## 3 Method
### Preliminary  
**Representation**. The NeRF [15] models the structure and appearance of a scene by using a limited collection of images, representing the 3D scene as a continuous 5D function and taking as input the 3D position \(\mathbf{x}=(x,y,z)\) and the viewing direction \(\mathbf{d}=(\theta,\phi)\) and predicting the radiance \(\mathbf{c}(\mathbf{x},\mathbf{d})=(r,g,b)\) and the volume density \(\sigma(\mathbf{x})\). The color depends on both the 3D position \(\mathbf{x}\) and the viewing direction \(\mathbf{d}\) to capture view-dependent effects, while the density depends only on \(\mathbf{x}\) to maintain geometric consistency. The NeRF is typically parameterized by a multi-layer perceptron (MLP):  
\[(\mathbf{c},\sigma)=F_{\theta}(\gamma_{x}(\mathbf{x}),\gamma_{d}(\mathbf{d})) \tag{1}\]  
where \(F_{\theta}\) represents the MLP with weight \(\theta\). \(\gamma(\cdot)\) is the position encoding that maps each element of the vector to a higher-dimensional frequency \(L\) space:  
\[\gamma_{L}(x)=\left[\sin\pi x,\cos\pi x,\ldots,\sin 2^{L-1}\pi x,\cos 2^{L-1} \pi x\right]^{T} \tag{2}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Representation** & **Method** & **Taxonomy** & **Training Steps** & **Training Time** \\ \hline \multirow{4}{*}{NeRF} & S-NeRF [11] & Reconstruction & 100k & 8 h \\  & Sat-NeRF [16] & Reconstruction & 300k & \(\sim\)10–20 h \\  & Season-NeRF [37] & Editing & 50k & \(\sim\)8 h \\  & SpS-NeRF [38] & Few-shot, depth & 30k & 2 h \\ \multirow{2}{*}{SDF} & \multirow{2}{*}{Sat-Mesh [39]} & \multirow{2}{*}{Reconstruction} & \multirow{2}{*}{300k} & \multirow{2}{*}{\(\sim\)8 h} \\  & & & & \\ \end{tabular}
\end{table} 
Table 1: Comparison of training time of multi-view satellite image reconstruction methods based on neural radiance fields. This table shows the training time of multi-view satellite image reconstruction algorithms for different scene representation techniques and their diverse application scenarios. The training time data are obtained from the respective original papers. It should be noted that it is not fair to directly compare the training times of different methods due to different computing devices, but this does not affect the overall conclusion that the satellite radiance field method requires a lot of time to train. In the implementation, \(\mathbf{x}\) is first fed into the MLP, \(\sigma\) and intermediate features are output, and then the feature and position encoding of the viewing direction are input into an additional fully connected layer to predict the color \(\mathbf{c}\), as shown in Figure 2. Therefore, the volume density is determined by the spatial position \(\mathbf{x}\), while the color \(\mathbf{c}\) is determined by both the spatial position and the viewing direction:  
\[(\sigma,\mathbf{z})=F_{\theta_{1}}(\gamma_{\mathbf{x}}(\mathbf{x})) \tag{3}\]  
\[\mathbf{c}=F_{\theta_{2}}(\gamma_{\mathbf{d}}(\mathbf{d}),\mathbf{z}) \tag{4}\]  
where \(\theta_{1}\) and \(\theta_{2}\) are the MLP parameters and \(\gamma_{\mathbf{x}}\) and \(\gamma_{\mathbf{d}}\) are the position encoding functions applied to each value in \(\mathbf{x}\) and \(\mathbf{d}\), respectively. **Render. ** To render each pixel in the image, a camera ray \(\mathbf{r}(t)=\mathbf{o}+t\mathbf{d}\) is projected from the camera origin \(\mathbf{o}\) along the direction \(\mathbf{d}\) toward the scene, as shown in Figure 3. According to volume rendering [15], the NeRF computes samples along the ray and accumulates the color of the sampled points across the camera ray to visualize the scene from any given camera position.