# Neural Radiance Fields with Hash-Low-Rank Decomposition
## 2 Related Works
### Neural Radiance Field  
In recent years, NeRF [11] has emerged as a groundbreaking approach in the field of novel view synthesis (NVS), driving substantial progress in rendering high-quality images from unseen perspectives. By representing a 3D scene in a compact and implicit manner, NeRF offers a powerful alternative to traditional explicit methods that often struggle with balancing geometric detail and appearance realism. This advancement has set a new standard for how 3D scenes can be efficiently modeled and rendered.  
Figure 1: We proposed a novel VM decomposition-based matrix factorization method specifically designed for the field of 3D reconstruction. In our approach, the feature planes derived from VM decomposition are encoded using multi-hash techniques. These encoded planes are then combined with the feature vectors obtained from VM decomposition via matrix multiplication, ultimately forming a set of hash tables and vectors. Through this decomposition method, redundant parts in the original feature planes can be effectively filtered out by the hash algorithm.  
The core innovation of NeRF lies in its use of a coordinate-based implicit function to represent the radiance field of a scene. Specifically, it employs a multilayer perceptron (MLP) trained through positional encoding, which maps 3D coordinates and view directions to color and density values. Combined with a coarse-to-fine optimization strategy, NeRF effectively captures detailed geometric structures and photorealistic appearance while maintaining a compact model size. This approach enables the generation of high-quality novel views, even for scenes with complex lighting and occlusions.  
However, the reliance on an MLP introduces computational challenges. For every coordinate query, the MLP must be fully evaluated, and since rendering involves integrating over numerous sampled points along each ray, the process entails a massive number of MLP evaluations. This not only prolongs the rendering process, but also significantly increases the computational cost during training. As a result, while NeRF achieves impressive results in terms of quality, its efficiency remains a major bottleneck for practical applications.  
Due to this characteristic, many recent works have sought to transition NeRF's implicit scene representation to explicit formats. These methods increase NeRF's speed by explicitly storing spatial information on corresponding coordinates to reduce query time. For example, PlenOctree [6] employs an octree structure for efficient scene modeling using spherical harmonics to represent color and storing volumetric density features at the octree nodes, achieving ultra-fast explicit scene reconstruction without using any neural network. Similarly, DVGO [4] adopts a voxel grid as the scene representation, storing implicit scene features at the voxel grid nodes and decoding these features with activation function or neural network. These approaches replace NeRF's implicit scene representation with specialized spatial data structures, reducing redundant feature queries while leveraging the adaptive properties of spatial data structures to increase local precision and enhance reconstruction quality.