RF-NeRF Variants

WiNeRT (Orekondy et al., 2023) extends NeRF for wireless by incorporating multi-bounce ray tracing within the neural rendering framework:

1. Primary rays are cast from Tx to Rx (as in NeRF$^2$).
2. Reflection rays are generated at high-density surfaces using learned normal vectors: $\mathbf{d}_r = \mathbf{d}_i - 2(\mathbf{d}_i \cdot \hat{\mathbf{n}})\hat{\mathbf{n}}$.
3. Diffraction rays are added at edges detected by density gradients.

The total received signal is:

$$\hat{S} = \sum_{p \in \mathcal{P}} w_p \cdot \hat{S}_p \cdot e^{-j 2\pi f \ell_p / c},$$

where $\mathcal{P}$ is the set of traced paths, $\ell_p$ the path length, and $w_p$ a learned path weight.

R-NeRF extends the RF-NeRF framework to environments with reconfigurable intelligent surfaces (RIS). The RIS is modelled as a controllable reflecting layer with element-wise phase shifts $\boldsymbol{\phi} \in [0, 2\pi)^{N_{\mathrm{RIS}}}$:

$$\hat{S}_{\mathrm{RIS}} = \sum_{i=1}^{N} T_i \alpha_i \, \rho_i \sum_{n=1}^{N_{\mathrm{RIS}}} e^{j\phi_n}\, G(\mathbf{x}_i, \mathbf{x}_n^{\mathrm{RIS}})\, G(\mathbf{x}_n^{\mathrm{RIS}}, \mathbf{x}_{\mathrm{rx}}),$$

where $G$ denotes the free-space Green's function. The RIS phase configuration $\boldsymbol{\phi}$ becomes a controllable input to the neural field, enabling joint scene reconstruction and RIS optimisation.

VoxelRF replaces the MLP with a dense or sparse voxel grid storing density $\sigma$ and features at each voxel. For a query point $\mathbf{x}$, trilinear interpolation retrieves the local density and feature, which a small MLP (1--2 layers) decodes into the signal contribution.

Advantage: Eliminating the deep MLP evaluation at each sample reduces inference time by $10\times$--$100\times$.

Disadvantage: Memory scales as $\mathcal{O}(N_x N_y N_z)$. For a $50 \times 50 \times 10$ m indoor scene at 5 cm resolution, the grid has $10^6 \times 200 = 2 \times 10^8$ voxels --- requiring pruning or sparse storage.

NeRF-APT inverts the typical NeRF$^2$ workflow: instead of predicting RSS at known Rx locations given known Tx positions, it localises unknown Tx positions from RSS measurements at known Rx locations.

The approach treats the Tx position $\mathbf{x}_{\mathrm{tx}}$ as a learnable parameter and jointly optimises:

$$\min_{\theta, \{\mathbf{x}_{\mathrm{tx},k}\}} \sum_{k,j} \bigl|\hat{P}_{k,j}^{(\mathrm{dB})}(\theta, \mathbf{x}_{\mathrm{tx},k}) - P_{k,j}^{(\mathrm{dB})}\bigr|^2.$$

The scene representation $\theta$ and Tx positions are learned simultaneously, analogous to how COLMAP jointly estimates camera poses and scene structure in optical NeRF.

DART (Huang et al., 2024) extends NeRF for radar by incorporating the Doppler dimension:

1. Input: Radar returns $y(r, \theta, f_D)$ in range--angle--Doppler space, from multiple viewpoints.

2. Scene MLP: Predicts radar cross-section $\sigma_{\mathrm{RCS}}(\mathbf{x})$ and velocity $\mathbf{v}(\mathbf{x})$ at each 3D point.

3. Doppler rendering: The Doppler shift at point $\mathbf{x}$ along ray direction $\hat{\mathbf{d}}$ is:

$$f_D(\mathbf{x}) = \frac{2 \mathbf{v}(\mathbf{x}) \cdot \hat{\mathbf{d}}}{\lambda},$$

and the rendered radar cube is:

$$\hat{y}(r, \theta, f_D) = \sum_{i: t_i \approx r} T_i \alpha_i \, \sigma_{\mathrm{RCS},i} \, \delta\!\bigl(f_D - f_{D,i}\bigr).$$

ISAR-NeRF adapts neural fields for coherent SAR / inverse SAR imaging. The forward model maps the 3D scene reflectivity $\rho(\mathbf{x})$ to the measured phase history:

$$s(u, f) = \int \rho(\mathbf{x})\, \exp\!\bigl(-j \tfrac{4\pi f}{c} \|\mathbf{x}_{\mathrm{ap}}(u) - \mathbf{x}\|\bigr)\, d\mathbf{x},$$

where $u$ is the aperture coordinate. ISAR-NeRF parameterises $\rho(\mathbf{x}) = f_\theta(\gamma(\mathbf{x}))$ and minimises:

$$\mathcal{L}(\theta) = \sum_{u, f} \bigl|\hat{s}(u, f; \theta) - s_{\mathrm{meas}}(u, f)\bigr|^2 + \lambda_{\mathrm{TV}} \|\nabla \rho_\theta\|_1.$$

The MLP's spectral bias acts as an implicit regulariser, suppressing grating-lobe artifacts in sparse-aperture regimes.