跳转至

渲染方程

前置的内容为 辐射度量学。

Reflection at a Point

Radiance from direction \(\omega_i\) turns into the power \(E(\omega_i)\) that \(\mathrm{d}A\) receives. Then power \(E(\omega_i)\) will become the radiance to any other direction \(\omega_r\).

Reflection at a Point
Reflection at a Point

  • Differential irradiance incoming: \(\mathrm{d}E(\omega_i)=L_i(\omega_i) \cos \theta_i \mathrm{d} \omega_i\)
  • Differential radiance exiting due to \(\mathrm{d}E(\omega_i)\): \(\mathrm{d}L_r(\omega_r)\)

BRDF

The Bidirectional Reflectance Distribution Function (BRDF) represents how much light is reflected into each outgoing direction \(\omega_r\) from each incoming direction.

BRDF
BRDF

\[ f_r(\omega_i \rightarrow \omega_r)=\frac{\mathrm{d}L_r(\omega_r)}{\mathrm{d}E_i(\omega_i)}=\frac{\mathrm{d}L_r(\omega_r)}{L_i(\omega_i) \cos \theta_i \mathrm{d} \omega_i} \]

The unit is \(sr^{-1}\)。

The Reflection Equation

\[ L_r(p,\omega_r)=\int_{H^2} f_r(p,\omega_i \rightarrow \omega_r)L_i(p,\omega_i) \cos \theta_i \mathrm{d} \omega_i \]

The Rendering Equation

Kajiya 提出 The Rendering Equation，就是自发光 + 反射光。

\[ L_o(p, \omega_o) = L_e(p, \omega_o) + \int_{H^2} f_r(p,\omega_i \rightarrow \omega_o)L_i(p,\omega_i) (n \cdot \omega_i) \mathrm{d} \omega_i \]

其中 \(n \cdot \omega_i = \cos \theta_i\)。

The Rendering Equation (Kajiya 86)
The Rendering Equation (Kajiya 86)

Ray Tracing

渲染方程中，未知的量只有 \(L_o(p, \omega_o)\) 和 \(L_i(p,\omega_i)\)，但某一点的 \(L_i(p,\omega_i)\) 又依赖其他点的 \(L_o(p, \omega_o)\)，形成了递归。积分是线性运算，如果用某个线性算子 \(K\) 改写渲染方程，能得到

\[ L = E + KL \]

可以把 \(K\) 看作某种变换矩阵，解得

\[ L = (I-K)^{-1}E \]

对 \((I-K)^{-1}\) 用二项式定理，或者类似 \(\dfrac{1}{1-x}\) 的泰勒展开得到

\[ \begin{align} L &= (I + K + K^2 + K^3 + \cdots) E\\ &= E + KE + K^2E + K^3E + \cdots \end{align} \]
项 意义
\(E\) Emission directly From light sources
\(KE\) Direct illumination on surfaces
\(K^2E\) One bounce indirect illumination, e.g., Mirrors, Refraction
\(K^3E\) Two bounce indirect illumination
\(\cdots\) \(\cdots\)

光栅化渲染中，前两项比较容易，后面的项就困难了。后来，提出 光线追踪 来解决这个问题。

光线追踪
光线追踪

后面的项计算一定数量后就差不多收敛了，再计算差别也不大了。

参考

相关文章