## 回顾

$L_0(\mathbf{v})=\int_{\Omega} \rho(\mathbf{l},\mathbf{v}) \otimes {L}_{i}(\mathbf{l}) (\mathbf{n} \cdot \mathbf{l}) d \omega_{i}$

$\rho(\mathbf{l}, \mathbf{v})=\frac{F(\mathbf{l},\mathbf{h})G(\mathbf{l},\mathbf{v},\mathbf{h})D(\mathbf{h})}{4(\mathbf{n} \cdot \mathbf{l})(\mathbf{n} \cdot \mathbf{v})}$

## Ground truth

$L_0(\mathbf{v})=\int_{\Omega} \rho(\mathbf{l},\mathbf{v}) \otimes {L}_{i}(\mathbf{l}) (\mathbf{n} \cdot \mathbf{l}) d \omega_{i}$
$\approx \frac{1}{N}\sum_{k=1}^N\frac{L_i(\mathbf{l_k})\rho(\mathbf{l_k}, \mathbf{v})(\mathbf{n} \cdot \mathbf{l_k})}{pdf(\mathbf{l_k}, \mathbf{v})}$

$pdf(\mathbf{l_k}, \mathbf{v})=\frac{D(\mathbf{h})(\mathbf{h} \cdot \mathbf{n})}{4(\mathbf{l} \cdot \mathbf{h})}$

## 传统做法

$L_0(\mathbf{v})=\rho(\mathbf{l},\mathbf{v}) (\mathbf{n} \cdot \mathbf{l})\otimes \int_{\Omega} {L}_{i}(\mathbf{l}) d \omega_{i}$

## 基于物理的做法

$L_0(\mathbf{v}) \approx \int_{\Omega} {L}_{i}(\mathbf{l})d \omega_{i} \int_{\Omega} \rho(\mathbf{l},\mathbf{v}) (\mathbf{n} \cdot \mathbf{l}) d \omega_{i}$

$F_{Schlick}(\mathbf{c}_{spec}, \mathbf{l}, \mathbf{h})=\mathbf{c}_{spec}+(1-\mathbf{c}_{spec})(1-\mathbf{l} \cdot \mathbf{h})^5$
$= \mathbf{c}_{spec}(1-(1-\mathbf{l} \cdot \mathbf{h})^5)+(1-\mathbf{l} \cdot \mathbf{h})^5$

$\int_{\Omega} \rho(\mathbf{l},\mathbf{v}) (\mathbf{n} \cdot \mathbf{l}) d \omega_{i}$
$=\mathbf{c}_{spec}\int_{\Omega}\frac{\rho(\mathbf{l}, \mathbf{v})}{F(\mathbf{v},\mathbf{h})}(1-(1-\mathbf{v} \cdot \mathbf{h})^5)(\mathbf{n} \cdot \mathbf{l}) d \omega_{i}$
$+\int_{\Omega}\frac{\rho(\mathbf{l}, \mathbf{v})}{F(\mathbf{v},\mathbf{h})}(1-\mathbf{v} \cdot \mathbf{h})^5(\mathbf{n} \cdot \mathbf{l}) d \omega_{i}$