Math
- Fourier Series: 任意周期为 $1$ 的函数可表示为 $\frac{a_0}{2}+\sum_{n=1}^N(a_n\cos(2\pi nt)+b_n\sin(2\pi nt))$
- $f(x)=\sum_{n=-N}^Nc_ne^{2\pi int},c_n=\int_0^1e^{-2\pi int}f(t)dt$
- $(f*g)(t)=\int_{-\infty}^{\infty} g(t-x)f(x)dx$
- $\Pi(x)=[|x|\leq 1]$
- $(\Pi*\Pi)(x)=\Lambda(x)$
- Radical Inverse
- $n=a_k\cdots a_2a_1$
- $\Phi_b(n)=0.a_1a_2\cdots a_k$
- Van der Corput Sequence: $x_i=\Phi_2(i)$
- Halton Sequence: $x_i=(\Phi_2(i),\Phi_3(i),\Phi_5(i),\dots,\Phi_{p_d}(i))$
- Hammersley Sequence: $x_i=(\frac{i-\frac{1}{2}}{N},\Phi_2(i),\Phi_3(i),\Phi_5(i),\dots,\Phi_{p_{d-1}}(i))$
Reconstruction
- 时间域
- 冲激串 $\delta_T$
- 重建 $\widetilde f(x)=(\delta f)\otimes r$
- 重构核 $r$
- 频率域
- $\widetilde F=(F(\omega)\otimes \delta_{1/T})\Pi_T(\omega)$
- reconstruction filter
- ideal ones not exist
- Box Filter
- Triangle Filter
- Gaussian Filter
- Mitchell Filter
- Windowed Sinc Filter
- Denoising
Aliasing
- Small triangles
- Stairstepping(jaggies)
- Moire Patterns
- 车轮倒转
Source of High Frequencies
- Geometry
- Edges, Vertices, sharp boundaries
- silhouettes
- Texture
- Illumination
Antialiasing Techiques
- Nonuniform sampling: $\sum_{i=-\infty}^{\infty}\delta(x-(i+\frac{1}{2}-\xi)T)$
- noise better than aliasing
- Adaptive sampling: Taking more samples in high-frequency regions
- Prefiltering: mipmap
Evaluating
- Blue noise property
- 白噪：完全随机采样，处处有能量
- 蓝噪：低频无能量，低频完美重构，高频转化为噪声
- gittered grid
- Poisson Disk Sampling
- Dart Throwing: keep throwing darts into a domain with minimum distance constrain
- Lloyd’s Relaxation
- construct voronoi
- move to centroid
- Tiled
- Discrepany: how “uniform” the sampling pattern is
- $D_N(B,P)=|\sum_{b\in B}\frac{#{x_i\in b}}{N}-\text{Vol}(b)|$
采样方法
- Uniform Sampling
- Random Sampling
- Blue noise Sampling
- Stratified Sampling
- Uniform sample + random perturbation (jittering)
- Low-Discrepancy Sampling(quasi-random sampling)
- Sample with Van der Corput Sequence $D^*_N(P)=O(\frac{\log N}{N})$
- Sample with Halton Sequence: $D^*_N(P)=O(\frac{(\log N)^d}{N})$