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TSSSNI’S BLOG

Metatron Dev. XI: 相关抽样

·1924 字·4 分钟
Graphics Rendering Correlated Sampling Metatron
目录

Control Variates
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控制变量引入与ff相关的辅助函数hh, 积分H=∫Ωxh(x)dxH = \int_{\Omega_x} h(\mathbf{x})\mathrm{d}\mathbf{x}已知, 可改写估计量:

⟨Ix⟩=αH+(f(X)−αh(X))WX \begin{equation} \langle I_x \rangle = \alpha H + \left(f(X) - \alpha h(X)\right)W_X \end{equation}

图像空间控制变量以相邻像素yy为辅助函数, 要求IyI_y可低方差估计, 估计量如下:

⟨Ix⟩←y=α⟨Iy⟩+⟨Ix−αIy⟩ \begin{equation} \langle I_x \rangle_{\leftarrow y} = \alpha\langle I_y \rangle + \langle I_x - \alpha I_y \rangle \end{equation}

记A=f(X)WXA = f(X)W_X, B=f(Y)WYB = f(Y)W_Y, 分别无偏估计IxI_x与IyI_y, A−αBA - \alpha B的方差为:

Var[A−αB]=Var[A]+α2Var[B]−2αCov[A,B] \begin{equation} \mathrm{Var}[A - \alpha B] = \mathrm{Var}[A] + \alpha^2\mathrm{Var}[B] - 2\alpha\mathrm{Cov}[A, B] \end{equation}

对α\alpha求导得最优系数, 令AA, BB的相关系数为ρ=Cov[A,B]Var[A]Var[B]\rho = \frac{\mathrm{Cov}[A,B]}{\sqrt{\mathrm{Var}[A]\mathrm{Var}[B]}}, 代回得方差缩减与ρ\rho相关:

α∗=Cov[A,B]Var[B],Var[A−α∗B]=(1−ρ2)Var[A] \begin{equation} \alpha^* = \frac{\mathrm{Cov}[A, B]}{\mathrm{Var}[B]}, \quad \mathrm{Var}[A - \alpha^* B] = (1 - \rho^2)\mathrm{Var}[A] \end{equation}

若XX与YY独立则Cov[A,B]=0\mathrm{Cov}[A, B] = 0, 任意α≠0\alpha \neq 0均使方差增加. 对于α⟨Iy⟩+(A−αB)\alpha\langle I_y \rangle + (A - \alpha B), 若⟨Iy⟩\langle I_y \rangle为BB则退化为AA; 若为独立样本B′B', 方差为Var[A]+α2(Var[B]+Var[B′])\mathrm{Var}[A] + \alpha^2(\mathrm{Var}[B] + \mathrm{Var}[B']). 两种情况都不低于Var[A]\mathrm{Var}[A], 因此独立样本无法缩减方差, XX与YY必须相关.

令Y=Tx→y(X)Y = T_{x \to y}(X), 场景连续处f(x)≈f(Tx→y(x))f(\mathbf{x}) \approx f(T_{x \to y}(\mathbf{x})), 此时ρ→1\rho \to 1.

Ix−αIy=∫Ωxf(x)dx−α∫Ωyf(y)dy=∫Ωx(f(x)−αf(Tx→y(x))Jx→y)dx \begin{equation} \begin{aligned} I_x - \alpha I_y &= \int_{\Omega_x} f(\mathbf{x})\mathrm{d}\mathbf{x} - \alpha\int_{\Omega_y} f(\mathbf{y})\mathrm{d}\mathbf{y}\\ &= \int_{\Omega_x} \left(f(\mathbf{x}) - \alpha f(T_{x \to y}(\mathbf{x}))J_{\mathbf{x} \to \mathbf{y}}\right)\mathrm{d}\mathbf{x} \end{aligned} \end{equation}

不保证Tx→y(Ωx)⊇ΩyT_{x \to y}(\Omega_x) \supseteq \Omega_y, 需要MIS:

⟨Ix−αIy⟩=mx(X)(f(X)−αf(Tx→y(X))JX→Y)WX+ my(Y)(f(Ty→x(Y))JY→X−αf(Y))WY \begin{equation} \begin{aligned} \langle I_x - \alpha I_y \rangle &= m_x(X)\left(f(X) - \alpha f(T_{x \to y}(X))J_{X \to Y}\right)W_X\\ &+\ m_y(Y)\left(f(T_{y \to x}(Y))J_{Y \to X} - \alpha f(Y)\right)W_Y \end{aligned} \end{equation}

ReSTCV
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xx重投影到yy, XX为当前帧新样本, ⟨Ix⟩init=f(X)WX\langle I_x \rangle_\mathrm{init} = f(X)W_X. 时域估计为:

⟨Ix⟩=My⟨Ix⟩←y+Minit⟨Ix⟩initMy+Minit \begin{equation} \langle I_x \rangle = \frac{M_y\langle I_x \rangle_{\leftarrow y} + M_\mathrm{init}\langle I_x \rangle_\mathrm{init}}{M_y + M_\mathrm{init}} \end{equation}

N(x)\mathcal{N}(x)为包含xx自身的空域像素集合, 空域估计为:

⟨Ix⟩=∑y∈N(x)My⟨Ix⟩←y∑y∈N(x)My \begin{equation} \langle I_x \rangle = \frac{\sum_{y \in \mathcal{N}(x)} M_y\langle I_x \rangle_{\leftarrow y}}{\sum_{y \in \mathcal{N}(x)} M_y} \end{equation}

ReSTIR PT下⟨Ix⟩←y\langle I_x \rangle_{\leftarrow y}的参数可从蓄水池获取, GRIS估计量只根据无偏权重调整样本亮度, 控制变量估计量包含多个通道, 可有效降低复杂色彩光照或光谱渲染的方差.

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