## 3 Method
### Multi-Resolution Hash Encoding  
Due to the large amount of time required for training, it is difficult to extend the NeRF to handle large-scale scenarios such as cities. Recently, Instant-NGP proposes a learned parameterized multi-resolution hash encoding that is trained simultaneously with the MLP network of the NeRF model. Thanks to the representation capabilities of multi-resolution  
Figure 4: Illumination changes and transient objects in multi-date satellite images. The left half shows scene 214 of the DFC 2019 dataset. The right half clearly reveals illumination changes and the existence of transient objects including cars and shadows through the comparison of local areas of 214 scenes in different satellite images. The red box in the left figure indicates the local area. Figure 5: Network architecture. The inputs to the network are the position \((x,y,z)\) of the 3D point \(\mathbf{x}\), the observation direction \(\mathbf{d}\), the sun direction \(\mathbf{d_{s}}\), and the transient embedding \(\mathbf{t_{j}}\). The output of the network is the albedo \(\mathbf{c_{a}}\), density \(\sigma\), transient scalar \(t\), uncertainty \(\beta\), and ambient color \(\mathbf{a}\). hash encoding, Instant-NGP can accurately represent scenes by using tiny and efficient MLPs, which greatly improves the training and inference speed of the NeRF models. Our work uses multi-resolution hash coding to encode location coordinates to accelerate model training. Multi-resolution hash coding transforms each input 3D point into higher-dimensional concatenated features learned from each resolution level by defining a multi-scale network structure. This structure finds embeddings for input coordinates in O(1) time by using hash tables [41] and interpolates them to obtain feature values for each position coordinate. For different resolution levels, the hash table is graded according to the level from coarse to fine, and at each resolution level, find the corner points of the voxels containing the coordinates, obtain their encodings, and then they are interpolated to create a collection of feature vectors for the queried coordinates, which are concatenated to form a single input embedding. Specifically, multi-resolution hash encoding is structured into \(L\) levels, with each level comprising \(T\) feature vectors, each of dimension \(F\). These levels operate independently and hold the feature vectors at the vertices of a mesh. The resolution of this mesh at each level follows a geometric progression, ranging from the coarsest to the finest resolution:  
\[N_{l}=\lfloor N_{min}\cdot b^{l}\rfloor \tag{12}\]  
\[b=\exp\left(\frac{\ln N_{max}-\ln N_{min}}{L-1}\right) \tag{13}\]  
where \(N_{max}\) represents the maximum resolution and \(N_{min}\) represents the basic resolution. As shown in Figure 6, for each point \(x\in\mathbb{R}^{d}\), first find the grid unit where the point is located and then perform \(d\) linear interpolation on the features at the grid vertices to generate the \(F\)-dimensional feature vector of the point \(x\). Finally, features from multiple resolution levels are concatenated into a single vector. For a given level \(l\), consider that the input coordinates \(x\) belong to \(\mathbb{R}^{d}\), scaled according to the grid resolution and rounded down and up to map to a certain grid cell within level \(l\):  
\[\lfloor x_{l}\rfloor =\lfloor x\cdot N_{l}\rfloor \tag{14}\] \[\lceil x_{l}\rceil =\lceil x\cdot N_{l}\rceil \tag{15}\]  
Figure 6: Example of 2D multi-resolution hash encoding. For a given input coordinate, we find surrounding voxels at \(L=2\) resolution level and assign indices to their corner points by hashing their integer coordinates. For the generated angular index, we look up the corresponding \(F\)-dimensional feature vector from the hash table and perform linear interpolation based on the relative position of the coordinates within each voxel. Finally, the results of each level are concatenated to generate the input y of the MLP. Scene 068 in the DFC 2019 dataset is used as an example.