Concepts

光的直线传播定律
光的反射定律
- 法向量： $N$
- 入射光： $L$ （归一化）
- 反射光： $R=2(N\cdot L)N-L$
光的折射定律
- Snell's law: $\eta_L\sin\theta_L=\eta_T\sin\theta_T$
- $T=-N\sqrt{1-\frac{\eta^2_L}{\eta^2_T\sin^2\theta_L}\sin^2\theta_L}-\frac{\eta_L}{\eta_T}[L-(N\cdot L)N]=(\frac{\eta_L}{\eta_T}N\cdot L-\sqrt{1-\frac{\eta_L^2}{\eta_T^2}[1-(N\cdot L)^2]})N-\frac{\eta_L}{\eta_T}L$
- Total Internal Reflection
基本假设
- Linearity
- Energy conservation
- Steady state
- No polarization
- No fluorescence or phosphorescence

Radiometry: the science of measuring radiant energy transfer
Radiometric Quantities
- Energy
- Radiant power (total flux) $\Phi,P$ 功率 (J/s, W)
- Irradiance (flux density) $E$ $E$ : total radiant power per unit area incident onto a surface
  - Inverse square law: $E=\frac{\Phi}{4\pi r^2}$
  - Lambert's law: $E=\frac{\Phi\cos\theta}{A}$
- Radiosity (flux density) $B$ : total radiant power per unit area leaving a surface
- Intensity $I$ $I$ : flux density per solid angle $I=\frac{d\Phi}{d\omega}$ $I = \frac{d Φ}{d ω}$
  - isotropic point source: $\Phi=4\pi I$
- Radiance $L$ $L$ : total flux traveling at some point $x$ $x$ in a specified direction $\omega$ $ω$ , per unit area perpendicular to the direction of travel
  - $L(x,\omega)=\frac{d^2\Phi}{dA\cos\theta d\omega}$
  - 沿光线传输方向，radiance 不变（真空），sensors response is proportional to radiance
Solid Angles $\omega$ $ω$ : the solid angle subtended by an object is the surface area of its projection on to the unit spher
- $d\omega=\frac{dA\cos\theta}{r^2}$
- 球坐标系： $\sin\theta d\theta d\phi$
Spectrum: 370nm - 730nm
- SPD: Spectral Power Distribution
- photometry: 物理量转换为视觉感知量

Bidirectional Reflectance Distribution Function: $f_r(\omega_i\rightarrow\omega_r)\equiv\frac{L_r(\omega_r)}{L_i(\omega_i)\cos\theta_i d\omega_i}$ (辐射率 $L$ 和辐照度 $E$ 的比值，对入射光改变的能力，材质，六维函数)
The Reflection Equation(local direct illumination model): $L_r(\omega_r)=\int_{\Omega_i}f_r(\omega_i\rightarrow\omega_r)L_i(\omega_i)\cos\theta_id\omega_i$
The Rendering Equation: $L_o(x,\omega_o)=L_e(x,\omega_o)+\int_{\omega_i\in\Omega_+}f_r(\omega_i,x,\omega_o)L_i(x,\omega_i)\cos\theta_id\omega_i$ $L_{o} (x, ω_{o}) = L_{e} (x, ω_{o}) + \int_{ω_{i} \in Ω_{+}} f_{r} (ω_{i}, x, ω_{o}) L_{i} (x, ω_{i}) cos θ_{i} d ω_{i}$
- 迭代计算： $L_o(x,\omega_o)=L_e(x,\omega_o)+\int_{y\in S}f_r(\omega_i,x,\omega_o)L_o(y(x,\omega_i),-\omega_i)V(x,y)G(x,y)dA_y$
- Ligiting Simulation = Solving the Rendering Equation
  - Monte Carlo Methods (Ray tracing, path tracing...)
  - Finite element methods (Classic radiosity)

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