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Beyond Geometry: Estimating What Surfaces Are Made Of

Sections 25.1--25.2 focused on reconstructing where objects are (the geometry). But for RF propagation modelling, we also need to know what the surfaces are made of: concrete walls reflect most of the incident power; glass windows transmit much of it; metallic surfaces are near-perfect reflectors. These material properties --- reflectivity, roughness, permittivity --- determine how the electromagnetic wave interacts with the scene and are essential for accurate channel prediction.

This section develops the joint estimation framework: a neural SDF for geometry, coupled with material property networks, all trained end-to-end from multi-view RF measurements with Eikonal regularisation enforcing geometric validity.

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Definition:
Eikonal Regularization Loss

The Eikonal regularization loss penalises deviations of the gradient norm from unity:

$\mathcal{L}_{\text{eik}}(\theta) = \mathbb{E}_{\mathbf{p} \sim \mathcal{U}(\Omega)} \bigl[\bigl(\|\nabla_{\mathbf{p}} f_\theta(\mathbf{p})\| - 1\bigr)^2\bigr],$

where $\Omega$ is the scene volume and the expectation is approximated by sampling random points. In practice, the sample set $\mathcal{P}$ is a mixture:

Uniform samples in $\Omega$ (enforce the Eikonal equation globally).
Near-surface samples drawn within a band $\{\mathbf{p} : |f_\theta(\mathbf{p})| < \delta\}$ (enforce the equation where it matters most).

Gropp et al. (2020) showed that Eikonal regularisation alone --- without any ground-truth distance supervision --- can learn valid SDFs from unoriented point clouds. This "Implicit Geometric Regularization" (IGR) result means that for RF imaging, we do not need ground-truth 3D meshes; the radar measurements plus the Eikonal constraint suffice.

Theorem: Eikonal Self-Supervision for SDF Learning

Let $\mathcal{X} = \{\mathbf{p}_{i}\}$ be a set of points on or near a surface $\mathcal{S}$ , and let $f_\theta$ minimise the loss

$\mathcal{L}(\theta) = \frac{1}{|\mathcal{X}|}\sum_{i}|f_\theta(\mathbf{p}_{i})|^2 + \lambda \mathbb{E}_{\mathbf{p} \sim \mathcal{U}(\Omega)} \bigl[(\|\nabla f_\theta(\mathbf{p})\| - 1)^2\bigr].$

If $f_\theta$ is expressive enough, the minimiser satisfies $f_\theta(\mathbf{p}) \approx \pm d(\mathbf{p}, \mathcal{S})$ : the network learns the signed distance function from surface points alone, without distance labels.

The first term forces $f_\theta = 0$ at the surface points. The Eikonal term forces $\|\nabla f_\theta\| = 1$ everywhere. The only smooth function satisfying both constraints is the signed distance function (up to a global sign flip).

Proof

Zero at the surface

The first term $\sum |f_\theta(\mathbf{p}_{i})|^2 \to 0$ means $f_\theta(\mathbf{p}_{i}) \to 0$ for all surface points.

Unit gradient everywhere

The Eikonal term forces $\|\nabla f_\theta\| \to 1$ a.e. Combined with $f_\theta = 0$ on $\mathcal{S}$ , the function increases at rate $1$ as we move away from $\mathcal{S}$ , which is precisely the definition of a distance function.

Sign determination

The sign of $f_\theta$ (positive outside, negative inside) is determined by the initialisation and the bias of the network. In practice, geometric initialisation ( $f_\theta^{(0)} \approx \|\mathbf{p}\| - R_0$ ) ensures the correct sign convention. $\blacksquare$

Definition:
Material Property Networks

The joint scene model augments the neural SDF with material property networks:

Reflectivity: $\Gamma_\phi(\mathbf{p}) \in [0, 1]$ , the fraction of incident power reflected at the surface.
Roughness: $\rho_\psi(\mathbf{p}) \in [0, 1]$ , parameterising the scattering lobe width (smooth metal $\to 0$ , rough concrete $\to 1$ ).
Relative permittivity: $\epsilon_{r,\xi}(\mathbf{p}) \in [1, \infty)$ , governing penetration depth and Fresnel reflection coefficients.

All networks share the backbone feature extractor and positional encoding, with separate output heads. The total learnable parameters are $(\theta, \phi, \psi, \xi)$ .

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Definition:
Joint Geometry-Material Training Loss

The full training objective combines data fidelity, Eikonal regularisation, and optional material priors:

$\mathcal{L} = \underbrace{\frac{1}{Q}\sum_{q=1}^{Q} \bigl(P_{\text{MF}}^{(v)}(\mathbf{p}_{q}) - \hat{P}_{\text{MF}}^{(v)}(\mathbf{p}_{q})\bigr)^2}_{\text{multi-view data fidelity}} + \lambda_{\text{eik}}\,\mathcal{L}_{\text{eik}} + \lambda_{\Gamma}\,\mathcal{L}_{\Gamma} + \lambda_{\epsilon}\,\mathcal{L}_{\epsilon},$

where:

The sum over views $v = 1, \ldots, V$ uses MF power images from different radar positions.
$\mathcal{L}_{\Gamma} = \|\Gamma_\phi\|_{\text{TV}}$ encourages piecewise-constant reflectivity (walls have uniform material).
$\mathcal{L}_{\epsilon} = \sum_{\mathbf{p}} (\epsilon_{r,\xi}(\mathbf{p}) - \bar{\epsilon}_r)^2$ softly regularises permittivity toward typical building material values.

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Example: Multi-View Estimation of Room Geometry and Materials

A mmWave MIMO radar is mounted on a mobile robot that collects measurements from $V = 8$ positions in an indoor room with concrete walls ( $\epsilon_r \approx 5$ , $\Gamma \approx 0.7$ ) and a glass window ( $\epsilon_r \approx 6$ , $\Gamma \approx 0.2$ ). Describe the joint estimation procedure and what each view contributes.

Solution

Geometry from multiple views

Each radar position provides a different MF power image. Multiple views reduce the ambiguity inherent in single-view radar: a wall segment that is edge-on from one position (producing weak return) may be face-on from another (producing strong return). The neural SDF must simultaneously explain all $V$ power images, disambiguating the geometry.

Material disambiguation

The glass window reflects weakly ( $\Gamma \approx 0.2$ ) but is present in the geometry (visible from certain angles). From a single view, a weak return could indicate either a distant wall or a nearby glass surface. Multiple views at different angles provide sufficient geometric constraint to separate the reflectivity $\Gamma$ from the geometry $f_\theta$ : the SDF fixes the surface location, and $\Gamma_\phi$ then fits the observed power level.

Permittivity estimation

The permittivity $\epsilon_r$ affects the Fresnel reflection coefficient, which varies with incidence angle. Multi-view data at different incidence angles constrains $\epsilon_r$ : the reflectivity at near-normal incidence differs from grazing incidence in a way that depends on $\epsilon_r$ . With $V = 8$ views spanning a range of incidence angles, the permittivity is identifiable up to the noise level.

Eikonal Loss During Training

Observe how the Eikonal loss, data fidelity loss, and surface quality evolve during training. Increasing $\lambda_{\text{eik}}$ produces smoother surfaces (enforcing $\|\nabla f\| = 1$ ) but may slow convergence of the data fidelity term.

Parameters

\lambda_{\text{eik}}

0.1

Training epochs200

Joint Geometry and Material Estimation

Visualise the estimated reflectivity map overlaid on the reconstructed SDF surface. Different materials appear as different colours on the surface: high reflectivity (metal, concrete) in red, low reflectivity (glass, drywall) in blue.

Parameters

Number of radar views8

Noise level0.1

Theorem: Permittivity Identifiability from Multi-Angle RF Data

Consider a planar dielectric interface with relative permittivity $\epsilon_r$ . The Fresnel reflection coefficient for TE polarisation is

$R_{\perp}(\theta_i) = \frac{\cos\theta_i - \sqrt{\epsilon_r - \sin^2\theta_i}}{\cos\theta_i + \sqrt{\epsilon_r - \sin^2\theta_i}},$

where $\theta_i$ is the angle of incidence. Given noise-free measurements of $|R_{\perp}(\theta_i)|^2$ at two or more distinct angles $\theta_1 \neq \theta_2$ , the permittivity $\epsilon_r$ is uniquely determined.

The Fresnel coefficient is a nonlinear function of $\epsilon_r$ and $\theta_i$ . A single measurement at one angle leaves a one-parameter family of $(\epsilon_r, \Gamma)$ pairs consistent with the data. A second angle breaks this degeneracy because the angular dependence of $R_\perp$ is a signature of $\epsilon_r$ .

Proof

Single-angle ambiguity

At a single angle $\theta_1$ , the measured power $|R_\perp(\theta_1)|^2$ constrains $\epsilon_r$ to lie on a curve in the $(\epsilon_r, \Gamma)$ plane. Without additional information, $\epsilon_r$ is not identifiable.

Two-angle identification

At a second angle $\theta_2$ , we obtain a second constraint. The two curves $|R_\perp(\theta_1)|^2 = c_1$ and $|R_\perp(\theta_2)|^2 = c_2$ generically intersect at a unique point in $\epsilon_r$ , since $R_\perp(\theta_i)$ is a monotonic function of $\epsilon_r$ for fixed $\theta_i$ in the range $\epsilon_r > 1$ .

Extension to noisy data

With noise, exact identification is replaced by estimation. Multiple angles ( $V > 2$ ) provide overdetermination, enabling least-squares estimation of $\epsilon_r$ with $\text{MSE} \propto 1/V$ . $\blacksquare$

Common Mistake: Choosing the Eikonal Weight $\lambda_{\text{eik}}$

Mistake:

Setting $\lambda_{\text{eik}}$ too large (e.g., $\lambda_{\text{eik}} = 10$ ), causing the Eikonal term to dominate and the network to learn a trivial solution $f_\theta(\mathbf{p}) \approx \|\mathbf{p}\|$ (a sphere) that perfectly satisfies the Eikonal equation but ignores the data.

Correction:

Start with $\lambda_{\text{eik}} \in [0.01, 0.1]$ and tune via validation. A good heuristic: set $\lambda_{\text{eik}}$ so that the Eikonal loss and data fidelity loss are of comparable magnitude after the first few training epochs. Some implementations use a warm-up schedule, starting with $\lambda_{\text{eik}} = 0$ and gradually increasing it.

Joint Geometry-Material Training Algorithm

Complexity: Per iteration:

O(|\mathcal{P}| \cdot d)

for MLP forward + backward, where

d

is the MLP depth. Total:

O(T \cdot |\mathcal{P}| \cdot d)

.

Input: MF power images

\{P_{\text{MF}}^{(v)}\}_{v=1}^V

,

weights

\lambda_{\text{eik}}, \lambda_\Gamma, \lambda_\epsilon

Output: Trained parameters

(\theta^*, \phi^*, \psi^*, \xi^*)

1. Initialise

f_\theta

with geometric initialisation (large sphere)

2. Initialise

\Gamma_\phi, \rho_\psi, \epsilon_{r,\xi}

randomly

3. for

t = 1, \ldots, T

do

4.

\quad

Sample batch of 3D points

\mathcal{P}

(uniform + near-surface)

5.

\quad

Compute SDF:

f_\theta(\mathbf{p})

for all

\mathbf{p} \in \mathcal{P}

6.

\quad

Compute materials:

\Gamma_\phi, \rho_\psi, \epsilon_{r,\xi}

at surface points

7.

\quad

Render MF power:

\hat{P}_{\text{MF}}^{(v)}

for each view

v

8.

\quad

Compute

\mathcal{L}_{\text{data}} + \lambda_{\text{eik}} \mathcal{L}_{\text{eik}} + \lambda_\Gamma \mathcal{L}_\Gamma + \lambda_\epsilon \mathcal{L}_\epsilon

9.

\quad

Update

(\theta, \phi, \psi, \xi)

via Adam

10. end for

🔧Engineering Note

Building Material Databases for RF Estimation

The regulariser $\mathcal{L}_\epsilon$ requires knowledge of typical permittivity values. Common building materials at mmWave frequencies:

Material	$\epsilon_r$ (60 GHz)	$\Gamma$ (normal incidence)
Concrete	5.0--6.5	0.6--0.8
Glass	5.5--7.0	0.2--0.4
Drywall	2.5--3.0	0.3--0.5
Metal	$\gg 1$ (conductor)	$\approx 1.0$
Wood	2.0--3.0	0.2--0.4

These values can initialise the permittivity prior or serve as discrete classes for a classification-based material estimator.

Primitive-Based SDF Representations

An alternative to fully neural SDFs is to represent scenes as combinations of simple analytic primitives (planes, cylinders, spheres) whose SDFs are known in closed form. Boolean operations (union, intersection) compose primitives into complex scenes. This approach has several advantages for RF:

Surface normals are analytically available (no autodiff needed).
Only surface voxels contribute to scattering --- highly sparse.
Lambda-scale discretisation is feasible without excessive memory.
Interpretability: the scene is parameterised by a small number of geometric parameters (wall positions, column radii).

The cost is reduced expressiveness: scenes that do not decompose into simple primitives (e.g., furniture, vegetation) require the fully neural approach.

Quick Check

What is the primary purpose of the Eikonal regulariser $\mathcal{L}_{\text{eik}} = \mathbb{E}[(\|\nabla f_\theta\| - 1)^2]$ in neural SDF training?

To make the network converge faster.

To ensure the network output is a valid signed distance function.

To reduce the number of training parameters.

To prevent overfitting to noise.

Correction:

To ensure the network output is a valid signed distance function.

The Eikonal loss forces $\|\nabla f_\theta\| = 1$ everywhere, which is the defining property of a signed distance function. Without it, the network can fit the data with an arbitrary scalar field that has no geometric meaning.

Eikonal Regularization

A training loss term $\mathcal{L}_{\text{eik}} = \mathbb{E}[(\|\nabla f_\theta\| - 1)^2]$ that encourages a neural network to output a valid signed distance function by enforcing the Eikonal equation.

Relative Permittivity

The ratio $\epsilon_r = \epsilon / \epsilon_0$ of the material's permittivity to the vacuum permittivity. Governs the Fresnel reflection coefficient at a dielectric interface.

Related: Surface Reflectivity

Surface Reflectivity

The fraction $\Gamma(\mathbf{p}) \in [0, 1]$ of incident electromagnetic power reflected at a surface point $\mathbf{p}$ . Depends on material properties and incidence angle.

Historical Note: Implicit Geometric Regularization

2020

Gropp, Yariv, Haim, Atzmon, and Lipman at the Weizmann Institute introduced Implicit Geometric Regularization (IGR) in 2020, demonstrating that the Eikonal loss alone can supervise SDF learning from raw point clouds without distance labels. This result was surprising: the community had assumed that explicit distance supervision was necessary. IGR opened the door to learning SDFs from any data source that provides surface point locations --- including radar returns, which give approximate surface points via matched filtering.

Key Takeaway

Eikonal regularisation enforces valid SDF geometry without requiring ground-truth distance labels. Joint estimation of geometry, reflectivity, roughness, and permittivity from multi-view RF data is feasible because the angular dependence of the Fresnel reflection coefficient provides material-discriminating information. The full training pipeline optimises all parameters end-to-end, with the Eikonal constraint serving as the geometric "anchor" that prevents the optimisation from converging to physically meaningless solutions.

Eikonal Regularization and Material Estimation

Beyond Geometry: Estimating What Surfaces Are Made Of

Definition: Eikonal Regularization Loss

Theorem: Eikonal Self-Supervision for SDF Learning

Zero at the surface

Unit gradient everywhere

Sign determination

Definition: Material Property Networks

Definition: Joint Geometry-Material Training Loss

Example: Multi-View Estimation of Room Geometry and Materials

Geometry from multiple views

Material disambiguation

Permittivity estimation

Eikonal Loss During Training

Parameters

Joint Geometry and Material Estimation

Parameters

Theorem: Permittivity Identifiability from Multi-Angle RF Data

Single-angle ambiguity

Two-angle identification

Extension to noisy data

Common Mistake: Choosing the Eikonal Weight λeik\lambda_{\text{eik}}λeik​

Joint Geometry-Material Training Algorithm

Building Material Databases for RF Estimation

Primitive-Based SDF Representations

Quick Check

Eikonal Regularization

Relative Permittivity

Surface Reflectivity

Historical Note: Implicit Geometric Regularization

Key Takeaway

Definition:
Eikonal Regularization Loss

Definition:
Material Property Networks

Definition:
Joint Geometry-Material Training Loss

Common Mistake: Choosing the Eikonal Weight $\lambda_{\text{eik}}$