This paper presents a comprehensive survey of Neural Radiance Fields (NeRFs) for multi-view 3D reconstruction and novel view synthesis. NeRFs represent a 3D scene as a continuous volumetric function, typically modeled by a Multi-Layer Perceptron (MLP) denoted as $F_\Theta$. This MLP takes a 3D coordinate $\mathbf{x} = (x, y, z)$ and a 2D viewing direction $\mathbf{d} = (\theta, \phi)$ as input, and outputs a predicted color $\mathbf{c} = (r, g, b)$ and volume density $\sigma$.

The core methodology of the original NeRF, which forms the basis for many variants discussed, involves two main components:

1. Implicit Scene Representation: The scene is encoded by the MLP $F_\Theta$. Positionally encoded inputs are often used to enable the MLP to represent high-frequency details. For a 3D point $\mathbf{x}$, its position encoding is $\gamma(\mathbf{x}) = (\sin(2^0\pi\mathbf{x}), \cos(2^0\pi\mathbf{x}), \dots, \sin(2^{L-1}\pi\mathbf{x}), \cos(2^{L-1}\pi\mathbf{x}))$, where $L$ is the number of frequencies. Similarly for the viewing direction $\mathbf{d}$. The MLP takes $\gamma(\mathbf{x})$ as input to predict $\sigma$ and an intermediate feature vector, which is then concatenated with $\gamma(\mathbf{d})$ to predict $\mathbf{c}$.
2. Volume Rendering: To synthesize the color of a pixel along a ray $r(t) = \mathbf{o} + t\mathbf{d}$ originating from camera origin $\mathbf{o}$ in direction $\mathbf{d}$, volume rendering techniques are employed. The expected color $C(\mathbf{r})$ for a ray is computed by numerically integrating color and density values along the ray:

$C(\mathbf{r}) = \int_{t_n}^{t_f} T(t) \sigma(\mathbf{r}(t)) \mathbf{c}(\mathbf{r}(t), \mathbf{d}) dt$
where $T(t) = \exp(-\int_{t_n}^t \sigma(\mathbf{r}(s)) ds)$ is the accumulated transmittance along the ray from $t_n$ to $t$. In practice, this integral is approximated using numerical quadrature, typically stratified sampling and a hierarchical sampling strategy (coarse-to-fine). The ray is divided into $N$ bins, and a sample is taken uniformly from each bin $t_i \sim \mathcal{U}[t_{i-1}, t_i]$. The discrete rendering formula is:
$\hat{C}(\mathbf{r}) = \sum_{i=1}^{N} T_i (1 - \exp(-\sigma_i \delta_i)) \mathbf{c}_i$
where $T_i = \exp(-\sum_{j=1}^{i-1} \sigma_j \delta_j)$ and $\delta_i = t_{i+1} - t_i$.
The training objective is to minimize the squared photometric error between rendered and ground-truth pixel colors:
$\mathcal{L} = \sum_{\mathbf{r} \in \mathcal{R}} \|\hat{C}(\mathbf{r}) - C(\mathbf{r})\|^2_2$

The survey categorizes various advancements and extensions of the original NeRF into several themes:

• Efficiency Enhancement: Addresses slow training and rendering. Methods include explicit data structures (e.g., hash grids in Instant-NGP, voxel grids), knowledge distillation, and optimized sampling strategies.
• Representation Capacity and Generalization: Aims to improve the ability to model complex scenes and generalize to unseen views. This includes methods using different scene representations (e.g., Gaussian splats, tri-plane representations), conditioning on input images, or meta-learning approaches.
• Geometric Consistency and Quality: Focuses on improving the fidelity of the reconstructed 3D geometry and visual quality, often by incorporating geometric priors (e.g., depth maps, 3D meshes) or more robust rendering techniques.
• Dynamic and Deformable Scenes: Extends NeRF to handle scenes with motion, typically by adding a time dimension to the MLP input or by learning explicit deformation fields.
• Pose Estimation and Illumination: Addresses challenges in scenes with unknown camera poses or varying lighting conditions, often by jointly optimizing pose parameters or learning illumination models.
• Applications: Highlights diverse applications beyond novel view synthesis, such as 3D reconstruction, augmented reality, virtual reality, robotic navigation, and content creation.

The survey concludes by outlining future research directions, including improving generalization, robustness to pose errors, real-time performance, composability of scenes, and addressing the lack of interpretable semantic information in current NeRF models.