### 辐射度量学概念辨析

$Q$

$[J = Joule]$

$\phi \equiv \frac{dQ}{dt}$

$[W = Watt] [lm = lumen]$

Energy per unit time

$I(\omega) \equiv \frac{d\phi}{d\omega}$

$[cd = candela = \frac{lm}{sr}]$

Radiant flux per unit solid angle

$E(x) \equiv \frac{d\phi(x)}{dA}$

$[lux = \frac{lm}{m^2}]$

The flux per unit area incident on a surface point.

$L(p, \omega) \equiv \frac{d^2\phi(p, \omega)}{d\omega dA \cos\theta}$

$[nit = \frac{cd}{m^2} = \frac{lm}{sr \space m^2}]$

The flux emitted, reflected, transmitted or received by a surface, per unit solid angle, per projected unit area.

### 渲染方程简化

$$\left[ \begin{matrix} a_o \\ b_o \\ c_o \end{matrix} \right] = \left[ \begin{matrix} a_e \\ b_e \\ c_e \end{matrix} \right] + K_{3\times3} \left[ \begin{matrix} a_o \\ b_o \\ c_o \end{matrix} \right]$$

$$K= \left[ \begin{matrix} 0 & k_{ba} & k_{ca} \\ k_{ab} & 0 & k_{cb} \\ k_{ac} & k_{bc} & 0 \end{matrix} \right]$$

### Lambert Diffuse BRDF 推导

$$f_r(\omega_i, \omega_o) = \frac {dL_o}{dE_i} = \frac {dL_o}{L_i\cos\theta_id\omega_i}$$

$$dL_o = f_r(\omega_i, \omega_o)L_i\cos\theta_id\omega_i$$

$$dE_o = \int_\Omega dL_o\cos\theta_od\omega_o \ =\int_\Omega(f_r(\omega_i, \omega_o)L_i\cos\theta_id\omega_i)\cos\theta_od\omega_o \ =L_i\cos\theta_id\omega_i\int_\Omega f_r(\omega_i, \omega_o)\cos\theta_od\omega_o$$

$$dE_o \le dE_i = L_i\cos\theta_id\omega_i$$

$$\int_\Omega f_r(\omega_i, \omega_o)\cos\theta_od\omega_o \le 1$$

$$\int_\Omega C\cos\theta_od\omega_o = \alpha$$

$$f_r(\omega_i, \omega_o) = C = \frac\alpha\pi$$