Lecture 01 Overview
cg含义： 合成和操作视觉信息
应用 ： video games, movies, animations, design, visualization, virtual reality, digital illustration, simulation, graphical user interfaces, typography(the quick brown fox jumps over the lazy dog) ... ...
Rasterization 光栅化 把三维物体投影到平面； 实时地，达到30fpsRay Tracing 光线追踪 慢； 但是有实时光线追踪Animation/Simulation
与cv的区别：
IDE Integrated Development Environment 集成开发环境
Lecture 02 Linear Algebra
dot product a ⃗ = ( x 1 y 1 ) , b ⃗ = ( x 2 y 2 ) , 则 a ⃗ ⋅ b ⃗ = ∣ a ⃗ ∣ ∣ b ⃗ ∣ cos θ = x 1 x 2 + y 1 y 2 \vec{a} = \left( \begin{matrix} x_1\\ y_1 \end{matrix} \right) ,
\vec{b} = \left( \begin{matrix} x_2\\ y_2 \end{matrix} \right) , 则
\vec{a} \cdot \vec{b} = \lvert \vec{a} \rvert \lvert \vec{b} \rvert \cos\theta =
x_1x_2 + y_1y_2
a = ( x 1 y 1 ) , b = ( x 2 y 2 ) , 则 a ⋅ b = ∣ a ∣ ∣ b ∣ cos θ = x 1 x 2 + y 1 y 2 ⊥ \bot ⊥
cross product a ⃗ = ( x 1 y 1 ) , b ⃗ = ( x 2 y 2 ) , 则 ∣ a ⃗ × b ⃗ ∣ = ∣ a ⃗ ∣ ∣ b ⃗ ∣ sin θ \vec{a} = \left( \begin{matrix} x_1\\ y_1 \end{matrix} \right) ,
\vec{b} = \left( \begin{matrix} x_2\\ y_2 \end{matrix} \right) , 则
\lvert \vec{a} \times \vec{b} \rvert = \lvert \vec{a} \rvert \lvert \vec{b} \rvert \sin\theta
a = ( x 1 y 1 ) , b = ( x 2 y 2 ) , 则 ∣ a × b ∣ = ∣ a ∣ ∣ b ∣ sin θ a ⃗ × b ⃗ = ( 0 − z 1 y 1 z 1 0 − x 1 − y 1 x 1 0 ) ( x 2 y 2 z 2 ) \vec{a} \times \vec{b} =
\left( \begin {matrix}
0 & -z_1 & y_1 \\
z_1 & 0 & -x_1 \\
-y_1 & x_1 & 0
\end {matrix} \right)
\left( \begin{matrix}
x_2 \\ y_2 \\ z_2
\end {matrix}\right) a × b = ⎝ ⎛ 0 z 1 − y 1 − z 1 0 x 1 y 1 − x 1 0 ⎠ ⎞ ⎝ ⎛ x 2 y 2 z 2 ⎠ ⎞ a ⃗ × b ⃗ = − b ⃗ × a ⃗ \vec{a} \times \vec{b} = -\vec{b} \times \vec{a} a × b = − b × a a ⃗ × a ⃗ = 0 ⃗ \vec{a} \times \vec{a} = \vec{0} a × a = 0
叉乘应用A P ⃗ × A B ⃗ , B P ⃗ × B C ⃗ , C P ⃗ × C A ⃗ \vec{AP} \times \vec{AB}, \vec{BP} \times \vec{BC}, \vec{CP} \times \vec{CA} A P × A B , BP × BC , CP × C A
matrix ( A B ) T = B T A T , ( A B ) − 1 = B − 1 A − 1 (AB)^{T} = B^T A^T, (AB)^{-1} = B^{-1} A^{-1} ( A B ) T = B T A T , ( A B ) − 1 = B − 1 A − 1
Lecture 03 Transformation
2D变换
scale matrix ( x ′ y ′ ) = ( s 1 0 0 s 2 ) ( x y ) \left(\begin{matrix} x'\\y' \end{matrix}\right) =
\left(\begin{matrix} s_1&0 \\ 0&s_2 \end{matrix}\right)
\left(\begin{matrix} x\\y \end{matrix}\right)
( x ′ y ′ ) = ( s 1 0 0 s 2 ) ( x y ) reflection matrix ( x ′ y ′ ) = ( − 1 0 0 1 ) ( x y ) \left(\begin{matrix} x'\\y' \end{matrix}\right) =
\left(\begin{matrix} -1&0 \\ 0&1 \end{matrix}\right)
\left(\begin{matrix} x\\y \end{matrix}\right)
( x ′ y ′ ) = ( − 1 0 0 1 ) ( x y ) shear matrix ( x ′ y ′ ) = ( 1 a 0 1 ) ( x y ) \left(\begin{matrix} x'\\y' \end{matrix}\right) =
\left(\begin{matrix} 1&a \\ 0&1 \end{matrix}\right)
\left(\begin{matrix} x\\y \end{matrix}\right)
( x ′ y ′ ) = ( 1 0 a 1 ) ( x y ) rotation matrix ( x ′ y ′ ) = ( cos θ − sin θ sin θ cos θ ) ( x y ) \left(\begin{matrix} x'\\y' \end{matrix}\right) =
\left(\begin{matrix} \cos\theta&-\sin\theta \\ \sin\theta& \cos\theta \end{matrix}\right)
\left(\begin{matrix} x\\y \end{matrix}\right)
( x ′ y ′ ) = ( cos θ sin θ − sin θ cos θ ) ( x y )
以上是线性变换，但平移不属于线性变换 ，因而引入了齐次坐标系。
Homogenous Coordinates
齐次坐标系中，
( x y 1 ) \left(\begin{matrix} x\\y\\1 \end{matrix}\right)
⎝ ⎛ x y 1 ⎠ ⎞ ( x y 0 ) \left( \begin {matrix} x\\ y\\ 0 \end {matrix} \right)
⎝ ⎛ x y 0 ⎠ ⎞ 规律 ：
在齐次坐标系中，仿射变换可以写为： ( x ′ y ′ 1 ) = ( a b t x c d t y 0 0 1 ) ( x y 1 ) \left(\begin{matrix} x'\\y'\\1 \end{matrix}\right) =
\left(\begin{matrix} a& b &t_x\\ c& d &t_y \\ 0& 0 &1 \end{matrix}\right)
\left(\begin{matrix} x\\y\\1 \end{matrix}\right)
⎝ ⎛ x ′ y ′ 1 ⎠ ⎞ = ⎝ ⎛ a c 0 b d 0 t x t y 1 ⎠ ⎞ ⎝ ⎛ x y 1 ⎠ ⎞
组合变换，先线性变换， 后平移 , 写成矩阵：T ∗ R ∗ ( x y 1 ) T * R *
\left(\begin{matrix} x\\y\\1 \end{matrix}\right)
T ∗ R ∗ ⎝ ⎛ x y 1 ⎠ ⎞ T ( c ) ∗ R ∗ T ( − c ) ∗ ( x y 1 ) T(c) * R * T(-c) *
\left(\begin{matrix} x\\y\\1 \end{matrix}\right)
T ( c ) ∗ R ∗ T ( − c ) ∗ ⎝ ⎛ x y 1 ⎠ ⎞
Lecture 04 Transformation Cont.
旋转矩阵是正交矩阵： R T = R − 1 R^T = R^{-1} R T = R − 1
3D变换
scale( s x 0 0 0 0 s y 0 0 0 0 s z 0 0 0 0 1 ) \left( \begin {matrix}
s_x & 0 & 0 & 0 \\
0 & s_y & 0 &0 \\
0 & 0 & s_z & 0 \\
0 & 0 & 0& 1
\end {matrix} \right )
⎝ ⎛ s x 0 0 0 0 s y 0 0 0 0 s z 0 0 0 0 1 ⎠ ⎞
translationn( 1 0 0 t x 0 1 0 t y 0 0 1 t z 0 0 0 1 ) \left( \begin {matrix}
1 & 0 & 0 & t_x \\
0 & 1 & 0 & t_y \\
0 & 0 & 1 & t_z \\
0 & 0 & 0& 1
\end {matrix} \right )
⎝ ⎛ 1 0 0 0 0 1 0 0 0 0 1 0 t x t y t z 1 ⎠ ⎞
rotation( 1 0 0 0 0 cos θ − sin θ 0 0 sin θ cos θ 0 0 0 0 1 ) \left( \begin {matrix}
1 & 0 & 0 & 0 \\
0 & \cos\theta & -\sin\theta & 0 \\
0 & \sin\theta & \cos\theta & 0 \\
0 & 0 & 0& 1
\end {matrix} \right )
⎝ ⎛ 1 0 0 0 0 cos θ sin θ 0 0 − sin θ cos θ 0 0 0 0 1 ⎠ ⎞ 绕y : ( cos θ 0 sin θ t x 0 1 0 t y − sin θ 0 cos θ t z 0 0 0 1 ) \left( \begin {matrix}
\cos\theta & 0 & \sin\theta & t_x \\
0 & 1 & 0 & t_y \\
-\sin\theta & 0 & \cos\theta & t_z \\
0 & 0 & 0& 1
\end {matrix} \right )
⎝ ⎛ cos θ 0 − sin θ 0 0 1 0 0 sin θ 0 cos θ 0 t x t y t z 1 ⎠ ⎞ ( cos θ − sin θ 0 t x sin θ cos θ 0 t y 0 0 1 t z 0 0 0 1 ) \left( \begin {matrix}
\cos\theta & -\sin\theta & 0 & t_x \\
\sin\theta & \cos\theta & 0 & t_y \\
0 & 0 & 1 & t_z \\
0 & 0 & 0& 1
\end {matrix} \right )
⎝ ⎛ cos θ sin θ 0 0 − sin θ cos θ 0 0 0 0 1 0 t x t y t z 1 ⎠ ⎞ Ridrigues's formula : 用一个旋转轴和旋转角度来刻画旋转Quaternions 四元数
Viewing Transformation
视图变换
定义Camera:
视图变换→ \rightarrow → Rotate g to -Z → \rightarrow → M v i e w = R v i e w T v i e w M_{view} = R_{view} T_{view} M v i e w = R v i e w T v i e w T v i e w = ( 1 0 0 − x e 0 1 0 − y e 0 0 1 − z e 0 0 0 1 ) T_{view} =
\left( \begin{matrix}
1 & 0 & 0 & -x_e \\
0 & 1 & 0 & -y_e \\
0 & 0 & 1 & -z_e \\
0 & 0 & 0 & 1
\end {matrix} \right)
T v i e w = ⎝ ⎛ 1 0 0 0 0 1 0 0 0 0 1 0 − x e − y e − z e 1 ⎠ ⎞ R v i e w = ( x g ^ × t ^ y g ^ × t ^ z g ^ × t ^ 0 x t y t z t 0 x − g y − g z − g 0 0 0 0 1 )
R_{view} =
\left( \begin{matrix}
x_{\hat{g} \times \hat{t}} & y_{\hat{g} \times \hat{t}} & z_{\hat{g} \times \hat{t}} & 0 \\
x_t & y_t & z_t & 0 \\
x_{-g} & y_{-g} & z_{-g} & 0 \\
0 & 0 & 0 & 1
\end {matrix} \right)
R v i e w = ⎝ ⎛ x g ^ × t ^ x t x − g 0 y g ^ × t ^ y t y − g 0 z g ^ × t ^ z t z − g 0 0 0 0 1 ⎠ ⎞
投影变换
正交投影 Orthographic projection→ \rightarrow → M o r t h o = ( 2 r − l 0 0 0 0 2 t − b 0 0 0 0 2 n − f 0 0 0 0 1 ) ( 1 0 0 − r + l 2 0 1 0 − t + b 2 0 0 1 − n + f 2 0 0 0 1 ) M_{ortho} =
\left( \begin{matrix}
\frac{2}{r-l} & 0 & 0 & 0 \\
0 & \frac{2}{t-b} & 0 & 0 \\
0 & 0 & \frac{2}{n-f} & 0 \\
0 & 0 & 0 & 1
\end {matrix} \right)
\left( \begin{matrix}
1 & 0 & 0 & -\frac{r+l}{2} \\
0 & 1 & 0 & -\frac{t+b}{2} \\
0 & 0 & 1 & -\frac{n+f}{2} \\
0 & 0 & 0 & 1
\end {matrix} \right) M or t h o = ⎝ ⎛ r − l 2 0 0 0 0 t − b 2 0 0 0 0 n − f 2 0 0 0 0 1 ⎠ ⎞ ⎝ ⎛ 1 0 0 0 0 1 0 0 0 0 1 0 − 2 r + l − 2 t + b − 2 n + f 1 ⎠ ⎞
透视投影 Perspective projectionM p e r s p = M o r t h o M p e r s p → o r t h o M_{persp} = M_{ortho} M_{persp \rightarrow ortho} M p ers p = M or t h o M p ers p → or t h o M p e r s p → o r t h o = ( n 0 0 0 0 n 0 0 0 0 n + f − n f 0 0 1 0 ) M_{persp \rightarrow ortho} =
\left( \begin{matrix}
n & 0 & 0 & 0 \\
0 & n & 0 & 0 \\
0 & 0 & n+f & -nf \\
0 & 0 & 1 & 0
\end {matrix} \right) M p ers p → or t h o = ⎝ ⎛ n 0 0 0 0 n 0 0 0 0 n + f 1 0 0 − n f 0 ⎠ ⎞
Lecture 05 Rasterization
Triangles
Antialiasing Sampling Artifacts : Jaggies 锯齿 Moire 摩尔纹 Wagon wheel effect 车轮效应solution : Pre-filtering, blurring before sampling
图片中的低频信息，高频信息；大多数是低频信息，高频信息意味着边界、意味着剧烈变化
反走样Antialiasing 的寻常办法就是 先模糊， 即先过滤掉图像信息中的 高频信息，此时再对低频信息进行采样，就能起到反走样的效果。
MSAA MultiSampling Anti-Aliasing 多重采样抗锯齿
FXAA Fast approximate AA 快速近似抗锯齿
对一幅图，通过图像匹配的方法找到有锯齿的边界，然后换成没有锯齿的边界。属于图像后期处理
TAA Temporal AA
DLSS Deep Learning Super Sampling
Z-Buffer
for (each triangle T)
for (each sample (x,y,z) in T)
if (z < zBuffer[x,y])
zBuffer[x,y] = z;
frameBuffer[x,y] = rgb;
Lecture 06 Shading
Shading
Blinn-phong reflectance model 布林冯着色模型
Diffuse reflectionL d = k d ( I / r 2 ) m a x ( 0 , n ⋅ l ) L_d = k_d (I/r^2) max(0, \bold{n}\cdot\bold{l}) L d = k d ( I / r 2 ) ma x ( 0 , n ⋅ l ) L d L_d L d k d k_d k d I I I r 2 r^2 r 2 l ⃗ \vec{l} l n ⃗ \vec{n} n
SpecularL s = k s ( I / r 2 ) m a x ( 0 , n ⋅ h ) p L_s = k_s (I/r^2) max(0, \bold{n}\cdot\bold{h})^p L s = k s ( I / r 2 ) ma x ( 0 , n ⋅ h ) p h \bold{h} h h = l + v ∣ l + v ∣ \bold{h} = \frac{\bold{l}+\bold{v}}{|\bold{l}+\bold{v}|} h = ∣ l + v ∣ l + v k d k_d k d
AmbientL a = k a I a L_a = k_a I_a L a = k a I a
Shading Frequency : Face、Vertex、Pixel
逐点着色的时候，顶点法线方向的计算：N v = ∑ i N i ∣ ∑ i N i ∣ N_v = \frac{\sum_iN_i}{|\sum_iN_i|} N v = ∣ ∑ i N i ∣ ∑ i N i
Pipeline
Texture Mapping ( u , v ) (u,v) ( u , v )
重心坐标系 barycentric coordinates ( α , β , γ ) (\alpha,\beta,\gamma) ( α , β , γ ) α + β + γ = 1 \alpha + \beta + \gamma = 1 α + β + γ = 1 双线性插值 bilinear interpolation l e r p ( s , v 0 , v 1 ) = v 0 + s ( v 1 − v 0 ) lerp(s,v_0, v_1) = v_0 + s(v_1 - v_0) l er p ( s , v 0 , v 1 ) = v 0 + s ( v 1 − v 0 ) MipMap l o g ( 原图分辨率 ) log(原图分辨率) l o g ( 原图分辨率 )
用法： 像素点对应到纹理图上， 通过计算对应的texel长度，L = m a x ( ( d u d x ) 2 + ( d v d x ) 2 , ( d u d y ) 2 + ( d v d y ) 2 ) L = max(\sqrt{(\frac{du}{dx})^2 + (\frac{dv}{dx})^2},\sqrt{(\frac{du}{dy})^2 + (\frac{dv}{dy})^2} ) L = ma x ( ( d x d u ) 2 + ( d x d v ) 2 , ( d y d u ) 2 + ( d y d v ) 2 ) D = l o g L D = logL D = l o gL 三线性插值
Anisotropic Filtering EWA Filtering 
