# Neural Radiance Fields with Hash-Low-Rank Decomposition
## 4 Methods
### Density  
In this section, we introduce the theoretical model for density modeling in our design. Given a 3D spatial coordinate \(d=(x,y,z)\) and a feature grid \(G_{\sigma}\) with a feature dimension of \(R_{\sigma}\), our method queries the vector feature \(F_{\sigma,r_{i}}^{\text{v}}\) of each dimension \(r_{i}\) using the coordinates \(x\), \(y\), and \(z\) in the vectors \(v_{x}\), \(v_{y}\), and \(v_{z}\), respectively. Similarly, it queries the plane feature \(F_{\sigma,r_{i}}^{\text{m}}\) of each dimension using the coordinates \(yz\), \(xz\), and \(xy\) in the matrices \(M_{yz}\), \(M_{xz}\), and \(M_{xy}\), respectively. Subsequently, based on the spatial hash function \(h(x)\) in multi-hash encoding, we input the coordinates corresponding to the plane, retrieve the corresponding index, and extract the features from multiple resolutions in the hash table. These features are then concatenated to form the optimized feature representation \(F_{h(M)}\) for the plane vector. We then obtain the final feature by taking the outer product of these two features.  
The output of this process can be expressed as three sets of features: \(F_{\sigma,r_{i}}(V_{x},h(M_{yz}))\), \(F_{\sigma,r_{i}}(V_{y},h(M_{xz}))\), and \(F_{\sigma,r_{i}}(V_{z},h(M_{xy}))\). Each group represents a combination of vector and plane features corresponding to a specific axis. To simplify notation and better represent these groups, we define a unified expression:  
\[A_{\sigma,r_{i}}^{i}=F_{\sigma,r_{i}}(V_{i},h(M_{xyz-i})), \tag{2}\]  
where \(i\) represents the coordinate axis (\(x\), \(y\), or \(z\)), \(V_{i}\) is the vector feature along the axis, and \(M_{xyz-i}\) denotes the feature plane formed by the other two axes.  
This process essentially captures the relationship between the vector and plane features along each axis, as described in Equation (3).  
\[\begin{split} G_{\sigma}(d)&=\sum_{r=1}^{R_{\sigma }}F_{\sigma,r}^{v,x}\otimes F_{\sigma,r}^{m,yz}+F_{\sigma,r}^{v,y}\otimes F_{ \sigma,r}^{m,xz}+F_{\sigma,r}^{v,z}\otimes F_{\sigma,r}^{m,xy}\\ &=\sum_{r=1}^{R_{\sigma}}F_{\sigma,r}(V_{x},h(M_{yz}))+F_{ \sigma,r}(V_{y},h(M_{xz}))+F_{\sigma,r}(V_{z},h(M_{xy}))\\ &=\sum_{r=1}^{R_{\sigma}}\sum_{m\in\{XYZ\}}A_{\sigma,r}^{m}(x). \end{split} \tag{3}\]  
Figure 2: Overview of the HashRF training pipeline. Given a 3D coordinate \(x=(x,y,z)\), our method separates a single dimension from the other two dimensions for modeling. For the single dimension, we use implicit feature vectors to represent the space, while, for the other two dimensions, we employ a low-rank multi-hash table component as the spatial representation. These two types of representations are then combined after passing through our fine-tuned MLP, ultimately merging into the 3D scene features for the given point.  
We then directly summed the three fused features and introduced non-linearity into the result using an activation function (_softplus_), ultimately obtaining the density \(\sigma\). The overall process is described by Equation (4).  
\[\begin{split}\sigma&=softplus(G_{\sigma}(d))\\ &=\sum_{r=1}^{R_{\sigma}}softplus(\sum_{m\in\{XYZ\}}A_{\sigma,r}^{ m}(x)).\end{split} \tag{4}\]