Domains and measures

$\int_\mathcal{M}f(x)\ dA(x)$

$\sigma_x^\perp(D)=\int_D|\omega\cdot N(x)|\ d\sigma(\omega)$

$T_M(x)=\{y\in \mathcal{R}^3|y\cdot N(x)=0 \}$

$\mathcal{H}_+^2(x)=\{\omega\in \mathcal{S}^2|\omega\cdot N(x)>0\}$

$\mathcal{H}_-^2(x)=\{\omega\in \mathcal{S}^2|\omega\cdot N(x)<0\}$

$\sigma_x^\perp(\mathcal{H}_+^2)=\pi$

The phase space

$\psi=\mathcal{R}^3\times \mathcal{S}^2\times \mathcal{R}^+$

The trajectory space and photon events

$\Psi=\mathcal{R}\times\psi$

Power

Radiant flux (power) is the energy emitted reflected, transmitted or received, per unit time.

$\Phi={dQ\over dt}$

$D(t)=[0,t]\times \mathcal{S}\times \mathcal{S}^2\times \mathcal{R}^+$

$\Phi(t)={dQ(t)\over dt}$

The irradiance is the power per (perpendicular / projected) unit area incident on a surface point.

$E(x)={d\Phi(x)\over dA(x)}$

The radiance (luminance) is the power emitted, reflected, transmitted or received by a surface, per unit solid angle, per projected unit area.

$L(x,\omega)={d^2\Phi(x,\omega)\over dA_\omega^\perp(x)\ d\sigma(\omega)}$

$L(x,\omega)={d^2\Phi(x,\omega)\over|\omega\cdot N(x)|\ dA(x)\ d\sigma(\omega)}$

$dA_\omega^\perp(x)=|\omega\cdot N(x)|\ dA(x)$

$L(x,\omega)={d^2\Phi(x,\omega)\over dA(x)\ d\sigma_x^\perp(\omega)}$

$L_\lambda={dL\over d\lambda}$

$L_\lambda(x,\omega,\lambda)={d^3\Phi(x,\omega)\over dA(x)\ d\sigma_x^\perp(\omega)\ d\lambda}$

$L:\mathcal{M}\times \mathcal{S}^2\rightarrow \mathcal{R}$

$L:R^3\times \mathcal{S}^2\rightarrow \mathcal{R}$

$L_i(x,\omega)=L_o(x,-\omega)$

The bidirectional scattering distribution function

$dE(\omega_i)=L_i(\omega_i)\ d\sigma^\perp(\omega_i)$

$dL_o(\omega_o)\propto dE(\omega_i)$

$f_s(\omega_i\rightarrow\omega_o)={dL_o(\omega_o)\over dE(\omega_i)}={dL_o(\omega_o)\over L_i(\omega_i)\ d\sigma^\perp(\omega_i)}$

The scattering equation

$dL_o(\omega_o)=L_i(\omega_i)f_s(\omega_i\rightarrow\omega_o)\ d\sigma^\perp(\omega_i)$

$L_o(\omega_o)=\int_{S^2}L_i(\omega_i)f_s(\omega_i\rightarrow\omega_o)\ d\sigma^\perp(\omega_i)$

The BRDF and BTDF

BSDF并非辐射度量学中的标准概念。通常，散射光会被拆分为反射光(reflected)和透射光(transmitted)两部分，由此也就有了双向反射分布函数(bidirectional reflectance distribution function，BRDF) $$f_r$$ 和双向透射分布函数(bidirectional transmittance distribution function，BTDF) $$f_t$$

$f_r:\mathcal{H}_i^2\times\mathcal{H}_r^2\rightarrow\mathcal{R}$

$f_t:\mathcal{H}_i^2\times\mathcal{H}_t^2\rightarrow\mathcal{R}$

$f_r(\omega_i\rightarrow\omega_o)=f_r(\omega_o\rightarrow\omega_i),\forall\omega_i,\omega_o$

$\int_{\mathcal{H}_o^2}f_r(\omega_i\rightarrow\omega_o)\ d\sigma^\perp(\omega_o)\le1,\forall\omega_i\in\mathcal{H}_i^2$

Angular parameterizations of the BSDF

\begin{aligned} \cos\theta=\omega\cdot N\\ \cos\phi=\omega\cdot T \end{aligned}

$d\sigma(\omega)\equiv\ \sin\theta\ d\theta\ d\phi\equiv\ d(-\cos\theta)\ d\phi$

\begin{aligned} d\sigma^\perp(\omega) &\equiv|\cos\theta|\sin\theta\ d\theta\ d\phi\\ &\equiv|\cos\theta|\ d(-\cos\theta)\ d\phi\\ &\equiv\sin\theta\ d\sin\theta\ d\phi\\ &\equiv{1\over2}\ d(-\cos^2\theta)\ d\phi\\ &\equiv{1\over2}\ d\sin^2\theta\ d\phi \end{aligned}

$\int_{S^2}\ d\sigma^\perp(\omega_i)=\int_0^{2\pi}\int_0^\pi|\cos\theta|\sin\theta\ d\theta\ d\phi$

$L_o(\theta_o,\phi_o)=\int_0^{2\pi}\int_0^\pi L_i(\theta_i,\phi_i)f_s(\theta_i,\phi_i,\theta_o,\phi_o)|\cos\theta_i|\sin\theta\ d\theta_i\ d\phi_i$

• 首先 $$(\theta,\phi)$$​ 是一个局部的方向表示，因为这两个角度都依赖于表面法线
• 在涉及多个表面点的分析中，采用参数表示并不直观
• 参数表示引入了三角函数项，而这部分实际上又是由向量点积实现的
• 参数表示还依赖于切向量 $$T$$ 以规定一个初始的azimuthal angle，而这并没有任何物理意义
posted @ 2021-10-19 21:42  Vel'Koz  阅读(210)  评论(1编辑  收藏  举报