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From Optical to RF Rendering

The rendering equation and its inverse were developed for visible light, where wavelengths ( $\sim 500$ nm) are far smaller than scene features, justifying the ray-optics approximation. RF waves have wavelengths from millimetres (mmWave) to metres (HF radar), making wave effects (diffraction, interference, multipath) dominant.

Adapting differentiable rendering to RF requires replacing ray-based transport with wave-based transport, while preserving the key property: differentiability with respect to scene parameters. This section develops that bridge systematically.

Definition:
The RF Rendering Equation

The RF analog of the optical rendering equation is the volume integral equation derived from Maxwell's equations under the Born approximation (Chapter 6):

$E^s(\mathbf{r}) = \kappa^{2} \int_{\Omega} \chi(\mathbf{p})\,G(\mathbf{r}, \mathbf{p})\,E^i(\mathbf{p})\,dV',$

where:

$E^s(\mathbf{r})$ : scattered field at receiver position $\mathbf{r}$ .
$\chi(\mathbf{p}) = \epsilon_r(\mathbf{p}) - 1$ : contrast function (the "scene" parameter).
$G(\mathbf{r}, \mathbf{p})$ : Green's function (free-space: $G = e^{-j\kappa|\mathbf{r}-\mathbf{p}|} / (4\pi|\mathbf{r}-\mathbf{p}|)$ ).
$E^i(\mathbf{p})$ : incident field from transmitter at $\mathbf{s}$ .

This integral relates the scene parameters ( $\chi$ ) to the measurements ( $E^s$ ) in a way that is linear in $\chi$ under the Born approximation — unlike the nonlinear rendering equation of optics.

The linearity of the RF forward model under Born is a significant simplification over optical rendering, where the light-material interaction (BRDF) introduces nonlinearity. This is why many RF imaging algorithms can use linear algebraic tools (matched filter, LASSO, AMP) that have no direct optical analog.

Optical vs. RF Rendering: Key Differences

Property	Optical Rendering	RF Rendering
Wavelength	$\sim 500$ nm $\ll$ scene features	$\sim 1$ mm -- $10$ cm $\sim$ scene features
Transport model	Ray optics (incoherent)	Wave optics (coherent)
Phase	Ignored (intensity only)	Critical (complex-valued fields)
Interference	None (incoherent addition)	Constructive/destructive
Multipath	Ignored or 1 bounce	Multiple bounces essential
Forward model linearity	Nonlinear (BRDF interaction)	Linear under Born approximation
Camera/lens	Lens-based imaging	Lensless (computational imaging)
Scene representation	Mesh, SDF, radiance field	Reflectivity field $c$ , permittivity $\epsilon_r(\mathbf{p})$

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Definition:
Differentiable RF Renderer

A differentiable RF renderer computes the scattered field as a differentiable function of scene parameters $\theta$ :

$\hat{\mathbf{y}} = \mathcal{A}_\theta(\text{scene}), \qquad \frac{\partial \hat{\mathbf{y}}}{\partial \theta} \text{ computed via autodiff}.$

Implementation approaches:

Neural scene + physics engine: Parameterise $\chi_\theta(\mathbf{p}) = \text{MLP}_\theta(\gamma(\mathbf{p}))$ and evaluate the volume integral numerically, backpropagating through the discretised integral.
Method of moments (MoM) + adjoint: Discretise Maxwell's equations into $\mathbf{Z}\mathbf{I} = \mathbf{V}$ and differentiate through the solve via the adjoint method: $\partial \mathcal{L}/\partial \theta = -\boldsymbol{\lambda}^\mathsf{T} (\partial \mathbf{Z}/\partial \theta)\mathbf{I}$ , where $\mathbf{Z}^\mathsf{T} \boldsymbol{\lambda} = \partial \mathcal{L}/\partial \mathbf{I}$ .
Differentiable ray tracing (Sionna RT): Combine geometric-optics ray tracing for dominant paths with diffraction corrections (UTD/GTD), implemented in a differentiable framework.

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Theorem: Adjoint Method for Differentiable EM Simulation

Consider a loss function $\mathcal{L}(\mathbf{I}(\theta))$ where $\mathbf{I}$ is the solution of the discretised Maxwell system $\mathbf{Z}(\theta)\,\mathbf{I} = \mathbf{V}$ . The gradient with respect to scene parameters $\theta$ is:

$\frac{\partial \mathcal{L}}{\partial \theta} = -\boldsymbol{\lambda}^\mathsf{T}\,\frac{\partial \mathbf{Z}}{\partial \theta}\,\mathbf{I},$

where the adjoint variable $\boldsymbol{\lambda}$ satisfies:

$\mathbf{Z}^\mathsf{T}\,\boldsymbol{\lambda} = \left(\frac{\partial \mathcal{L}}{\partial \mathbf{I}}\right)^\mathsf{T}.$

The adjoint method computes the gradient in $O(N)$ (one additional linear solve), regardless of the dimension of $\theta$ .

The adjoint method is the continuous analog of backpropagation. Instead of storing all intermediate states (which would require $O(N \cdot |\theta|)$ memory), it solves a single "backward" system to compute all gradients simultaneously.

Proof

Chain rule

By the chain rule: $\frac{\partial \mathcal{L}}{\partial \theta} = \frac{\partial \mathcal{L}}{\partial \mathbf{I}} \frac{\partial \mathbf{I}}{\partial \theta}$ .

Implicit differentiation

Differentiating $\mathbf{Z}\mathbf{I} = \mathbf{V}$ with respect to $\theta$ : $\mathbf{Z}\frac{\partial \mathbf{I}}{\partial \theta} = -\frac{\partial \mathbf{Z}}{\partial \theta}\mathbf{I}$ (assuming $\mathbf{V}$ does not depend on $\theta$ ).

Substitution

Substituting: $\frac{\partial \mathcal{L}}{\partial \theta} = -\frac{\partial \mathcal{L}}{\partial \mathbf{I}}\mathbf{Z}^{-1}\frac{\partial \mathbf{Z}}{\partial \theta}\mathbf{I} = -\boldsymbol{\lambda}^\mathsf{T}\frac{\partial \mathbf{Z}}{\partial \theta}\mathbf{I}$ , where $\boldsymbol{\lambda}^\mathsf{T} = \frac{\partial \mathcal{L}}{\partial \mathbf{I}}\mathbf{Z}^{-1}$ , or equivalently $\mathbf{Z}^\mathsf{T}\boldsymbol{\lambda} = (\partial \mathcal{L}/\partial \mathbf{I})^\mathsf{T}$ . $\blacksquare$

Optical vs. RF Rendering: Wavelength Effects

Compare how optical (incoherent, $\lambda/d \ll 1$ ) and RF (coherent, $\lambda/d \sim 1$ ) rendering produce different images of the same scene. At small $\lambda/d$ , the image is sharp (ray optics); as $\lambda/d$ increases, diffraction effects blur the image and interference fringes appear. The complex-valued RF field shows constructive and destructive interference that is absent in the intensity-only optical case.

Parameters

\lambda / d

(wavelength / feature size)0.5

Number of scatterers3

Example: Differentiable EM Simulation for RF Lens Design

Describe how to optimise the geometry of an RF lens (e.g., a Luneburg lens) using a differentiable EM simulator.

Solution

Scene representation

Parameterise the lens permittivity as a neural field: $\epsilon_r(\mathbf{p}) = 1 + \text{softplus}(f_\theta(\mathbf{p}))$ , where the softplus ensures $\epsilon_r > 1$ (physical constraint).

Forward simulation

Discretise the domain on a grid (FDTD or FEM).
Simulate the incident plane wave and compute the total field.
Extract the far-field pattern $E_{\text{far}}(\phi)$ at the output plane.

Optimisation

$\min_\theta \; \|E_{\text{far}}(\phi; \theta) - E_{\text{target}}(\phi)\|^2 + \lambda\,\mathcal{R}(\epsilon_r),$ $where$ E_{\text{target}} $is the desired focusing pattern and$ \mathcal{R}$ regularises the permittivity distribution (e.g., TV penalty for manufacturability). Gradients are computed via the adjoint method through the FDTD solver.

🔧Engineering Note

Sionna RT: Differentiable Ray Tracing for 6G

NVIDIA's Sionna RT is a differentiable ray tracer built on TensorFlow/JAX that models wireless channel propagation in 3D environments. It computes channel impulse responses as differentiable functions of scene geometry (building positions, materials) and transceiver placement, enabling gradient-based optimisation of:

Network planning: Optimise base station placement to maximise coverage while satisfying SINR constraints.
Digital twins: Calibrate a digital twin of a building by matching simulated channels to measured data.
ISAC waveform design: Jointly optimise communication and sensing performance by differentiating through the channel model.

Sionna RT supports up to $\sim 10$ reflections and $\sim 1$ diffraction per path, with GPU-accelerated ray tracing achieving $\sim 10^6$ paths/second.

Practical Constraints

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Ray tracing assumes geometric optics; accuracy degrades for $\lambda \sim$ feature size
•
Diffraction modelling (UTD/GTD) is approximate and not validated for all geometries

Why This Matters: RF Inverse Rendering Applications

Differentiable RF rendering enables several applications directly relevant to the CommIT research programme:

Microwave tomography: Recover the 3D permittivity map $\epsilon_r(\mathbf{p})$ of a biological tissue from multi-angle scattering measurements, with the neural field providing implicit regularisation.
SAR autofocus: Jointly optimise the scene reflectivity and platform motion errors by differentiating through the SAR formation process (Chapter 12).
Antenna/metasurface design: Optimise the geometry and material distribution of RF devices to achieve desired radiation patterns.
Digital twins: Build differentiable RF models of indoor environments for wireless channel prediction, optimised to match measured channel data.

See full treatment in Chapter 39، Section 1

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Common Mistake: Born Approximation Breaks for Strong Scatterers

Mistake:

Applying the linear Born-approximation RF renderer to scenes with high-contrast objects ( $|\chi| \gg 1$ ) or objects much larger than the wavelength, and trusting the gradient-based reconstruction.

Correction:

The Born approximation is first-order perturbation theory: it assumes the total field inside the scatterer equals the incident field. For strong scatterers (e.g., metal objects, dense tissue), multiple scattering makes the forward model nonlinear. The gradient computed from the Born renderer is then biased, and the optimisation may converge to an incorrect solution.

Remedies: (i) Use the Rytov approximation (better for smooth phase variations). (ii) Use iterative Born (distorted Born iteration). (iii) Use a full-wave solver (MoM, FDTD) as the forward model and differentiate via the adjoint method.

Historical Note: From Ray Tracing to Differentiable RF Simulation

1992--2022

Ray tracing for wireless channel prediction dates to the early 1990s (Schaubach, Davis, and Rappaport, 1992), motivated by indoor wireless system design. For two decades, ray tracers were used as "black box" simulators — their outputs could be evaluated but not differentiated. The breakthrough came with the rise of differentiable programming (2017--2020): tools like JAX and PyTorch enabled automatic differentiation through complex simulation pipelines. Sionna RT (Hoydis et al., 2022) was the first GPU-accelerated differentiable ray tracer for wireless channels, opening the door to gradient-based optimisation of network deployments and ISAC systems.

Contrast Function

The function $\chi(\mathbf{p}) = \epsilon_r(\mathbf{p}) - 1$ representing the deviation of the relative permittivity from free space. The scene parameter in the RF rendering equation.

Related: The RF Rendering Equation

Key Takeaway

RF rendering replaces ray optics with wave optics: the forward model is the volume integral equation, not the rendering equation. Phase and coherence are critical in RF, unlike incoherent optical rendering. The Born approximation makes the RF forward model linear in the scene parameters — a simplification with no optical analog. Differentiable RF renderers (neural fields + physics engine, adjoint MoM, Sionna RT) enable the same gradient-based inverse problems that have revolutionised computer vision.

From Optical to RF Differentiable Rendering