，先设想一个最简单的情况，材质只有漫反射，也就是$k_d=1,k_s=0$,入射光均匀遍布所有的方向，而且每个方向的亮度都是相同的常数。因为这样$f_d$和$L_i$都是常数，可以提到积分外面。反射方程变成
$$
L_o=f_dL_i\int_{\Omega}\cos\theta_id\omega_i
$$
计算这个半球积分
$$
\displaystyle{\int_{\Omega}\cos\theta d\omega=\int_{-\pi}^{\pi}\int_{0}^{\pi/2}}\sin(\theta)\cos(\theta)d\theta d\phi=\pi
$$
得到
$$
L_o = f_dL_i\pi
$$
考虑一下，如果所有方向都有射向球面的光，而且每个方向的光强度都等于$L_i$，那么小平面受到的入射光照度，应该等于光强度乘以半球面积。
$$
E_i=2\pi L_i
$$
根据能量守恒，光出射度$M_o=E_i$
$$
M_o=2\pi L_o=2\pi ^2 f_dL_i=2\pi L_i
$$
可知
$$
f_d=\frac{1}{\pi}
$$
考虑材质的颜色，在这里颜色可定义为对三原色的吸收比例，比如白色(1,1,1)表示不吸收任何光，红色(1,0,0)表示将绿色和蓝色完全吸收掉，所以最终漫反射的BRDF为
$$
f_d=\dfrac{c}{\pi}
$$
## 4.最终形式
Cook-Torrance模型其实就是一种材质的高光模型，公式
$$
f_s=\frac{DGF}{4(\vec{n}\cdot \vec{\omega}_i)(\vec{n}\cdot\vec{\omega}_o)}
$$
中的$F$就是高光反射的分量比例$k_s$，所以结合在一起，得到最终的渲染方程为
$$
L_o(\vec{\omega_o})=L_{e}(\vec{\omega_o})+\int_{\Omega}\left(k_d\dfrac{c}{\pi}+\dfrac{DGF}{4(\vec{\omega}_o\vec{n})(\vec{\omega}_i\vec{n})}\right)L_i(\vec{\omega_i})(\vec{\omega}_i\vec{n})d\omega_i
$$
其中
$$
k_d=(1-F)(1-M)
$$