Occupancy Networks for RF

Occupancy Networks: A Simpler Alternative to SDFs

Signed distance functions encode rich geometric information --- the distance to the surface, the surface normal, and the inside/outside classification --- all in a single scalar field. But this richness comes at a cost: training requires either ground-truth distance values or the Eikonal regulariser, and the distance property can be difficult to enforce exactly with neural networks.

Occupancy networks offer a simpler alternative: a binary classifier that predicts whether each 3D point is inside or outside an object. The surface is the decision boundary. For RF imaging, occupancy networks are attractive when the goal is obstruction mapping (is this region blocked?) rather than precise surface reconstruction.

Definition:

Occupancy Network

An occupancy network maps a 3D point to a probability of occupancy:

oθ(p)=σ(MLPθ(γ(p)))∈[0,1],o_\theta(\mathbf{p}) = \sigma\bigl(\text{MLP}_\theta(\gamma(\mathbf{p}))\bigr) \in [0, 1],

where σ(⋅)\sigma(\cdot) is the sigmoid function. The interpretation: oθ(p)=1o_\theta(\mathbf{p}) = 1 means p\mathbf{p} is inside the object; oθ(p)=0o_\theta(\mathbf{p}) = 0 means outside.

The surface is the decision boundary: S={p:oθ(p)=0.5}\mathcal{S} = \{\mathbf{p} : o_\theta(\mathbf{p}) = 0.5\}.

Training loss (binary cross-entropy):

L=−1∣P∣∑p∈P[o∗(p)log⁡oθ(p)+(1−o∗(p))log⁡(1−oθ(p))],\mathcal{L} = -\frac{1}{|\mathcal{P}|}\sum_{\mathbf{p} \in \mathcal{P}} \bigl[o^*(\mathbf{p})\log o_\theta(\mathbf{p}) + (1 - o^*(\mathbf{p}))\log(1 - o_\theta(\mathbf{p}))\bigr],

where o∗(p)∈{0,1}o^*(\mathbf{p}) \in \{0, 1\} is the ground-truth occupancy at sampled points.

Definition:

RF-Supervised Occupancy Training

When ground-truth occupancy labels are unavailable (the typical case in RF imaging), the occupancy network can be trained from radar measurements by defining a differentiable rendering loss analogous to GeRaF:

P^MF(pq;θ)=∑m,nwmn(pq) oθ(pq),\hat{P}_{\text{MF}}(\mathbf{p}_{q}; \theta) = \sum_{m,n} w_{mn}(\mathbf{p}_{q})\,o_\theta(\mathbf{p}_{q}),

with the training loss:

L(θ)=1Q∑q=1Q(PMF(pq)−P^MF(pq;θ))2.\mathcal{L}(\theta) = \frac{1}{Q}\sum_{q=1}^{Q} \bigl(P_{\text{MF}}(\mathbf{p}_{q}) - \hat{P}_{\text{MF}}(\mathbf{p}_{q}; \theta)\bigr)^2.

Here the occupancy field directly modulates the MF power, replacing the SDF-based surface delta function. No Eikonal regularisation is needed since no distance-function structure is assumed.

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Theorem: Occupancy and SDF: Decision Boundary Equivalence

Let f(p)f(\mathbf{p}) be a valid SDF and define the occupancy field

o(p)=σ(−α⋅f(p))o(\mathbf{p}) = \sigma(-\alpha \cdot f(\mathbf{p}))

for some α>0\alpha > 0. Then:

  1. The decision boundary {o=0.5}={f=0}\{o = 0.5\} = \{f = 0\}: the surfaces coincide.
  2. As α→∞\alpha \to \infty, o(p)→1[f(p)<0]o(\mathbf{p}) \to \mathbf{1}[f(\mathbf{p}) < 0]: the occupancy field approaches the indicator function of the object.

Every SDF induces an occupancy field (via the sigmoid), but the converse is not true: an occupancy network does not provide distance information. This is why SDFs are strictly more informative than occupancy networks.

Proof
Decision boundary

o(p)=0.5o(\mathbf{p}) = 0.5 iff σ(−αf(p))=0.5\sigma(-\alpha f(\mathbf{p})) = 0.5 iff −αf(p)=0-\alpha f(\mathbf{p}) = 0 iff f(p)=0f(\mathbf{p}) = 0. Hence {o=0.5}={f=0}=S\{o = 0.5\} = \{f = 0\} = \mathcal{S}.

Convergence to indicator

For f(p)<0f(\mathbf{p}) < 0 (inside): −αf>0-\alpha f > 0 and grows with α\alpha, so σ(−αf)→1\sigma(-\alpha f) \to 1.

For f(p)>0f(\mathbf{p}) > 0 (outside): −αf<0-\alpha f < 0 and σ(−αf)→0\sigma(-\alpha f) \to 0.

Hence o→1[f<0]o \to \mathbf{1}[f < 0] pointwise as α→∞\alpha \to \infty. ■\blacksquare

SDF vs. Occupancy Network for RF Imaging

PropertyNeural SDFOccupancy Network
OutputSigned distance f(p)∈Rf(\mathbf{p}) \in \mathbb{R}Occupancy probability o(p)∈[0,1]o(\mathbf{p}) \in [0, 1]
Surface definitionZero level set {f=0}\{f = 0\}Decision boundary {o=0.5}\{o = 0.5\}
Surface normals∇f/∥∇f∥\nabla f / \|\nabla f\| (analytic via autodiff)Requires numerical differentiation of binary-like field
Ray intersectionSphere tracing (efficient)Dense sampling along ray (expensive)
Regularisation neededEikonal loss ∥∥∇f∥−1∥2\| \|\nabla f\| - 1 \|^2None (binary cross-entropy suffices)
Training supervisionDistance values or Eikonal self-supervisionBinary labels (easier to obtain)
RF use casePrecise surface reconstruction (GeRaF)Obstruction mapping, coarse scene layout
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Example: Occupancy Network for Wall Detection

A mmWave radar system observes an L-shaped room with two walls meeting at a right angle. Compare the occupancy network and SDF reconstructions of the wall geometry.

Solution
Occupancy reconstruction

The occupancy network produces a smooth transition from o≈1o \approx 1 (inside the walls) to o≈0o \approx 0 (in the room). The decision boundary at o=0.5o = 0.5 correctly identifies the wall surfaces, but the transition width varies with the sigmoid sharpness, producing rounded corners at the L-junction.

SDF reconstruction

The neural SDF produces f≈0f \approx 0 at the wall surfaces with ∥∇f∥≈1\|\nabla f\| \approx 1. The corner is better preserved because the distance field naturally has a sharp kink at the intersection of two planes (the gradient changes direction discontinuously). The SDF also provides the distance to the nearest wall at every point in the room --- useful for path-loss prediction.

Comparison

For simple obstruction detection ("is there a wall here?"), both methods perform comparably. For geometric operations that require distance information (path-loss computation, ray tracing for beam management), the SDF is superior.

Occupancy vs. SDF Reconstruction

Compare occupancy network and SDF reconstructions of a 2D scene. The left panel shows the occupancy field (green = inside, white = outside); the right panel shows the SDF (blue = outside, red = inside, white = surface). Observe how the SDF provides distance information while the occupancy field is binary.

Parameters
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Quick Check

What is the primary advantage of occupancy networks over SDFs for RF imaging applications?

They provide surface normals.

They require simpler supervision (binary labels instead of distance values).

They render faster than SDFs.

They produce higher-resolution surfaces.

Correction:
They require simpler supervision (binary labels instead of distance values).

Occupancy networks require only inside/outside labels, which are easier to obtain than signed distance values. However, they provide less geometric information than SDFs.

Quick Check

The surface extracted from an occupancy network with threshold 0.50.5 is equivalent to which SDF-derived surface?

The surface where f(p)=1f(\mathbf{p}) = 1.

The zero level set {p:f(p)=0}\{\mathbf{p} : f(\mathbf{p}) = 0\}, via the sigmoid transform.

The surface of maximum gradient magnitude.

There is no equivalence; the representations are fundamentally different.

Correction:
The zero level set {p:f(p)=0}\{\mathbf{p} : f(\mathbf{p}) = 0\}, via the sigmoid transform.

If o(p)=σ(−αf(p))o(\mathbf{p}) = \sigma(-\alpha f(\mathbf{p})), then o=0.5o = 0.5 when f=0f = 0. The two surface definitions coincide.

🔧Engineering Note

Occupancy Maps for Beam Blockage Prediction

In 5G NR and beyond, beam blockage by pedestrians, vehicles, and furniture is a major cause of link failure. An occupancy network trained from periodic radar sweeps can maintain a real-time 3D occupancy map of the environment. A simple ray-tracing query through the occupancy field predicts whether a given beam direction is blocked, enabling proactive beam switching before the blockage event occurs.

Since blockage prediction requires only binary information (blocked / not blocked), the occupancy representation suffices --- the additional distance information from an SDF is not needed for this application.

Occupancy Network

A neural network that maps a 3D coordinate p\mathbf{p} to a probability oθ(p)∈[0,1]o_\theta(\mathbf{p}) \in [0, 1] indicating whether the point is inside (o=1o = 1) or outside (o=0o = 0) an object.

Related: Signed Distance Function (SDF)

Key Takeaway

Occupancy networks provide a simple, binary implicit representation for 3D scenes. They require only inside/outside labels for supervision, making them easier to train than SDFs. However, they lack distance information and surface normals, limiting their use to coarse scene understanding tasks like obstruction mapping. For RF imaging, occupancy networks complement SDFs: use occupancy for fast blockage prediction, SDFs for precise surface reconstruction.

GeRaF: Geometry from mmWave RadarEikonal Regularization and Material Estimation