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The RF Rendering Equation

We now derive the complete RF volume rendering equation from first principles, connecting the neural NeRF framework to the physics-based forward model of Chapters 5--6. The central question is: under what conditions does neural volumetric rendering agree with the Born-approximation forward model $\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}$ ?

The answer reveals both the power and the limitations of the NeRF approach for RF imaging.

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Theorem: RF Volume Rendering Equation

For a monochromatic RF signal at frequency $f$ , the received complex baseband signal along a ray $\mathbf{r}(t) = \mathbf{o} + t\mathbf{d}$ from the transmitter at origin $\mathbf{o}$ is:

$\hat{S}(\mathbf{r}, f) = \int_{t_n}^{t_f} T(t, f)\,\sigma(\mathbf{r}(t))\, \rho(\mathbf{r}(t), \mathbf{d}, f)\, e^{-j\kappa \cdot 2t}\,dt,$

where $\rho(\mathbf{x}, \mathbf{d}, f) \in \mathbb{C}$ is the complex reflectivity, the factor $e^{-j\kappa \cdot 2t}$ captures round-trip propagation phase ( $\kappa = 2\pi f/c$ ), and the RF transmittance is:

$T(t, f) = \exp\!\left(-\int_{t_n}^{t} \alpha(\mathbf{r}(s), f)\,ds\right),$

with $\alpha(\mathbf{x}, f)$ the frequency-dependent attenuation coefficient.

Each point along the ray contributes a complex phasor to the received signal. The transmittance $T$ accounts for cumulative absorption (energy lost traversing walls). The reflectivity $\rho$ determines how much and at what phase the material scatters. The exponential carries the round-trip propagation delay.

Proof

Physics derivation

At each point $\mathbf{r}(t)$ , the incident wave is: (1) attenuated by $\alpha$ (absorption), (2) scattered with complex coefficient $\rho$ , and (3) phase-shifted by the round-trip path delay $2t/c$ . Integrating along the ray and enforcing causality (the transmittance $T$ accounts for attenuation of all preceding segments) yields the result.

Discrete approximation

With $N$ samples $\{t_i\}_{i=1}^N$ :

$\hat{S} \approx \sum_{i=1}^{N} T_i\,\alpha_i\,\rho_i\, e^{-j \kappa \cdot 2t_i},$

where $\alpha_i = 1 - \exp(-\sigma_i \delta_i)$ and $T_i = \prod_{j<i}(1 - \alpha_j)$ . This is differentiable with respect to all MLP parameters. $\square$

Definition:
Frequency-Dependent RF Transmittance

Unlike optical transmittance, the RF transmittance depends on frequency:

$T(t, f) = \exp\!\left(-\int_{t_n}^{t} \alpha(\mathbf{r}(s), f)\,ds\right).$

For a material with linear frequency dependence $\alpha(\mathbf{x}, f) = \alpha_0(\mathbf{x}) + \alpha_1(\mathbf{x}) f$ , the transmittance factors as:

$T(t, f) = T_0(t) \cdot T_1(t, f), \qquad T_0(t) = e^{-\int \alpha_0\,ds}, \quad T_1(t, f) = e^{-f\int \alpha_1\,ds}.$

This factorisation is useful for multi-frequency training: the geometry MLP learns $\alpha_0$ (frequency-independent loss), while the signal MLP learns $\alpha_1$ (frequency-dependent dispersion).

Definition:
View-Dependent RF Reflectivity

The complex reflectivity $\rho(\mathbf{x}, \mathbf{d}, f)$ depends on position, viewing direction, and frequency. For a planar surface with normal $\hat{\mathbf{n}}$ at $\mathbf{x}$ , the Fresnel reflection coefficient gives:

$\rho(\mathbf{x}, \mathbf{d}, f) = \Gamma(\theta_i, \varepsilon_r(f))\, \delta(\mathbf{d} - \mathbf{d}_r),$

where $\theta_i = \arccos(|\mathbf{d} \cdot \hat{\mathbf{n}}|)$ is the incidence angle, $\mathbf{d}_r = \mathbf{d} - 2(\mathbf{d} \cdot \hat{\mathbf{n}})\hat{\mathbf{n}}$ is the specular reflection direction, and $\Gamma$ is the Fresnel coefficient.

In the neural field, the MLP implicitly learns an approximation to this directional dependence. The stronger the specular component, the more capacity the MLP must allocate to directional features.

Theorem: Connection to the Born Forward Model

Under the Born approximation (weak scattering, single scattering), the RF volume rendering equation reduces to the linear forward model:

$\hat{S}(\mathbf{r}, f) \approx \int_\Omega \rho(\mathbf{p})\, e^{-j\kappa(\|\mathbf{p} - \mathbf{s}\| + \|\mathbf{p} - \mathbf{r}\|)}\, d\mathbf{p},$

which is the continuous form of $\mathbf{y} = \mathbf{A}\mathbf{c}$ . Specifically, when:

$T(t, f) \approx 1$ everywhere (negligible attenuation --- weak scattering);
$\sigma(\mathbf{x})$ is an indicator function of the scatterer support;
$\rho$ is view-independent (isotropic scattering);

then the neural rendering integral becomes the Born integral with the sensing kernel given by the free-space Green's function.

Proof

Setting $T = 1$ (weak scattering)

If attenuation is negligible, $T(t, f) = 1$ for all $t$ . The rendering integral becomes $\hat{S} = \int \sigma \cdot \rho \cdot e^{-j\kappa \cdot 2t}\,dt$ .

Isotropic scattering

If $\rho(\mathbf{x}, \mathbf{d}, f) = \rho(\mathbf{x})$ (no view dependence), the product $\sigma \cdot \rho$ reduces to the reflectivity function $c(\mathbf{p})$ from Chapter 5.

Bistatic geometry

For separated Tx at $\mathbf{s}$ and Rx at $\mathbf{r}$ , the round-trip phase becomes $\kappa(\|\mathbf{p} - \mathbf{s}\| + \|\mathbf{p} - \mathbf{r}\|)$ , recovering the Born forward model. The neural field $\rho_\theta(\mathbf{x})$ replaces the discrete vector $\mathbf{c}$ , and the rendering integral replaces the matrix-vector product $\mathbf{A}\mathbf{c}$ . $\square$

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Example: 1D RF Volume Rendering with Two Walls

A ray traverses two walls at distances $t_1 = 3$ m and $t_2 = 7$ m. Each wall has density $\sigma = 5$ m $^{-1}$ and thickness $0.2$ m. Free space has $\sigma = 0$ . The reflectivities are $\rho_1 = 0.8 e^{j\pi/4}$ and $\rho_2 = 0.3 e^{-j\pi/6}$ . At $f = 5$ GHz, compute the rendered received signal $\hat{S}$ using $N = 4$ representative samples.

Solution

Sample positions and densities

Sample 1 (free space, $t = 1.5$ m): $\sigma_1 = 0$ , $\delta_1 = 1.5$ m. Sample 2 (wall 1, $t = 3.0$ m): $\sigma_2 = 5$ , $\delta_2 = 0.2$ m. Sample 3 (free space, $t = 5.0$ m): $\sigma_3 = 0$ , $\delta_3 = 2.0$ m. Sample 4 (wall 2, $t = 7.0$ m): $\sigma_4 = 5$ , $\delta_4 = 0.2$ m.

Alpha values and transmittances

$\alpha_1 = 1 - e^{0} = 0$ (free space contributes nothing). $\alpha_2 = 1 - e^{-5 \times 0.2} = 1 - e^{-1} = 0.632$ . $\alpha_3 = 0$ . $\alpha_4 = 0.632$ . $T_1 = 1$ , $T_2 = 1$ , $T_3 = 1 - 0.632 = 0.368$ , $T_4 = 0.368$ .

Phase terms

$\kappa = 2\pi \times 5 \times 10^9 / (3 \times 10^8) = 104.7$ rad/m. Phase at wall 1: $e^{-j \times 104.7 \times 2 \times 3} = e^{-j628.3}$ . Phase at wall 2: $e^{-j \times 104.7 \times 2 \times 7} = e^{-j1466.1}$ .

Rendered signal

$\hat{S} = T_2 \alpha_2 \rho_1 e^{-j628.3} + T_4 \alpha_4 \rho_2 e^{-j1466.1}$ $= 1 \times 0.632 \times 0.8 e^{j\pi/4} e^{-j628.3} + 0.368 \times 0.632 \times 0.3 e^{-j\pi/6} e^{-j1466.1}$ .

Wall 1 contributes $|T_2 \alpha_2 \rho_1| = 0.506$ and wall 2 contributes $|T_4 \alpha_4 \rho_2| = 0.070$ . Wall 1 dominates because it is encountered first (higher transmittance). $\square$

Discrete RF Volume Rendering

Complexity:

\mathcal{O}(N)

per ray, where

N

is the number of samples. Each sample requires one MLP evaluation:

\mathcal{O}(N \cdot C_{\mathrm{MLP}})

total, where

C_{\mathrm{MLP}}

is the cost of a forward pass.

Input: ray samples {(x_i, d)}, MLP F_θ, frequency f

Output: complex received signal Ŝ

1. For i = 1 to N:

2. (σ_i, ρ_i) ← F_θ(γ(x_i), γ(d), f)

3. δ_i ← t_{i+1} - t_i

4. α_i ← 1 - exp(-σ_i · δ_i)

5. phase_i ← exp(-j · 2κ · t_i) // κ = 2πf/c

6. T_1 ← 1

7. Ŝ ← 0

8. For i = 1 to N:

9. Ŝ ← Ŝ + T_i · α_i · ρ_i · phase_i

10. T_{i+1} ← T_i · (1 - α_i)

11. Return Ŝ

The algorithm is identical to optical volume rendering except for lines 2 (complex output), 5 (round-trip phase), and 9 (complex accumulation). Backpropagation through this algorithm is straightforward because all operations are differentiable.

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When NeRF Disagrees with Born

The connection between neural volumetric rendering and the Born forward model breaks down when:

Strong scattering: Attenuation is significant ( $T \ll 1$ ), violating the Born single-scattering assumption. The NeRF transmittance $T(t, f)$ captures this naturally, while the Born model does not.
Multiple scattering: The Born model assumes single scattering; NeRF's single-ray rendering likewise misses multi-bounce paths. Both fail in the same way.
View dependence: The Born model assumes isotropic scattering ( $\rho$ independent of $\mathbf{d}$ ). NeRF can model view-dependent reflectivity, which is necessary for specular surfaces but adds model complexity.

The point is that NeRF extends the Born model by adding attenuation and view dependence, but both share the limitation of single-scattering. Multi-bounce extensions (WiNeRT) address this at the cost of training complexity.

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Example: Born Model as a Special Case of RF-NeRF

Show explicitly that for a 2D scene with $Q$ point scatterers at positions $\{\mathbf{p}_{q}\}_{q=1}^Q$ with reflectivities $\{c_q\}$ , and a single Tx at $\mathbf{s}$ and Rx at $\mathbf{r}$ , the Born forward model $y = \sum_q c_q\, e^{-j\kappa(\|\mathbf{p}_{q} - \mathbf{s}\| + \|\mathbf{p}_{q} - \mathbf{r}\|)} + w$ is recovered from the RF-NeRF rendering equation under the Born assumptions.

Solution

Set $T = 1$

Negligible attenuation: $\alpha(\mathbf{x}, f) = 0$ everywhere, so $T(t, f) = 1$ for all $t$ .

Point scatterer density

The density $\sigma(\mathbf{x}) = \sum_q \delta(\mathbf{x} - \mathbf{p}_{q})$ . In the discrete approximation, $\alpha_q \approx 1$ at scatterer positions and $\alpha_q = 0$ elsewhere.

Isotropic scattering

$\rho(\mathbf{p}_{q}, \mathbf{d}, f) = c_q$ (view-independent). The rendering sum becomes $\hat{S} = \sum_q c_q\, e^{-j\kappa \cdot 2t_q}$ . For a bistatic geometry, $2t_q$ is replaced by the total path length $\|\mathbf{p}_{q} - \mathbf{s}\| + \|\mathbf{p}_{q} - \mathbf{r}\|$ .

Conclude

This is exactly the Born forward model for point scatterers. The sensing matrix $\mathbf{A}$ has entries $[\mathbf{A}]_{(i,j,k),q} = e^{-j\kappa_{k}(\|\mathbf{p}_{q} - \mathbf{s}_i\| + \|\mathbf{p}_{q} - \mathbf{r}_j\|)}$ , and the NeRF rendering integral implements each row of this matrix continuously. $\square$

⚠️Engineering Note

Computational Cost of RF Volume Rendering

The computational bottleneck in RF-NeRF is the MLP evaluation at each sample point. For a scene with $R$ Tx-Rx pairs, $N$ samples per ray, and $K$ frequency subcarriers, the total number of MLP evaluations per training iteration is:

$C = B \cdot N \cdot K,$

where $B$ is the batch size (number of Tx-Rx pairs per batch).

Practical numbers: $B = 256$ , $N = 128$ , $K = 234$ (80 MHz Wi-Fi CSI) gives $C \approx 7.7 \times 10^6$ MLP evaluations per iteration. With an 8-layer, 256-unit MLP, this takes $\sim 50$ ms on an A100 GPU. For 100k iterations: $\sim 83$ minutes of training.

Mitigation: Hash encoding reduces MLP cost by $10\times$ . Frequency-sharing (evaluating the geometry MLP once per spatial point and only the signal MLP per frequency) reduces cost by $\sim K/2$ .

Practical Constraints

•
GPU memory limits batch size for large $N \cdot K$ products
•
Multi-frequency rendering requires $K$ signal MLP evaluations per sample
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Hash encoding memory: $\sim 64$ MB for $T = 2^{19}$ , $F = 2$ , $L = 16$

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RF Volume Rendering vs Born Forward Model

Compare the Born forward model (no attenuation, single scattering) with full RF volume rendering (with attenuation) for a 1D scene. When $\alpha_0 = 0$ , both agree perfectly. As attenuation increases, the Born model overestimates contributions from distant scatterers (it ignores shadowing by nearer objects), while the volume rendering correctly applies transmittance.

Parameters

Number of scatterers3

Attenuation

\alpha_0

(Np/m)0

Frequency (GHz)5

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Common Mistake: Assuming Born Validity for Strong Scatterers

Mistake:

Using the Born forward model $\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}$ for scenes with thick concrete walls or metallic objects, where through-wall attenuation exceeds 10 dB.

Correction:

The Born approximation assumes negligible attenuation ( $T \approx 1$ ). For strong scatterers, the NeRF transmittance model naturally handles shadowing, while the Born model produces physically incorrect predictions. Use the full RF volume rendering equation, or at minimum validate Born predictions against measurements.

Open Problems in RF Volume Rendering

Several fundamental challenges remain:

Multi-bounce rendering: Single-ray NeRF captures only direct paths. WiNeRT adds reflections but at $3\times$ cost. Efficient differentiable multi-bounce rendering remains open.
Diffraction: Volume rendering handles absorption and specular reflection but not diffraction around edges. Incorporating the uniform theory of diffraction (UTD) into neural rendering is an active research direction.
Generalisation: Current RF-NeRFs train per-scene. Foundation models that generalise across environments (trained on diverse RF scenes) would mirror the trajectory of optical NeRF generalisation.
Phase coherence: Many practical systems measure only power (no phase). Training from power-only measurements introduces a phase-retrieval problem within the NeRF framework (see Chapter 16).

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Quick Check

Under which condition does the RF volume rendering equation reduce to the Born forward model?

When the MLP has infinite capacity

When $T(t, f) \approx 1$ , scattering is isotropic, and density is supported on the scatterer locations

When the frequency is above 100 GHz

When hash encoding is used instead of positional encoding

Correction:

When

T(t, f) \approx 1

, scattering is isotropic, and density is supported on the scatterer locations

Correct. These three conditions eliminate attenuation, view dependence, and the continuous density field, recovering the discrete Born model.

RF Reflectivity

The complex-valued function $\rho(\mathbf{x}, \mathbf{d}, f)$ describing how a material at position $\mathbf{x}$ scatters an incident RF wave from direction $\mathbf{d}$ at frequency $f$ . The magnitude $|\rho|$ determines the scattered power; the phase $\angle \rho$ determines the phase shift upon reflection. For specular surfaces, $\rho$ is strongly peaked around the specular direction.

Related: Volume Density

Key Takeaway

The RF volume rendering equation integrates complex-valued reflectivity, frequency-dependent attenuation, and round-trip propagation phase along each ray. Under the Born-approximation assumptions (no attenuation, isotropic scattering), it reduces exactly to the linear forward model of Chapters 5--6. The NeRF framework extends the Born model by naturally handling shadowing (via transmittance) and specular scattering (via view-dependent reflectivity), at the cost of per-scene neural network training.

Volume Rendering Adapted for RF

The RF Rendering Equation

Theorem: RF Volume Rendering Equation

Physics derivation

Discrete approximation

Definition: Frequency-Dependent RF Transmittance

Definition: View-Dependent RF Reflectivity

Theorem: Connection to the Born Forward Model

Setting $T = 1$ (weak scattering)

Isotropic scattering

Bistatic geometry

Example: 1D RF Volume Rendering with Two Walls

Sample positions and densities

Alpha values and transmittances

Phase terms

Rendered signal

Discrete RF Volume Rendering

When NeRF Disagrees with Born

Example: Born Model as a Special Case of RF-NeRF

Set $T = 1$

Point scatterer density

Isotropic scattering

Conclude

Computational Cost of RF Volume Rendering

RF Volume Rendering vs Born Forward Model

Parameters

Common Mistake: Assuming Born Validity for Strong Scatterers

Open Problems in RF Volume Rendering

Quick Check

RF Reflectivity

Key Takeaway

Definition:
Frequency-Dependent RF Transmittance

Definition:
View-Dependent RF Reflectivity