体散射过程
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吸收, 自发光与散射影响传输介质中辐亮度的分布, 这些系数与位置和方向有关, pbrt认为介质中粒子的位置是独立的.
吸收
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吸收由介质的吸收系数σ a \sigma_a σ a
d L o ( p , ω ) = − σ a ( p , ω ) L i ( p , − ω ) d t
\begin{equation}
\mathrm{d}L_o(p,\omega)=-\sigma_a(p,\omega)L_i(p,-\omega)\mathrm{d}t
\end{equation}
d L o ( p , ω ) = − σ a ( p , ω ) L i ( p , − ω ) d t 自发光
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介质中的化学, 热或核过程使得经过它的光线的辐亮度增加, 其形式如下. 自发光是吸收的逆过程, 因此二者系数相同. 注意到pbrt中L e L_e L e L i L_i L i
d L o ( p , ω ) = σ a ( p , ω ) L e ( p , ω ) d t
\begin{equation}
\mathrm{d}L_o(p,\omega)=\sigma_a(p,\omega)L_e(p,\omega)\mathrm{d}t
\end{equation}
d L o ( p , ω ) = σ a ( p , ω ) L e ( p , ω ) d t 外散射与衰减
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粒子间的碰撞导致粒子的运动方向被改变, 从而导致辐亮度的衰减, 这被称为外散射. 将吸收与外散射系数结合即可得到衰减系数σ t ( p , ω ) = σ a ( p , ω ) + σ s ( p , ω ) \sigma_t(p,\omega)=\sigma_a(p,\omega)+\sigma_s(p,\omega) σ t ( p , ω ) = σ a ( p , ω ) + σ s ( p , ω ) ρ ( p , ω ) = σ s ( p , ω ) σ t ( p , ω ) \rho(p,\omega)=\frac{\sigma_s(p,\omega)}{\sigma_t(p,\omega)} ρ ( p , ω ) = σ t ( p , ω ) σ s ( p , ω ) 1 σ t ( p , ω ) \frac{1}{\sigma_t(p,\omega)} σ t ( p , ω ) 1 σ t ( p , ω ) \sigma_t(p,\omega) σ t ( p , ω )
d L o ( p , ω ) = − σ s ( p , ω ) L i ( p , − ω ) d t
\begin{equation}
\mathrm{d}L_o(p,\omega)=-\sigma_s(p,\omega)L_i(p,-\omega)\mathrm{d}t
\end{equation}
d L o ( p , ω ) = − σ s ( p , ω ) L i ( p , − ω ) d t 内散射
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其它光线中的粒子通过外散射成为当前光线中的粒子, 使得辐亮度增加, 这被称为内散射. 定义相位方程p ( ω , ω ′ ) p(\omega,\omega') p ( ω , ω ′ ) ω \omega ω ω ′ \omega' ω ′
d L o ( p , ω ) = σ s ( p , ω ) ∫ Θ p ( p , ω i , ω ) L i ( p , ω i ) d ω i d t
\begin{equation}
\mathrm{d}L_o(p,\omega)=\sigma_s(p,\omega)\int_\Theta p(p,\omega_i,\omega)L_i(p,\omega_i)\mathrm{d}\omega_idt
\end{equation}
d L o ( p , ω ) = σ s ( p , ω ) ∫ Θ p ( p , ω i , ω ) L i ( p , ω i ) d ω i d t 此时可以得到辐亮度总增加量.
d L o ( p , ω ) = σ t ( p , ω ) L s ( p , ω ) d t = σ t ( p , ω ) ( σ a ( p , ω ) σ t ( p , ω ) L e ( p , ω ) + σ s ( p , ω ) σ t ( p , ω ) ∫ Θ p ( p , ω i , ω ) L i ( p , ω i ) d ω i ) d t
\begin{equation}
\begin{aligned}
\mathrm{d}L_o(p,\omega)
&=\sigma_t(p,\omega)L_s(p,\omega)\mathrm{d}t\\
&=\sigma_t(p,\omega)(\frac{\sigma_a(p,\omega)}{\sigma_t(p,\omega)}L_e(p,\omega)+\frac{\sigma_s(p,\omega)}{\sigma_t(p,\omega)}\int_\Theta p(p,\omega_i,\omega)L_i(p,\omega_i)\mathrm{d}\omega_i)\mathrm{d}t
\end{aligned}
\end{equation}
d L o ( p , ω ) = σ t ( p , ω ) L s ( p , ω ) d t = σ t ( p , ω ) ( σ t ( p , ω ) σ a ( p , ω ) L e ( p , ω ) + σ t ( p , ω ) σ s ( p , ω ) ∫ Θ p ( p , ω i , ω ) L i ( p , ω i ) d ω i ) d t 透射率
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根据微分方程可以得到透射率, 本节不考虑辐亮度增益.
T r ( p → p ′ ) = e − ∫ 0 d σ t ( p + t ω , ω ) d t
\begin{equation}
T_r(p \to p') = e^{-\int_0^d \sigma_t(p + t\omega, \omega)\mathrm{d}t}
\end{equation}
T r ( p → p ′ ) = e − ∫ 0 d σ t ( p + t ω , ω ) d t 根据光线的传播过程可知透射率是可逆的.
T r ( p → p ′ ) = T r ( p ′ → p )
\begin{equation}
T_r(p \to p')=T_r(p' \to p)
\end{equation}
T r ( p → p ′ ) = T r ( p ′ → p ) 透射率可以被分解, 这是积分的性质决定的.
T r ( p → p ′ ′ ) = T r ( p → p ′ ) T r ( p ′ → p ′ ′ )
\begin{equation}
T_r(p \to p'')=T_r(p \to p')T_r(p' \to p'')
\end{equation}
T r ( p → p ′′ ) = T r ( p → p ′ ) T r ( p ′ → p ′′ ) 衰减系数的积分被称为光学厚度. 根据Jensen不等式可知, E [ e − X ] ≠ e − E [ X ] E[e^{-X}] \ne e^{-E[X]} E [ e − X ] = e − E [ X ]
τ ( p → p ′ ) = ∫ 0 d σ t ( p + t ω , ω ) d t
\begin{equation}
\tau(p \to p')=\int_0^d \sigma_t(p+t\omega,\omega)\mathrm{d}t
\end{equation}
τ ( p → p ′ ) = ∫ 0 d σ t ( p + t ω , ω ) d t 路径追踪任务是光线传播的逆过程, 因此− d L ( p ) = − σ t ( p , ω ) L ( p ) d t -\mathrm{d}L(p)=-\sigma_t(p,\omega)L(p)\mathrm{d}t − d L ( p ) = − σ t ( p , ω ) L ( p ) d t Integral formulations of volumetric transmittance .
根据微分方程可以得到L ( p , ω ) − L ( p + d ω , ω ) = ∫ 0 d − σ t ( p + t ω ) L ( p + t ω , ω ) d t L(p,\omega)-L(p+\mathrm{d}\omega,\omega)=\int_0^d -\sigma_t(p+t\omega)L(p+t\omega,\omega)\mathrm{d}t L ( p , ω ) − L ( p + d ω , ω ) = ∫ 0 d − σ t ( p + t ω ) L ( p + t ω , ω ) d t T r ( p ← p + d ω ) = L ( p , ω ) L ( p + d ω , ω ) T_r(p \leftarrow p + \mathrm{d}\omega)=\frac{L(p,\omega)}{L(p + \mathrm{d}\omega,\omega)} T r ( p ← p + d ω ) = L ( p + d ω , ω ) L ( p , ω )
T r ( p ← p + d ω ) = 1 − ∫ 0 d σ t ( p + t ω ) T r ( p + t ω ← p + d ω ) d t
\begin{equation}
T_r(p \leftarrow p + \mathrm{d}\omega)=1 - \int_0^d \sigma_t(p+t\omega)T_r(p + t\omega \leftarrow p+\mathrm{d}\omega)\mathrm{d}t
\end{equation}
T r ( p ← p + d ω ) = 1 − ∫ 0 d σ t ( p + t ω ) T r ( p + t ω ← p + d ω ) d t 空散射
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定义一个σ maj ( p , ω ) \sigma_{\text{maj}}(p,\omega) σ maj ( p , ω ) σ maj ( p , ω ) = σ a ( p , ω ) + σ s ( p , ω ) + σ n ( p , ω ) \sigma_{\text{maj}}(p,\omega)=\sigma_a(p,\omega)+\sigma_s(p,\omega)+\sigma_n(p,\omega) σ maj ( p , ω ) = σ a ( p , ω ) + σ s ( p , ω ) + σ n ( p , ω )
σ n ( p , ω ) = σ maj ( p , ω ) − σ t ( p , ω )
\begin{equation}
\sigma_n(p,\omega)=\sigma_{\text{maj}}(p,\omega)-\sigma_t(p,\omega)
\end{equation}
σ n ( p , ω ) = σ maj ( p , ω ) − σ t ( p , ω ) 此时微分方程转为如下形式.
− d L ( p , ω ) d t + σ maj ( p , ω ) L ( p , ω ) = σ n ( p , ω ) L ( p , ω )
\begin{equation}
-\frac{\mathrm{d}L(p,\omega)}{\mathrm{d}t}+\sigma_{\text{maj}}(p,\omega)L(p,\omega)=\sigma_n(p,\omega)L(p,\omega)
\end{equation}
− d t d L ( p , ω ) + σ maj ( p , ω ) L ( p , ω ) = σ n ( p , ω ) L ( p , ω ) 定义h ( t ) h(t) h ( t ) g ( t ) g(t) g ( t )
h ( t ) = e − ∫ 0 t σ maj ( p + t ′ ω , ω ) d t ′ d h ( t ) d t = − h ( t ) σ maj ( p + t ω , ω ) g ( t ) = σ n ( p + t ω , ω ) L ( p + t ω , ω )
\begin{equation}
\begin{aligned}
h(t)&=e^{-\int_0^t \sigma_{\text{maj}}(p+t'\omega,\omega) \mathrm{d}t'}\\
\frac{dh(t)}{\mathrm{d}t}&=-h(t)\sigma_{\text{maj}}(p+t\omega,\omega)\\
g(t)&=\sigma_n(p+t\omega,\omega)L(p+t\omega,\omega)
\end{aligned}
\end{equation}
h ( t ) d t d h ( t ) g ( t ) = e − ∫ 0 t σ maj ( p + t ′ ω , ω ) d t ′ = − h ( t ) σ maj ( p + t ω , ω ) = σ n ( p + t ω , ω ) L ( p + t ω , ω ) 微分方程两边同乘h ( t ) h(t) h ( t )
− h ( t ) d L ( p + t ω , ω ) d t − L ( p + t ω , ω ) d h ( t ) d t = h ( t ) g ( t ) h ( 0 ) L ( p , ω ) − h ( d ) L ( p + d ω , ω ) = ∫ 0 d h ( t ) g ( t ) d t
\begin{equation}
\begin{aligned}
-h(t)\frac{\mathrm{d}L(p+t\omega,\omega)}{\mathrm{d}t}-L(p+t\omega,\omega)\frac{dh(t)}{\mathrm{d}t}&=h(t)g(t)\\
h(0)L(p,\omega)-h(d)L(p+\mathrm{d}\omega,\omega)&=\int_0^d h(t)g(t)\mathrm{d}t
\end{aligned}
\end{equation}
− h ( t ) d t d L ( p + t ω , ω ) − L ( p + t ω , ω ) d t d h ( t ) h ( 0 ) L ( p , ω ) − h ( d ) L ( p + d ω , ω ) = h ( t ) g ( t ) = ∫ 0 d h ( t ) g ( t ) d t 两边同除h ( 0 ) L ( p + d ω , ω ) h(0)L(p+\mathrm{d}\omega,\omega) h ( 0 ) L ( p + d ω , ω )
T r ( p ← p + d ω ) = h ( d ) h ( 0 ) + ∫ 0 d h ( t ) g ( t ) h ( 0 ) L ( p + d ω , ω ) = e − ∫ 0 d σ maj ( p + t ω , ω ) d t + ∫ 0 d e − ∫ 0 t σ maj ( p + t ′ ω , ω ) d t ′ σ n ( p + t ω , ω ) T r ( p + t ω ← p + d ω ) d t
\begin{equation}
\begin{aligned}
T_r(p \leftarrow p+\mathrm{d}\omega)
&=\frac{h(d)}{h(0)}+\int_0^d \frac{h(t)g(t)}{h(0){L(p+\mathrm{d}\omega,\omega)}}\\
&=e^{-\int_0^d \sigma_\text{maj}(p+t\omega,\omega)\mathrm{d}t}+\int_0^d e^{-\int_0^t \sigma_\text{maj}(p+t'\omega,\omega)\mathrm{d}t'}\sigma_n(p+t\omega,\omega)T_r(p+t\omega \leftarrow p+\mathrm{d}\omega)\mathrm{d}t
\end{aligned}
\end{equation}
T r ( p ← p + d ω ) = h ( 0 ) h ( d ) + ∫ 0 d h ( 0 ) L ( p + d ω , ω ) h ( t ) g ( t ) = e − ∫ 0 d σ maj ( p + t ω , ω ) d t + ∫ 0 d e − ∫ 0 t σ maj ( p + t ′ ω , ω ) d t ′ σ n ( p + t ω , ω ) T r ( p + t ω ← p + d ω ) d t 计算上式的Monte Carlo需要使用一个与积分项成比例的分布, pbrt使用p maj ( t ) ∝ e ∫ 0 t − σ maj ( p + t ′ ω , ω ) d t ′ p_{\text{maj}}(t) \propto e^{\int_0^t -\sigma_{\text{maj}}(p+t'\omega, \omega)\mathrm{d}t'} p maj ( t ) ∝ e ∫ 0 t − σ maj ( p + t ′ ω , ω ) d t ′
p maj ( t ) = σ maj ( p , ω ) e − ∫ 0 t σ maj ( p + t ′ ω , ω ) d t ′
\begin{equation}
p_{\text{maj}}(t)=\sigma_{\text{maj}}(p,\omega)e^{-\int_0^t \sigma_{\text{maj}}(p+t'\omega,\omega) \mathrm{d}t'}
\end{equation}
p maj ( t ) = σ maj ( p , ω ) e − ∫ 0 t σ maj ( p + t ′ ω , ω ) d t ′ 将积分项视为t ′ ≥ d t' \ge d t ′ ≥ d t ′ ≥ d t' \ge d t ′ ≥ d
T r ( p ← p + d ω ) ≈ e − ∫ 0 d σ maj ( p + t ω , ω ) d t + { σ n ( p + t ′ ω , ω ) σ maj ( p + t ′ ω , ω ) T r ( p + t ′ ω ← p + d ω ) t ′ < d 0 otherwise
\begin{equation}
T_r(p \leftarrow p+\mathrm{d}\omega) \approx e^{-\int_0^\mathrm{d}\sigma_{\text{maj}}(p+t\omega,\omega)\mathrm{d}t}+
\begin{cases}
\frac{\sigma_n(p+t'\omega,\omega)}{\sigma_{\text{maj}}(p+t'\omega,\omega)}T_r(p+t'\omega \leftarrow p+\mathrm{d}\omega) & t'<d\\
0 & \text{otherwise}
\end{cases}
\end{equation}
T r ( p ← p + d ω ) ≈ e − ∫ 0 d σ maj ( p + t ω , ω ) d t + { σ maj ( p + t ′ ω , ω ) σ n ( p + t ′ ω , ω ) T r ( p + t ′ ω ← p + d ω ) 0 t ′ < d otherwise 随机选取其中一项也可以进行Monte Carlo, 由于CDF具有P ( t ′ ≥ d ) = e − ∫ 0 d σ maj d t P(t' \ge d)=e^{-\int_0^\mathrm{d}\sigma_{\text{maj}}\mathrm{d}t} P ( t ′ ≥ d ) = e − ∫ 0 d σ maj d t ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) ( 0 , d ) (0,d) ( 0 , d ) 1 − e − ∫ 0 d σ maj ( p + t ω , ω ) d t 1-e^{-\int_0^\mathrm{d}\sigma_{\text{maj}}(p+t\omega,\omega)\mathrm{d}t} 1 − e − ∫ 0 d σ maj ( p + t ω , ω ) d t 1 1 − p e \frac{1}{1-p_e} 1 − p e 1
T r ( p ← p + d ω ) ≈ { e − ∫ 0 d σ maj ( p + t ω , ω ) d t p e with probability p e 1 1 − p e ∫ 0 d e − ∫ 0 t σ maj ( p + t ′ ω , ω ) d t ′ σ n ( p + t ω , ω ) T r ( p + t ω ← p + d ω ) d t otherwise ≈ { 1 t ′ > d σ n ( p + t ′ ω , ω ) σ maj ( p + t ′ ω , ω ) T r ( p + t ′ ω ← p + d ω ) otherwise ≈ ∏ i = 1 n σ n ( p + t i ω , ω ) σ maj ( p + t i ω , ω )
\begin{equation}
\begin{aligned}
T_r(p \leftarrow p+\mathrm{d}\omega)
&\approx
\begin{cases}
\frac{e^{-\int_0^\mathrm{d}\sigma_{\text{maj}}(p+t\omega,\omega)\mathrm{d}t}}{p_e} & \text{with}\ \text{probability}\ p_e\\
\frac{1}{1-p_e}\int_0^d e^{-\int_0^t\sigma_{\text{maj}}(p+t'\omega,\omega)\mathrm{d}t'}\sigma_n(p+t\omega,\omega)T_r(p+t\omega \leftarrow p+\mathrm{d}\omega)\mathrm{d}t & \text{otherwise}
\end{cases}\\
&\approx
\begin{cases}
1 & t' > d\\
\frac{\sigma_n(p+t'\omega,\omega)}{\sigma_{\text{maj}}(p+t'\omega,\omega)}T_r(p+t'\omega \leftarrow p+\mathrm{d}\omega) & \text{otherwise}
\end{cases}\\
&\approx
\prod_{i=1}^n \frac{\sigma_n(p+t_i\omega,\omega)}{\sigma_{\text{maj}}(p+t_i\omega,\omega)}
\end{aligned}
\end{equation}
T r ( p ← p + d ω ) ≈ { p e e − ∫ 0 d σ maj ( p + t ω , ω ) d t 1 − p e 1 ∫ 0 d e − ∫ 0 t σ maj ( p + t ′ ω , ω ) d t ′ σ n ( p + t ω , ω ) T r ( p + t ω ← p + d ω ) d t with probability p e otherwise ≈ { 1 σ maj ( p + t ′ ω , ω ) σ n ( p + t ′ ω , ω ) T r ( p + t ′ ω ← p + d ω ) t ′ > d otherwise ≈ i = 1 ∏ n σ maj ( p + t i ω , ω ) σ n ( p + t i ω , ω ) 比率跟踪估计器在透射率已经很小时仍然会继续采样, 可以通过俄罗斯轮盘解决该问题, 将俄罗斯轮盘概率设置为σ n ( p + t ′ ω , ω ) σ maj ( p + t ′ ω , ω ) \frac{\sigma_n(p+t'\omega,\omega)}{\sigma_{\text{maj}}(p+t'\omega,\omega)} σ maj ( p + t ′ ω , ω ) σ n ( p + t ′ ω , ω )
T r ( p ← p + d ω ) ≈ { 1 t ′ > d T r ( p + t ′ ω ← p + d ω ) t ′ ≤ d and with probability σ n ( p + t ′ ω , ω ) σ maj 0 otherwise
\begin{equation}
\begin{aligned}
T_r(p \leftarrow p+\mathrm{d}\omega)&\approx
\begin{cases}
1 & t' > d\\
T_r(p+t'\omega \leftarrow p+\mathrm{d}\omega) & t' \le d\ \text{and}\ \text{with}\ \text{probability}\ \frac{\sigma_n(p+t'\omega,\omega)}{\sigma_{\text{maj}}}\\
0 & \text{otherwise}
\end{cases}
\end{aligned}
\end{equation}
T r ( p ← p + d ω ) ≈ ⎩ ⎨ ⎧ 1 T r ( p + t ′ ω ← p + d ω ) 0 t ′ > d t ′ ≤ d and with probability σ maj σ n ( p + t ′ ω , ω ) otherwise 相位函数
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只与入射方向和出射方向的夹角有关的相位函数被称为对称相位函数, 否则就是需要用四个维度表示的非对称相位函数. 各个方向均匀分布的相位函数为各向同性相位函数, 只有p ( ω i , ω o ) = 1 ∫ 0 2 π ∫ 0 π sin θ d θ d ϕ = 1 4 π p(\omega_i,\omega_o)=\frac{1}{\int_0^{2\pi}\int_0^\pi \sin\theta \mathrm{d}\theta \mathrm{d}\phi}=\frac{1}{4\pi} p ( ω i , ω o ) = ∫ 0 2 π ∫ 0 π s i n θ d θ d ϕ 1 = 4 π 1
pbrt的相位函数定义在PhaseFunction中, 目前只有HGPhaseFunction一种实现.
class PhaseFunction : public TaggedPointer<HGPhaseFunction> {
public:
// PhaseFunction Interface
using TaggedPointer::TaggedPointer;
std::string ToString() const;
PBRT_CPU_GPU inline Float p(Vector3f wo, Vector3f wi) const;
PBRT_CPU_GPU inline pstd::optional<PhaseFunctionSample> Sample_p(Vector3f wo,
Point2f u) const;
PBRT_CPU_GPU inline Float PDF(Vector3f wo, Vector3f wi) const;
};
Henyey–Greenstein相位函数
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Henyey-Greenstein相位函数来自于对测量散射数据的拟合, 其形式如下. 原文中cos θ \cos\theta cos θ
p H G ( cos θ ) = 1 4 π 1 − g 2 ( 1 + g 2 + 2 g cos θ ) 3 2
\begin{equation}
p_{HG}(\cos\theta)=\frac{1}{4\pi}\frac{1-g^2}{(1+g^2+2g\cos\theta)^{\frac{3}{2}}}
\end{equation}
p H G ( cos θ ) = 4 π 1 ( 1 + g 2 + 2 g cos θ ) 2 3 1 − g 2 Henyey-Greenstein中的g g g ( − 1 , 1 ) (-1,1) ( − 1 , 1 ) 1 1 1 ω \omega ω − 1 -1 − 1 − ω -\omega − ω
g = ∫ Θ p ( − ω ⋅ ω ′ ) ( ω ⋅ ω ′ ) d ω ′ = 2 π ∫ 0 π p ( − cos θ ) cos θ sin θ d θ
\begin{equation}
g=\int_\Theta p(-\omega\cdot\omega')(\omega\cdot\omega')\mathrm{d}\omega'=2\pi\int_0^\pi p(-\cos\theta)\cos\theta\sin\theta \mathrm{d}\theta
\end{equation}
g = ∫ Θ p ( − ω ⋅ ω ′ ) ( ω ⋅ ω ′ ) d ω ′ = 2 π ∫ 0 π p ( − cos θ ) cos θ sin θ d θ 逆变换法得到的cos θ \cos\theta cos θ ϵ \epsilon ϵ
cos θ = { − 1 2 g ( 1 + g 2 − ( 1 − g 2 1 + g − 2 g ϵ ) 2 ) g ≠ 0 1 − 2 ϵ g = 0
\begin{equation}
\begin{aligned}
\cos\theta=
\begin{cases}
-\frac{1}{2g}(1+g^2-(\frac{1-g^2}{1+g-2g\epsilon})^2) & g \ne 0\\
1-2\epsilon & g=0
\end{cases}
\end{aligned}
\end{equation}
cos θ = { − 2 g 1 ( 1 + g 2 − ( 1 + g − 2 g ϵ 1 − g 2 ) 2 ) 1 − 2 ϵ g = 0 g = 0 介质
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Medium类定义介质, NanoVDBMedium这一派生类用于NanoVDB格式的介质, 本节不讨论第三方API的调用.
class Medium
: public TaggedPointer< // Medium Types
HomogeneousMedium, GridMedium, RGBGridMedium, CloudMedium, NanoVDBMedium
> {
public:
// Medium Interface
using TaggedPointer::TaggedPointer;
static Medium Create(const std::string &name, const ParameterDictionary ¶meters,
const Transform &renderFromMedium, const FileLoc *loc,
Allocator alloc);
std::string ToString() const;
PBRT_CPU_GPU
bool IsEmissive() const;
PBRT_CPU_GPU
MediumProperties SamplePoint(Point3f p, const SampledWavelengths &lambda) const;
// Medium Public Methods
RayMajorantIterator SampleRay(Ray ray, Float tMax, const SampledWavelengths &lambda,
ScratchBuffer &buf) const;
};
SamplePoint返回介质在空间中某个点处的性质.
struct MediumProperties {
SampledSpectrum sigma_a, sigma_s;
PhaseFunction phase;
SampledSpectrum Le;
};
SampleRay返回多个RayMajorantSegment, 用于存储每一段的σ maj \sigma_{\text{maj}} σ maj RayMajorantIterator可以每次只返回一段, 减小内存开销.
struct RayMajorantSegment {
Float tMin, tMax;
SampledSpectrum sigma_maj;
};
pbrt通过在各个类中组合Medium对象来表示当前介质, 例如Camera中的介质表示相机当前所处位置的介质, Light中的介质初始化时与相机的介质相同, 随着路径追踪的推进而修改. 若介质为空指针则代表位于真空中.
pbrt通过Primitive表示介质的边界, 它会存储MediumInterface用于表示图元两侧的介质. pbrt不会检查介质是否有效, 例如将相机的介质设置为与实际所处的Primitive的某一侧的介质不同是被允许的. 实际只有透明物体需要在相交时将MediumInterface传入SurfaceInteraction.
struct MediumInterface {
// ...
// MediumInterface Public Members
Medium inside, outside;
};
pbrt中类似云的介质也需要一个Primitive来表示范围, 但是并不会改变光线传播路径. 这可以通过将BSDF设置为两侧IOR相同的完美透射实现, 但为了降低复杂度, pbrt通过将材质设置为空指针来表示不改变光线路径.
均匀介质
#
介质中σ a \sigma_a σ a σ s \sigma_s σ s L e L_e L e HomogeneousMedium表示.
class HomogeneousMedium {
public:
// HomogeneousMedium Public Type Definitions
using MajorantIterator = HomogeneousMajorantIterator;
// ...
private:
// HomogeneousMedium Private Data
DenselySampledSpectrum sigma_a_spec, sigma_s_spec, Le_spec;
HGPhaseFunction phase;
};
均匀介质不需要进行空散射, 因此HomogeneousMajorantIterator只存储一段RayMajorantSegment.
PBRT_CPU_GPU
pstd::optional<RayMajorantSegment> Next() {
if (called)
return {};
called = true;
return seg;
}
DDA主值迭代器
#
为保证平均自由程不会过大, 不均匀介质需要分段的σ maj \sigma_{\text{maj}} σ maj
struct MajorantGrid {
// ...
// MajorantGrid Public Members
Bounds3f bounds;
pstd::vector<Float> voxels;
Point3i res;
};
类似于数字微分分析器(digital differential analyzer, DDA)画线算法, pbrt通过DDAMajorantIterator来迭代当前所经过的网格.
PBRT_CPU_GPU
pstd::optional<RayMajorantSegment> Next() {
if (tMin >= tMax)
return {};
// Find _stepAxis_ for stepping to next voxel and exit point _tVoxelExit_
int bits = ((nextCrossingT[0] < nextCrossingT[1]) << 2) +
((nextCrossingT[0] < nextCrossingT[2]) << 1) +
((nextCrossingT[1] < nextCrossingT[2]));
const int cmpToAxis[8] = {2, 1, 2, 1, 2, 2, 0, 0};
int stepAxis = cmpToAxis[bits];
Float tVoxelExit = std::min(tMax, nextCrossingT[stepAxis]);
// Get _maxDensity_ for current voxel and initialize _RayMajorantSegment_, _seg_
SampledSpectrum sigma_maj = sigma_t * grid->Lookup(voxel[0], voxel[1], voxel[2]);
RayMajorantSegment seg{tMin, tVoxelExit, sigma_maj};
// Advance to next voxel in maximum density grid
tMin = tVoxelExit;
if (nextCrossingT[stepAxis] > tMax)
tMin = tMax;
voxel[stepAxis] += step[stepAxis];
if (voxel[stepAxis] == voxelLimit[stepAxis])
tMin = tMax;
nextCrossingT[stepAxis] += deltaT[stepAxis];
return seg;
}
网格介质
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pbrt通过GridMedium表示网格介质. 网格介质中的σ a \sigma_a σ a σ s \sigma_s σ s SampledGrid在不通位置对其施加缩放, s i g m a maj sigma_{\text{maj}} s i g m a maj
for (int z = 0; z < majorantGrid.res.z; ++z)
for (int y = 0; y < majorantGrid.res.y; ++y)
for (int x = 0; x < majorantGrid.res.x; ++x) {
Bounds3f bounds = majorantGrid.VoxelBounds(x, y, z);
majorantGrid.Set(x, y, z, densityGrid.MaxValue(bounds));
}
自发光有两种实现方式, 第一种为设置空间上各个位置的色温, 第二种为设置L e L_e L e
pstd::optional<SampledGrid<Float>> temperatureGrid;
DenselySampledSpectrum Le_spec;
SampledGrid<Float> LeScale;
RGB网格介质
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RGBGridMedium通过RGB网格存储介质信息. 选取区域内的RGB的最大值后转为RGBUnboundedSpectrum, 然后取最大值作为主值, 这样的结果与直接取区域内RGBUnboundedSpectrum的最大值是不同的, 因此pbrt采用RGBUnboundedSpectrum类型而非RGB类型的网格.
class RGBGridMedium {
public:
// RGBGridMedium Public Type Definitions
using MajorantIterator = DDAMajorantIterator;
// ...
private:
// RGBGridMedium Private Members
Bounds3f bounds;
Transform renderFromMedium;
pstd::optional<SampledGrid<RGBIlluminantSpectrum>> LeGrid;
Float LeScale;
HGPhaseFunction phase;
pstd::optional<SampledGrid<RGBUnboundedSpectrum>> sigma_aGrid, sigma_sGrid;
Float sigmaScale;
MajorantGrid majorantGrid;
};
主值通过取区域内各个点的分布中的最大值实现. RGBUnboundedSpectrum本身不支持比较, 因此通过传入lambda函数max来将其转为float.
for (int z = 0; z < majorantGrid.res.z; ++z)
for (int y = 0; y < majorantGrid.res.y; ++y)
for (int x = 0; x < majorantGrid.res.x; ++x) {
Bounds3f bounds = majorantGrid.VoxelBounds(x, y, z);
// Initialize _majorantGrid_ voxel for RGB $\sigmaa$ and $\sigmas$
auto max = [] PBRT_CPU_GPU(RGBUnboundedSpectrum s) {
return s.MaxValue();
};
Float maxSigma_t =
(sigma_aGrid ? sigma_aGrid->MaxValue(bounds, max) : 1) +
(sigma_sGrid ? sigma_sGrid->MaxValue(bounds, max) : 1);
majorantGrid.Set(x, y, z, sigmaScale * maxSigma_t);
}
