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From Optical to Radio Splatting

The success of 3DGS in computer vision raises a natural question: can we replace the optical colour $\mathbf{c}_k$ with an RF attribute --- received power, path loss, or complex channel gain --- and use the same splatting framework to reconstruct radio environments?

Zhang et al. (2024) answered this affirmatively with RF-3DGS, a method that adapts 3D Gaussian Splatting to model "radio radiance fields." The key modifications are: (1) replacing RGB colour with dB-scale received power, (2) using image-based initialisation instead of structure-from-motion point clouds, and (3) adapting the loss function and density control for the RF domain where measurements are sparse and the "image" is a power map.

Definition:
Radio Radiance Field

A radio radiance field maps every 3D point $\mathbf{p} \in \ntn{tgt_region}$ and direction $\hat{\mathbf{d}} \in \mathbb{S}^2$ to a received power $P(\mathbf{p}, \hat{\mathbf{d}})$ (in dBm). In the RF-3DGS framework, this field is represented by a collection of Gaussians:

$P_{\text{dB}}(\mathbf{p}, \hat{\mathbf{d}}) = 10\log_{10}\!\left(\sum_{k=1}^N p_k(\hat{\mathbf{d}})\,\alpha_k\,G_k(\mathbf{p})\,T_k(\mathbf{p})\right),$

where $p_k(\hat{\mathbf{d}})$ is the directional power pattern of the $k$ -th Gaussian (replacing colour), and $T_k(\mathbf{p}) = \prod_{j<k}(1 - \alpha_j G_j(\mathbf{p}))$ is the accumulated transmittance (attenuation from preceding scatterers).

Definition:
RF Gaussian Primitive

An RF Gaussian primitive extends the optical 3DGS primitive with RF-specific attributes:

$(\boldsymbol{\mu}_k, \boldsymbol{\Sigma}_k, \alpha_k, p_k, \phi_k),$

where:

$\boldsymbol{\mu}_k, \boldsymbol{\Sigma}_k, \alpha_k$ retain their geometric meaning (position, shape, opacity),
$p_k \in \mathbb{R}_+$ is the RF power contribution (linear scale),
$\phi_k \in [0, 2\pi)$ is the phase (optional, needed for coherent channel reconstruction but not for power-map prediction).

For power-only prediction, the feature $\mathbf{f}_k = p_k$ is a scalar. For full channel reconstruction, $\mathbf{f}_k = (p_k, \phi_k)$ or equivalently the complex gain $h_k = \sqrt{p_k}\,e^{j\phi_k}$ .

Theorem: RF Rendering via Gaussian Splatting

Given a set of RF Gaussians $\mathcal{G} = \{(\boldsymbol{\mu}_k, \boldsymbol{\Sigma}_k, \alpha_k, p_k)\}_{k=1}^N$ , the predicted received power at receiver location $\ntn{rx_pos}$ from transmitter $\ntn{tx_pos}$ is:

$\hat{P}(\ntn{rx_pos} \mid \ntn{tx_pos}) = \sum_{k=1}^N p_k \, \alpha_k \, \mathcal{K}(\ntn{tx_pos}, \boldsymbol{\mu}_k) \, G_k'(\ntn{rx_pos}) \, T_k(\ntn{rx_pos}),$

where $\mathcal{K}(\ntn{tx_pos}, \boldsymbol{\mu}_k)$ encodes the Tx-to-scatterer contribution (a function of distance and the Gaussian's directional pattern) and $G_k'$ is the projected Gaussian at the receiver "image plane."

This is a direct analog of the optical splatting equation (Definition DDifferentiable Rasterisation (Splatting)) with:

Optical 3DGS	RF-3DGS
Camera	Receiver
RGB colour $\mathbf{c}_k$	Power $p_k$
View direction	Tx-Rx link direction
Photometric loss	Power prediction loss

Each Gaussian acts as a virtual scatterer that intercepts energy from the transmitter and re-radiates it toward the receiver. The alpha-compositing handles occlusion: a Gaussian behind an opaque obstacle contributes less because the transmittance $T_k$ is small.

Proof

Scatterer contribution

The received power from the $k$ -th Gaussian is proportional to (1) the power it intercepts from the Tx, encoded in $\mathcal{K}(\ntn{tx_pos}, \boldsymbol{\mu}_k)$ , (2) its opacity $\alpha_k$ and spatial footprint $G_k'(\ntn{rx_pos})$ , and (3) the fraction of power that reaches the receiver without being absorbed by intermediate Gaussians, $T_k(\ntn{rx_pos})$ .

Summation

Summing over all Gaussians yields the total received power. The front-to-back ordering ensures that occluded scatterers contribute less. When all $\alpha_k$ are small, $T_k \approx 1$ and the model reduces to a simple superposition of scatterers.

dB conversion

The predicted power in dBm is $\hat{P}_{\text{dB}} = 10\log_{10}(\hat{P})$ , which is the quantity compared to measurements. $\blacksquare$

Definition:
Image-Based Initialisation for RF-3DGS

Unlike optical 3DGS which initialises Gaussians from a structure-from-motion (SfM) point cloud, RF-3DGS uses image-based initialisation:

Measurement grid: Place initial Gaussians at a regular grid covering the environment of interest.
Power-weighted initialisation: Set the initial power $p_k^{(0)}$ proportional to the interpolated measurement power at $\boldsymbol{\mu}_k$ .
Uniform scale: Initialise all scales $\mathbf{s}_k$ to the grid spacing, and opacities $\alpha_k$ to a small uniform value.

This initialisation avoids the need for a visual SfM pipeline, which is unavailable in RF-only settings. When camera images ARE available (as in RFCanvas, Section 26.3), SfM can provide a better initialisation.

Example: RF-3DGS Training for Indoor Power Map

Consider an indoor office environment of $20 \times 15$ m with a single transmitter at $f_0 = 3.5$ GHz. We have $M = 200$ received power measurements at known locations. Describe how RF-3DGS reconstructs the full power map.

Solution

Initialisation

Place $N_0 = 2000$ Gaussians on a $50 \times 30$ grid (spacing $\approx 0.4$ m). Initialise power attributes from nearest-neighbour interpolation of measurements. Set $\alpha_k = 0.1$ , $\mathbf{s}_k = (0.4, 0.4, 0.4)$ m.

Training

For each training iteration, select a batch of measurement locations. Render the predicted power at those locations by evaluating the splatting equation. Compute the loss:

$\mathcal{L} = \frac{1}{|\mathcal{B}|}\sum_{m \in \mathcal{B}} \bigl(\hat{P}_{\text{dB}}(\ntn{rx_pos}_m) - P_{\text{dB}}^{\text{meas}}(\ntn{rx_pos}_m)\bigr)^2.$

Update all Gaussian parameters via Adam. Every 50 iterations, apply adaptive density control.

Result

After $\sim 3000$ iterations (a few minutes on a GPU), the optimised $\sim 5000$ Gaussians predict power at arbitrary locations with mean absolute error $\sim 3$ -- $5$ dB. The Gaussians naturally cluster around walls, furniture, and other scatterers, providing an interpretable "scattering map."

RF-3DGS Training Convergence

Observe how the training loss (MSE in dB) decreases as the number of Gaussians adapts through densification and pruning. Compare with a fixed-Gaussian baseline.

Parameters

Initial Gaussians

N_0

200

Number of measurements

M

200

Adaptive density control

Common Mistake: Loss Function in dB vs Linear Scale

Mistake:

Computing the training loss in linear power scale ( $\hat{P} - P_{\text{meas}})^2$ ) when measurements span a large dynamic range.

Correction:

RF measurements typically span 40--80 dB of dynamic range. A loss function in linear scale is dominated by the few strongest measurements and effectively ignores weak signals. The loss MUST be computed in dB scale: $\mathcal{L} = (\hat{P}_{\text{dB}} - P_{\text{dB}}^{\text{meas}})^2$ . This ensures that a 3 dB error at $-90$ dBm is penalised equally to a 3 dB error at $-50$ dBm. Zhang et al. report that dB-scale loss reduces the mean prediction error by 5--8 dB compared to linear-scale loss.

⚠️Engineering Note

Measurement Requirements for RF-3DGS

RF-3DGS requires spatially distributed power measurements with known positions. In practice, these come from:

Drive tests with GPS-equipped receivers (outdoor),
Robot-mounted or drone-mounted receivers scanning a grid (indoor),
Crowdsourced smartphone measurements (urban, but with position uncertainty).

The spatial sampling density determines the achievable resolution. A rule of thumb from Zhang et al.: the measurement spacing should be at most $\sim 2\lambda$ for good reconstruction. At $f_0 = 3.5$ GHz ( $\lambda \approx 8.6$ cm), this means measurements every $\sim 17$ cm --- feasible with robotic platforms but challenging with manual data collection.

Practical Constraints

•
Receiver position accuracy must be better than $\lambda/4$ for coherent reconstruction
•
Minimum 100 measurements per $10 \times 10$ m area for 5 dB accuracy

Quick Check

Why does RF-3DGS use image-based initialisation (regular grid) rather than SfM point cloud initialisation used in optical 3DGS?

SfM is too slow for real-time applications

RF measurements do not produce visual features needed by SfM

A regular grid provides more Gaussians than SfM

SfM point clouds are always too noisy for Gaussian initialisation

Correction:

RF measurements do not produce visual features needed by SfM

SfM relies on matching visual features (SIFT, SuperPoint) across camera images to triangulate 3D points. RF power measurements are scalar values at known locations --- they contain no visual features to match. Without camera images, SfM cannot produce a point cloud. When cameras ARE available (as in RFCanvas), SfM-based initialisation can and should be used.

Radio Radiance Field

A continuous function mapping 3D spatial coordinates and propagation direction to received RF power (or complex channel gain). The RF analog of the optical radiance field that NeRF and 3DGS model. In RF-3DGS, the radio radiance field is represented by a set of Gaussian primitives rather than a neural network.

Related: Splatting

Why This Matters: Channel Prediction from Gaussian Scene Models

RF-3DGS has direct applications in channel prediction for 5G/6G systems. Once a Gaussian scene model is trained from measurement data, it can predict the received power (and potentially the full channel response) at arbitrary Tx-Rx locations --- without ray tracing or additional measurements.

This is valuable for:

Network planning: Predicting coverage maps for base station placement.
Beam management: Predicting which beam direction maximises received power at a moving user.
Digital twins: Maintaining an up-to-date RF model of the environment for simulation and optimisation.

The key advantage over ray tracing is that RF-3DGS does NOT require a detailed geometric model of the environment (CAD drawings, material properties). It learns the effective scattering directly from measurements, capturing effects (diffuse scattering, furniture, vegetation) that ray tracers often miss.

See full treatment in ISAC Fundamentals

Key Takeaway

RF-3DGS adapts 3D Gaussian Splatting to radio propagation by replacing optical colour with dB-scale received power. Key modifications include image-based initialisation (since SfM point clouds are unavailable without cameras), dB-scale loss functions (to handle the large dynamic range of RF measurements), and adapted density control. The trained Gaussian model provides an interpretable, explicit representation of the RF environment that enables real-time channel prediction at novel locations.

RF-3DGS: Gaussian Splatting for Radio Fields