## 3 Method
### Loss Function  
Traditional NeRF training paradigms usually only optimize point-wise MSE loss, ignoring the rich structural information contained in pixel groups. High-resolution remote sensing images contain rich structural textures, and different types of targets, such as airports, ports, buildings, green spaces, sea surfaces, etc., have different texture structures. Inspired by the above, in addition to MSE loss, we introduce a new structural supervision loss \(\mathcal{L}_{\text{SSIM}}\) based on stochastic structural similarity [42] to capture the structural information in the input satellite images. The stochastic structural similarity (S3 IM) index measures the similarity between two sets of pixels and captures non-local structural similarity information from randomly sampled pixels. S3 IM is model-independent and can be universally applied to all types of neural field methods with limited coding and computational costs. It has been demonstrated in [42] that S3 IM is very effective in improving the NeRF learning of correct colors and geometric structures and is robust even when the input is sparse and the scene content is complex. S3 IM builds on SSIM and is a stochastic variant of SSIM to preserve positional relationships in local blocks in random training. SSIM is a commonly used image quality metric used to evaluate local structural similarities between images. SSIM is believed to have a good correlation with the quality perception of the human visual system and is widely used to evaluate the quality of NeRF-rendered images. Assuming that \(x\) and \(y\) represent two different image blocks, SSIM consists of three indicators: luminance, contrast, and structure:  
\[\text{SSIM}(x,y)=l(x,y)c(x,y)s(x,y) \tag{17}\]  
In the proposed method, we quantify the image quality by evaluating three key components: luminance \(l(x,y)\), contrast \(c(x,y)\), and structure \(s(x,y)\). These components are defined with respect to the local statistics of an image as follows:  
The luminance component \(l(x,y)\) is calculated by using the equation  
\[l(x,y)=\frac{2\mu_{x}\mu_{y}+C_{1}}{\mu_{x}^{2}+\mu_{y}^{2}+C_{1}} \tag{18}\]  
where \(\mu_{x}\) and \(\mu_{y}\) represent the mean luminance values of two images (or two regions within an image) being compared, and \(C_{1}\) is a constant to avoid division by zero. The contrast component \(c(x,y)\) is represented by  
\[c(x,y)=\frac{2\sigma_{x}\sigma_{y}+C_{2}}{\sigma_{x}^{2}+\sigma_{y}^{2}+C_{2}} \tag{19}\]  
where \(\sigma_{x}\) and \(\sigma_{y}\) are the standard deviations of the luminance values, indicating the image contrast levels, and \(C_{2}\) is a constant for stabilization. Lastly, the structure component \(s(x,y)\) is defined as  
\[s(x,y)=\frac{\sigma_{xy}+C_{3}}{\sigma_{x}\sigma_{y}+C_{3}} \tag{20}\]  
with \(\sigma_{xy}\) denoting the covariance between the two images (or regions), indicating the structural similarity, and \(C_{3}\) is a constant to ensure mathematical stability. S3 IM is computed on a local block containing some position information:  
\[\text{S3 IM}\big{(}\hat{R},R\big{)}=\frac{1}{M}\sum_{m=1}^{M}\text{SSIM}\Big{(}P ^{(m)}\big{(}\hat{\mathbf{C}}\big{)},P^{(m)}(\mathbf{C})\Big{)} \tag{21}\]  
where \(P\big{(}\hat{\mathbf{C}}\big{)}\) and \(P(\mathbf{C})\) denote a rendering block and the corresponding ground truth image block randomly formed by \(B\) rays/pixels in batch \(R\). \(\text{SSIM}\Big{(}P^{(m)}\big{(}\hat{\mathbf{C}}\big{)},P^{(m)}(\mathbf{C}) \Big{)}\) denotes the computation of SSIM with kernel size of \(\text{K}\times\text{K}\) and step size of s on the rendering block and the corresponding ground truth patches. The final S3 IM is obtained by averaging the M-estimated SSIM values. Although S3 IM requires multiple computations of SSIM over multiple random patches, it only needs to be backpropagated once per iteration, and thus it does not incur significant additional computational costs.