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# 环境光渲染(一）
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----
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## 1. 定义
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使用一张HDR环境图片作为环境光源，每个像素表示一个光源和入射方向，回顾一下渲染方程
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$$
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L_o(\vec{\omega_o})=\int_{\Omega}\left(k_d\dfrac{c}{\pi}+f_s(\vec{\omega_i}, \vec{\omega_o})\right)L_i(\vec{\omega_i})(\vec{\omega}_i\vec{n})d\omega_i
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$$
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显然在实时渲染系统中，针对环境贴图做完整的积分运算代价非常昂贵，所以需要做预处理
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## 2.漫反射部分
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### 2.1 原理
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只考虑渲染方程中的漫反射部分
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$$
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L_{od}(\vec{\omega_o})=k_d\dfrac{c}{\pi}\int_{\Omega}L_i(\vec{\omega_i})(\vec{\omega}_i\vec{n})d\omega_i
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$$
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将其中之和光照贴图相关的部分做一个预计算，针对所有球面方向，每个方向做一个半球积分
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$$
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\dfrac{1}{\pi}\int_{\Omega}L_i(\vec{\omega_i})(\vec{\omega}_i\vec{n})d\omega_i
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$$
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将结果存成一张新的环境贴图Irradiance,那么最终实时渲染时，漫反射部分可以这个贴图来计算
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$$
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L_{od}(\vec{\omega_o})=\text{TexCube}(\text{IrrMap}, \vec{n})*k_d*c
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$$
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### 2.2 漫反射环境贴图的生成
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#### 2.2.1 球坐标积分
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将立体角转换为球坐标
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$$\begin{split}
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&\dfrac{1}{\pi}\int_{\Omega}L_i(\vec{\omega_i})(\vec{\omega}_i\vec{n})d\omega_i\\
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=&\dfrac{1}{\pi}\int_{\phi=0}^{2\pi}\int_{\theta=0}^{\pi/2}L_i(\phi,\theta)\cos(\theta)\sin(\theta)d\theta d\phi\\
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\approx&\dfrac{1}{\pi}\dfrac{2\pi}{n_1}\dfrac{\pi}{2n_2}\sum_{j=0}^{n_1}\sum_{i=0}^{n_2}L_i(\phi_j,\theta_i)\cos(\theta_i)\sin(\theta_i) \\
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=&\dfrac{\pi}{n_1n_2}\sum_{j=0}^{n_1}\sum_{i=0}^{n_2}L_i(\phi_j,\theta_i)\cos(\theta_i)\sin(\theta_i)
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\end{split}$$
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其中：$\phi_j=\dfrac{j}{n_1}2\pi \\ \theta_i=\dfrac{i}{n_2}\dfrac{\pi}{2}$
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代码:
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````cpp :no-line-numbers
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vec3 irradiance = vec3(0.0);
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// tangent space calculation from origin point(x'=up, y'=N, z'=left)(left-hand)
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vec3 up = vec3(0.0, 1.0, 0.0); //x'
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vec3 left = cross(up, N); //z'
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up = cross(N, left); //x'
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float sampleDelta = 0.025;
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float nrSamples = 0.0;
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for(float phi = 0.0; phi < 2.0 * PI; phi += sampleDelta)
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{
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for(float theta = 0.0; theta < 0.5 * PI; theta += sampleDelta)
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{
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float cosTheta = cos(theta);
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float sinTheta = sin(theta);
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// spherical to cartesian (in tangent space)
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vec3 tangentSample = vec3(cos(phi) * sinTheta, cosTheta, sin(phi) * sinTheta);
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// tangent space to world
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vec3 sampleVec = tangentSample.x * up + tangentSample.y * N + tangentSample.z * left;
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irradiance += texture(environmentMap, sampleVec).rgb * cosTheta * sinTheta;
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nrSamples++;
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}
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}
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irradiance = PI * irradiance * float(nrSamples);
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````
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#### 2.2.2 蒙特卡洛估算
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使用蒙特卡洛采样估算积分
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$$
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\displaystyle\int_{a}^{b}f(x)dx\approx\dfrac{1}{N}\sum_{i=1}^{N}\dfrac{f(X_i)}{\text{pdf}(X_i)}
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$$
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其中$\text{pdf}(x)$函数是采样时使用的概率分布函数。
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方法1：
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对于要计算的这个二维积分,如果使用的随机采样是$\phi$在$[0,2\pi]$之间均匀分布，$\theta$在$[0,\pi/2]$之间均匀分布，那么
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$$
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\text{pdf}(\phi)=1/(2\pi) , \text{pdf}(\theta)=2/\pi
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$$
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$$\begin{split}
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&\dfrac{1}{\pi}\int_{\phi=0}^{2\pi}\int_{\theta=0}^{\pi/2}L_i(\phi,\theta)\cos(\theta)\sin(\theta)d\theta d\phi \\
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=&\dfrac{1}{\pi}\dfrac{2\pi}{n_1}\sum_{j=0}^{n_1}\left(\dfrac{\pi}{2n_2}\sum_{i=0}^{n_2}L_i(\phi_j,\theta_i)\cos(\theta_i)\sin(\theta_i)
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\right) \\
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=&\dfrac{\pi}{n_1n_2}\sum_{j=0}^{n_1}\sum_{i=0}^{n_2}L_i(\phi_j,\theta_i)\cos(\theta_i)\sin(\theta_i)\\
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=&\dfrac{\pi}{N}\sum_{i=0}^{N}L_i(\phi_i,\theta_i)\cos(\theta_i)\sin(\theta_i)
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\end{split}$$
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其中
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$$
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\phi_i=2\pi\xi, \theta_i=\dfrac{\pi}{2}\xi
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$$
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$\xi$表示均匀分布在[0,1]之间的随机变量
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方法2：
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上面的采样方法在极点位置会比较密集，在赤道位置比较稀疏，由于漫反射是均匀分布的，如果采样点也是均匀分布在半球面上的，收敛速度会比较快，这种情况下
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$$
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\text{pdf}(\phi)=1/(2\pi) , \text{pdf}(\theta)=\sin(\theta)
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$$
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$$\begin{split}
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&\dfrac{1}{\pi}\int_{\phi=0}^{2\pi}\int_{\theta=0}^{\pi/2}L_i(\phi,\theta)\cos(\theta)\sin(\theta)d\theta d\phi \\
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=&\dfrac{1}{\pi}\dfrac{2\pi}{n_1}\sum_{j=0}^{n_1}\left(\dfrac{1}{n_2}\sum_{i=0}^{n_2}L_i(\phi_j,\theta_i)\cos(\theta_i)
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\right) \\
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=&\dfrac{2}{n_1n_2}\sum_{j=0}^{n_1}\sum_{i=0}^{n_2}L_i(\phi_j,\theta_i)\cos(\theta_i)\\
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=&\dfrac{2}{N}\sum_{i=0}^{N}L_i(\phi_i,\theta_i)\cos(\theta_i)
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\end{split}$$
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其中
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$$
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\phi_i=2\pi\xi, \theta_i=\arccos(1-\xi)
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$$
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````cpp :no-line-numbers
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vec3 irradiance = vec3(0.0);
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vec3 up = vec3(0.0, 1.0, 0.0); //x'
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vec3 left = cross(up, N); //z'
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up = cross(N, left); //x'
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for(uint i=0; i<sampleCounts; i++)
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{
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//http://holger.dammertz.org/stuff/notes_HammersleyOnHemisphere.html
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vec2 uv = Hammersley(i, totalCounts);
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// tangent space sample point
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float phi = uv.y * 2.0 * PI;
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float cosTheta = 1.0 - uv.x;
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float sinTheta = sqrt(1-cosTheta*cosTheta);
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vec3 tangentSample = vec3(cos(phi) * sinTheta, cosTheta, sin(phi) * sinTheta);
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// tangent space to world
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vec3 sampleVec = tangentSample.x * up + tangentSample.y * N + tangentSample.z * left;
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irradiance += texture2D(s_texSkybox, sampleVec).rgb * cosTheta;
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}
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irradiance = 2*irradiance / float(sampleCounts);
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````
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