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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/luminosity/luminosity.md
+4 -1
View File
@@ -134,7 +134,10 @@ 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}}
\begin{aligned}
d^2\Phi&=\displaystyle{L_sdA_s\cos\theta_s\frac{dA_r\cos\theta_r}{r^2}}\\
&=\displaystyle{L_rdA_r\cos\theta_r\frac{dA_s\cos\theta_s}{r^2}}
\end{aligned}
$$
整理可得
$$
docs/note/graphics/illumination_model/microfacets/microfacet_3.svg
+129 -119
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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}
$$
## 8. 菲尼尔反射方程
docs/note/graphics/image_based_lighting/ibl_01.md
+8 -2
View File
@@ -83,7 +83,10 @@ $$\begin{split}
\end{split}$$
其中
$$
\phi_i=2\pi\xi, \theta_i=\dfrac{\pi}{2}\xi
\begin{aligned}
\phi_i&=2\pi\xi\\
\theta_i&=\dfrac{\pi}{2}\xi
\end{aligned}
$$
$\xi$表示均匀分布在[0,1]之间的随机变量
@@ -101,7 +104,10 @@ $$\begin{split}
\end{split}$$
其中
$$
\phi_i=2\pi\xi, \theta_i=\arccos(1-\xi)
\begin{aligned}
\phi_i&=2\pi\xi\\
\theta_i&=\arccos(1-\xi)
\end{aligned}
$$
````cpp :no-line-numbers
vec3 irradiance = vec3(0.0);
docs/note/graphics/image_based_lighting/ibl_02.md
+15 -5
View File
@@ -8,11 +8,17 @@ L_{os}(\vec{\omega_o})=\int_{\Omega}f_s(\vec{\omega_i}, \vec{\omega_o})L_i(\vec{
$$
同样使用蒙特卡洛进行近似积分
$$
\int_{\Omega}f_s(\vec{\omega_i}, \vec{\omega_o})L_i(\vec{\omega_i})(\vec{\omega}_i\vec{n})d\omega_i \approx \dfrac{1}{N}\sum_{k=1}^{N}\dfrac{f_s(\vec{\omega_k}, \vec{\omega_o})L_i(\vec{\omega_k})(\vec{\omega_k}\cdot\vec{n})}{\text{pdf}(\vec{\omega_k})}
\begin{aligned}
&\int_{\Omega}f_s(\vec{\omega_i}, \vec{\omega_o})L_i(\vec{\omega_i})(\vec{\omega}_i\vec{n})d\omega_i \\
\approx &\dfrac{1}{N}\sum_{k=1}^{N}\dfrac{f_s(\vec{\omega_k}, \vec{\omega_o})L_i(\vec{\omega_k})(\vec{\omega_k}\cdot\vec{n})}{\text{pdf}(\vec{\omega_k})}
\end{aligned}
$$
EPIC在此基础上提出了一种近似[计算方法](http://blog.selfshadow.com/publications/s2013-shading-course/karis/s2013_pbs_epic_notes_v2.pdf)
$$
\dfrac{1}{N}\sum_{k=1}^{N}\dfrac{f_s(\vec{\omega_k}, \vec{\omega_o})L_i(\vec{\omega_k})(\vec{\omega_k}\cdot\vec{n})}{\text{pdf}(\vec{\omega_k}, \vec{\omega_o})}\approx\left(\dfrac{1}{N}\sum_{k=1}^N L_i(\vec{\omega_k})\right)\left(\dfrac{1}{N}\sum_{k=1}^{N}\dfrac{f_s(\vec{\omega_k}, \vec{\omega_o})(\vec{\omega_k}\cdot\vec{n})}{\text{pdf}(\vec{\omega_k})}\right)
\begin{aligned}
&\dfrac{1}{N}\sum_{k=1}^{N}\dfrac{f_s(\vec{\omega_k}, \vec{\omega_o})L_i(\vec{\omega_k})(\vec{\omega_k}\cdot\vec{n})}{\text{pdf}(\vec{\omega_k}, \vec{\omega_o})}\\
\approx&\left(\dfrac{1}{N}\sum_{k=1}^N L_i(\vec{\omega_k})\right)\left(\dfrac{1}{N}\sum_{k=1}^{N}\dfrac{f_s(\vec{\omega_k}, \vec{\omega_o})(\vec{\omega_k}\cdot\vec{n})}{\text{pdf}(\vec{\omega_k})}\right)
\end{aligned}
$$
这种方法的好处是可以通过分开预计算，尽量减少实际渲染时的计算量，这个公式可以理解为“Env. lighting x BRDF”，左边的第一部分是小平面受到的所有环境光的采样值，第二部分只跟材质的BRDF属性相关。
@@ -71,7 +77,10 @@ $$\begin{split}
\end{split}$$
使用Mathmatica计算可得
$$
\text{cdf}(\theta)=\dfrac{\sin^2\theta}{\cos^2\theta(\alpha^2-1)+1}=\dfrac{1-\cos^2\theta}{\cos^2\theta(\alpha^2-1)+1}
\begin{aligned}
\text{cdf}(\theta)&=\dfrac{\sin^2\theta}{\cos^2\theta(\alpha^2-1)+1}\\
&=\dfrac{1-\cos^2\theta}{\cos^2\theta(\alpha^2-1)+1}
\end{aligned}
$$
根据前面的概率知识，可以得到微表面的法线的球坐标$\theta$和$\phi$的采样函数为
$$\begin{split}
@@ -85,7 +94,7 @@ $$
\dfrac{1}{N}\sum_{k=1}^N L_i(\vec{\omega}_k)\approx\dfrac{1}{\sum_{k=1}^{N}(\vec{n}\cdot\vec{\omega}_k)}\sum_{k=1}^{N}{L_i(\vec{\omega}_k)(\vec{n}\cdot\vec{\omega}_k)}
$$
```cpp
```cpp :no-line-numbers
// make the simplyfying assumption that V equals R equals the normal
vec3 R = N;
vec3 V = R;
@@ -96,7 +105,8 @@ float totalWeight = 0.0;
for(uint i = 0u; i < SAMPLE_COUNT; ++i)
{
// generates a sample vector that's biased towards the preferred alignment direction (importance sampling).
// generates a sample vector that's biased towards the preferred
//alignment direction (importance sampling).
vec2 Xi = Hammersley(i, SAMPLE_COUNT);
vec3 H = ImportanceSampleGGX(Xi, N, roughness);
vec3 L = normalize(2.0 * dot(V, H) * H - V);
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docs/note/graphics/math/math_01.md
+5 -4
View File
@@ -142,13 +142,14 @@ $$
### 2.5叉积
#### 2.5.1 定义
对于两个三维矢量$\vec{a}, \vec{b}$，定义它们的叉积
$$
\vec{\boldsymbol a}\times\vec{\boldsymbol b}=\begin{vmatrix}
$$\begin{aligned}
\vec{\boldsymbol a}\times\vec{\boldsymbol b}&=\begin{vmatrix}
\vec{\boldsymbol i}& \vec{\boldsymbol j} & \vec{\boldsymbol k}\cr
a_x & a_y & a_z \cr
b_x & b_y & b_z
\end{vmatrix}=
(a_yb_z-a_zb_y)\vec{\boldsymbol{i}}+(a_zb_x-a_xb_z)\vec{\boldsymbol{j}}+(a_xb_y-a_yb_x)\vec{\boldsymbol{k}}
\end{vmatrix}\\
&=(a_yb_z-a_zb_y)\vec{\boldsymbol{i}}+(a_zb_x-a_xb_z)\vec{\boldsymbol{j}}+(a_xb_y-a_yb_x)\vec{\boldsymbol{k}}
\end{aligned}
$$
#### 2.5.2 叉积的几何意义
在三维几何中，向量a和向量b的叉乘结果是一个向量，更为熟知的叫法是法向量，该向量垂直于a和b向量构成的平面。
docs/note/graphics/math/math_02.md
+16 -7
View File
@@ -7,19 +7,23 @@ $\boldsymbol A_{ij}$表示第$i$行第$j$列的元素，
$\boldsymbol A_{i,*}$表示第$i$行所有元素，
$\boldsymbol A_{*,j}$表示第$j$列所有元素，
$$
\boldsymbol A=\begin{bmatrix}
\begin{aligned}
\boldsymbol A&=\begin{bmatrix}
A_{11}&A_{12}&A_{13}\\
A_{21}&A_{22}&A_{23}\\
A_{31}&A_{32}&A_{33}
\end{bmatrix}=\begin{bmatrix}
\end{bmatrix}\\
&=\begin{bmatrix}
\leftarrow & A_{1,*} & \rightarrow \\
\leftarrow & A_{2,*} & \rightarrow \\
\leftarrow & A_{3,*} & \rightarrow
\end{bmatrix}=\begin{bmatrix}
\end{bmatrix}\\
&=\begin{bmatrix}
\uparrow & \uparrow & \uparrow \\
A_{*,1} & A_{*,2} & A_{*,3} \\
\downarrow & \downarrow & \downarrow
\end{bmatrix}
\end{aligned}
$$
### 3.2 加法
如果矩阵$\boldsymbol A$是一个$m\times n$的矩阵，矩阵$\boldsymbol B$是一个$m\times n$的矩阵，那么$\boldsymbol A +\boldsymbol B$是一个$m\times n$的矩阵$\boldsymbol C$，并且
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#### 3.4.2 举例
$$
\boldsymbol A=\begin{bmatrix}
\begin{aligned}
\boldsymbol A&=\begin{bmatrix}
2&-1&8\\3&6&-4\end{bmatrix}\\
\boldsymbol A^T=\begin{bmatrix}
\boldsymbol A^T&=\begin{bmatrix}
2&3\\-1&6\\8&-4
\end{bmatrix}
\end{aligned}
$$
### 3.4.3 正交矩阵
@@ -140,10 +146,13 @@ $$
#### 3.8.2 举例
$$
\det{\begin{bmatrix}
\begin{aligned}
&\det{\begin{bmatrix}
A_{11}&A_{12}\\
A_{21}&A_{22}
\end{bmatrix}}=A_{11}\det[A_{22}]-A_{12}\det[A_{21}]=A_{11}A_{22}-A_{12}A_{21}
\end{bmatrix}}\\
&=A_{11}\det[A_{22}]-A_{12}\det[A_{21}]=A_{11}A_{22}-A_{12}A_{21}
\end{aligned}
$$
$$\begin{split}
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docs/note/graphics/math/proj_transform_3.svg
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docs/note/graphics/math/transform_01.md
+8 -6
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@@ -58,9 +58,11 @@ $$
#### 3.2.1 **结果**
围绕$x,y,z$轴旋转$\theta$的旋转矩阵
$$
\boldsymbol{R_x}=\begin{bmatrix}1&0&0\\0&\cos(\theta)&-\sin(\theta)\\0&\sin(\theta)&\cos(\theta)\end{bmatrix}
\boldsymbol{R_y}=\begin{bmatrix}\cos(\theta)&0&\sin(\theta)\\ 0&1&0\\ -\sin(\theta)&0&\cos(\theta)\end{bmatrix}
\boldsymbol{R_z}=\begin{bmatrix}\cos(\theta)&-\sin(\theta)&0 \\ \sin(\theta)&\cos(\theta)&0\\ 0&0&1\end{bmatrix}
\begin{aligned}
\boldsymbol{R_x}&=\begin{bmatrix}1&0&0\\0&\cos(\theta)&-\sin(\theta)\\0&\sin(\theta)&\cos(\theta)\end{bmatrix}\\
\boldsymbol{R_y}&=\begin{bmatrix}\cos(\theta)&0&\sin(\theta)\\ 0&1&0\\ -\sin(\theta)&0&\cos(\theta)\end{bmatrix}\\
\boldsymbol{R_z}&=\begin{bmatrix}\cos(\theta)&-\sin(\theta)&0 \\ \sin(\theta)&\cos(\theta)&0\\ 0&0&1\end{bmatrix}
\end{aligned}
$$
#### 3.2.2 **求解过程**
以$R_z$为例
@@ -134,7 +136,7 @@ R_n(\vec{v})&=\vec{v}-\vec{v}+\vec{v}\cos(\theta)+(1-\cos(\theta))(\vec{v}\cdot\
&=\vec{v}+(1-\cos(\theta))[(\vec{v}\cdot\vec{n})\vec{n}-\vec{v}]+(\vec{n}\times\vec{v})\sin(\theta)\\
&=\vec{v}+(1-\cos(\theta))[(\vec{v}\cdot\vec{n})\vec{n}-(\vec{n}\cdot\vec{n})\vec{v}]+(\vec{n}\times\vec{v})\sin(\theta)
\end{split}$$
根据矢量[三重积公式](/graphics/math/math_01.html#_2-7-%E4%B8%89%E9%87%8D%E7%A7%AF)，可以得到
根据矢量[三重积公式](/note/graphics/math/math_01.html#_2-7-%E4%B8%89%E9%87%8D%E7%A7%AF)，可以得到
$$
(\vec{v}\cdot\vec{n})\vec{n}-(\vec{n}\cdot\vec{n})\vec{v}=\vec{n}\times(\vec{n}\times\vec{v})
$$
@@ -142,7 +144,7 @@ $$
$$
R_n(\vec{v})=\vec{v}+(1-\cos(\theta))[\vec{n}\times(\vec{n}\times\vec{v})]+(\vec{n}\times\vec{v})\sin(\theta)
$$
使用向量$\vec{n}$的[叉乘矩阵](./graphics/math/math_01.html#_2-5-4-%E5%8F%89%E7%A7%AF%E7%9A%84%E7%9F%A9%E9%98%B5%E5%BD%A2%E5%BC%8F)$[\boldsymbol{n}]_{\times}$来代替叉乘，这里记矩阵$\boldsymbol{M_n}$为向量$\vec{n}$的叉积矩阵
使用向量$\vec{n}$的[叉乘矩阵](/note/graphics/math/math_01.html#_2-5-4-%E5%8F%89%E7%A7%AF%E7%9A%84%E7%9F%A9%E9%98%B5%E5%BD%A2%E5%BC%8F)$[\boldsymbol{n}]_{\times}$来代替叉乘，这里记矩阵$\boldsymbol{M_n}$为向量$\vec{n}$的叉积矩阵
$$
\boldsymbol{M_n}=\begin{bmatrix}
0&-n_z&n_y\\
@@ -168,7 +170,7 @@ R_n(\vec{\boldsymbol{i}})&=c\vec{i}+(1-c)(\vec{i}\cdot\vec{n})\vec{n}
R_n(\vec{\boldsymbol{j}})&=[(1-c)xy-sz,c+(1-c)y^2,(1-c)yz+sx]^T\\
R_n(\vec{\boldsymbol{k}})&=[(1-c)xz+sy,(1-c)yz-sx,c+(1-c)z^2]^T
\end{split}$$
代入公式[1.3.1](/graphics/math/transform_01.html#_1-3-使用矩阵表示)得到
代入公式[1.3.1](/note/graphics/math/transform_01.html#_1-3-使用矩阵表示)得到
$$
\boldsymbol{R_n}=\begin{bmatrix}
c+(1-c)x^2& (1-c)xy-sz&(1-c)xz+sy\\
docs/note/graphics/math/transform_02.md
+12 -2
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@@ -22,7 +22,14 @@ $$
0&0&1
\end{bmatrix}\begin{bmatrix}x\\y\\0\end{bmatrix}=\begin{bmatrix}x\\y\\0\end{bmatrix}
$$
两个矢量的和或者差$\vec{\boldsymbol{u}}\pm\vec{\boldsymbol{v}}=[u_x,u_y,u_z,0]\pm[v_x,v_y,v_z,0]=[u_x\pm v_x,u_y\pm v_y,u_z\pm v_z,0]$仍然是一个矢量，而两个点的差也是矢量，点和矢量的和或者差也是矢量
两个矢量的和或者差
$$
\begin{aligned}
\vec{\boldsymbol{u}}\pm\vec{\boldsymbol{v}}&=[u_x,u_y,u_z,0]\pm[v_x,v_y,v_z,0]\\
&=[u_x\pm v_x,u_y\pm v_y,u_z\pm v_z,0]
\end{aligned}
$$
仍然是一个矢量，而两个点的差也是矢量，点和矢量的和或者差也是矢量
$$\begin{split}
{vector}+{vector}&={vector} \\
{point}-{point}&={vector} \\
@@ -31,7 +38,10 @@ $$\begin{split}
\end{split}$$
其次坐标表示的两个点的和也是有意义的，表示两个点的中点
$$
[x_1,y_1,1]+[x_2,y_2,1]=[x_1+x_2,y_1+y_2,2]=[\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, 1]
\begin{aligned}
[x_1,y_1,1]+[x_2,y_2,1]&=[x_1+x_2,y_1+y_2,2]\\
&=[\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, 1]
\end{aligned}
$$
### 4.2 齐次矩阵表达
docs/note/graphics/math/transform_03.md
+1 -1
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@@ -11,7 +11,7 @@
### 7.2 法线变换矩阵
以上图为例，设$\vec{\boldsymbol{u}}=\vec{ab}$，那么$\vec{\boldsymbol{u}}\cdot\vec{\boldsymbol{n}}=0$，用矩阵的方式表达，$\mathrm{u}=[u_x, u_y, u_z]$, $\mathrm{n}=[n_x, n_y, n_z]$，就是$\mathrm{u}\mathrm{n}^T=0$
设从$\vec{ab}$到$\vec{a'b'}$的变换矩阵为$A$，根据转置矩阵的[基本运算规则](/graphics/math/math_02.html#_3-5-基本运算规则) ，可以得到
设从$\vec{ab}$到$\vec{a'b'}$的变换矩阵为$A$，根据转置矩阵的[基本运算规则](/note/graphics/math/math_02.html#_3-5-基本运算规则) ，可以得到
$$\begin{split}
0&=\mathrm{u}\mathrm{n}^T\\
&=\mathrm{u}(\mathrm{AA^{-1}})\mathrm{n}^T \\
docs/note/graphics/math/transform_04.md
+5 -2
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@@ -26,7 +26,7 @@ $$\begin{split}
\end{split}$$
### 9.2 计算
根据[坐标系变换公式](/graphics/math/transform_02.html#_6-2-3-%E7%BB%84%E5%90%88%E5%90%8E%E7%9A%84%E5%9D%90%E6%A0%87%E8%BD%AC%E6%8D%A2%E7%9F%A9%E9%98%B5)，设摄像机在世界坐标系中的位置为$e$，摄像机的三个轴的单位轴向量为$\vec{\boldsymbol{i}'}, \vec{\boldsymbol{j}'}, \vec{\boldsymbol{k}'}$视图变换矩阵为
根据[坐标系变换公式](/note/graphics/math/transform_02.html#_6-2-3-%E7%BB%84%E5%90%88%E5%90%8E%E7%9A%84%E5%9D%90%E6%A0%87%E8%BD%AC%E6%8D%A2%E7%9F%A9%E9%98%B5)，设摄像机在世界坐标系中的位置为$e$，摄像机的三个轴的单位轴向量为$\vec{\boldsymbol{i}'}, \vec{\boldsymbol{j}'}, \vec{\boldsymbol{k}'}$视图变换矩阵为
$$
M_{view}=\begin{bmatrix}
i_x'&i_y'&i_z'&-\vec{\boldsymbol{e}}\cdot\vec{\boldsymbol{i}}' \\
@@ -161,12 +161,15 @@ $$\begin{split}
#### 10.2.3 合并后的矩阵
最终得到合并后的投影矩阵为
$$
\boldsymbol{M}_{\text{persp}\rightarrow\text{ccv}}=\boldsymbol{M}_{\text{ortho}\rightarrow\text{ccv}}\boldsymbol{M}_{\text{persp}\rightarrow\text{ortho}}=\begin{bmatrix}
\begin{aligned}
\boldsymbol{M}_{\text{persp}\rightarrow\text{ccv}}&=\boldsymbol{M}_{\text{ortho}\rightarrow\text{ccv}}\boldsymbol{M}_{\text{persp}\rightarrow\text{ortho}}\\
&=\begin{bmatrix}
\frac{2n}{w}&0&0&0\\
0&\frac{2n}{h}&0&0\\
0&0&-\frac{n+f}{n-f}&\frac{2nf}{n-f}\\
0&0&1&0
\end{bmatrix}
\end{aligned}
$$
在大部分工程项目中，摄像机的输入参数一般是如下几个
* 宽高比 $aspect=w/h$