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Why Signed Distance Functions for RF Imaging?

Classical RF imaging (Chapters 13--14) reconstructs the reflectivity $c(\mathbf{p})$ on a discrete voxel grid. This approach has two fundamental limitations: (1) the memory cost scales as $O(N^3)$ for a 3D grid of side $N$ , and (2) the representation cannot express sub-voxel geometry.

Signed distance functions offer an alternative: a continuous scalar field whose zero level set defines the surface, independent of any grid. When parameterised by a neural network, the SDF can represent complex 3D shapes with far fewer parameters than a voxel grid. More importantly for RF, the SDF provides surface normals (via the gradient) and interior/exterior classification --- both essential for physically-grounded scattering models.

This section develops the SDF formalism and its neural parameterisation as preparation for GeRaF (Section 25.2), which applies SDFs to mmWave radar reconstruction.

Definition:
Signed Distance Function

A signed distance function (SDF) $f \colon \mathbb{R}^3 \to \mathbb{R}$ assigns to each point $\mathbf{p} \in \mathbb{R}^3$ the signed distance to the nearest surface $\mathcal{S}$ :

$f(\mathbf{p}) = \begin{cases} -d(\mathbf{p}, \mathcal{S}) & \text{if } \mathbf{p} \text{ is inside the object,} \\ \;\; 0 & \text{if } \mathbf{p} \in \mathcal{S}, \\ +d(\mathbf{p}, \mathcal{S}) & \text{if } \mathbf{p} \text{ is outside,} \end{cases}$

where $d(\mathbf{p}, \mathcal{S}) = \min_{\mathbf{q} \in \mathcal{S}} \|\mathbf{p} - \mathbf{q}\|$ .

The surface is the zero level set: $\mathcal{S} = \{\mathbf{p} : f(\mathbf{p}) = 0\}$ .

Theorem: The Eikonal Equation

If $f$ is a valid signed distance function, then the gradient norm equals unity almost everywhere:

$\|\nabla f(\mathbf{p})\| = 1, \quad \text{a.e. } \mathbf{p} \in \mathbb{R}^3.$

This is the Eikonal equation. At any point on the surface $\mathcal{S}$ , the gradient $\nabla f(\mathbf{p})/\|\nabla f(\mathbf{p})\|$ gives the outward-pointing surface normal.

The SDF measures distance; moving one unit in any direction changes the distance by at most one unit (Lipschitz constant $= 1$ ). Furthermore, the gradient points in the direction of steepest ascent (away from the surface), and by the distance property the rate of increase is exactly $1$ .

Proof

Lipschitz property

By the triangle inequality, for any $\mathbf{p}, \mathbf{q} \in \mathbb{R}^3$ : $|f(\mathbf{p}) - f(\mathbf{q})| \leq \|\mathbf{p} - \mathbf{q}\|$ . Hence $f$ is Lipschitz with constant $1$ , and by Rademacher's theorem it is differentiable a.e. with $\|\nabla f\| \leq 1$ .

Lower bound on the gradient

Let $\mathbf{p}^* = \arg\min_{\mathbf{q} \in \mathcal{S}} \|\mathbf{p} - \mathbf{q}\|$ be the closest surface point. The function $t \mapsto f(\mathbf{p} + t(\mathbf{p}^* - \mathbf{p})/\|\mathbf{p}^* - \mathbf{p}\|)$ decreases from $f(\mathbf{p})$ to $0$ over distance $\|\mathbf{p} - \mathbf{p}^*\| = |f(\mathbf{p})|$ . The directional derivative in this direction has magnitude $1$ , so $\|\nabla f(\mathbf{p})\| \geq 1$ .

Combining

From the two bounds, $\|\nabla f(\mathbf{p})\| = 1$ wherever $f$ is differentiable. By Rademacher's theorem, this holds almost everywhere. $\blacksquare$

,

Definition:
Neural SDF (DeepSDF)

A neural SDF parameterises $f$ as a multi-layer perceptron (MLP):

$f_\theta(\mathbf{p}) = \text{MLP}_\theta(\gamma(\mathbf{p})),$

where $\gamma(\mathbf{p})$ is a positional encoding:

$\gamma(\mathbf{p}) = \bigl(\sin(2^0 \pi \mathbf{p}),\, \cos(2^0 \pi \mathbf{p}),\, \ldots,\, \sin(2^{L-1} \pi \mathbf{p}),\, \cos(2^{L-1} \pi \mathbf{p})\bigr).$

The positional encoding maps the 3D coordinate to a higher-dimensional space, enabling the MLP to represent high-frequency spatial detail despite its inherent spectral bias toward smooth functions.

Without positional encoding, standard MLPs with smooth activations (ReLU, Softplus) converge slowly to high-frequency targets. With $L = 10$ , the encoding provides frequencies up to $2^9 \approx 500$ cycles per unit length, sufficient for sub-wavelength RF imaging detail.

Definition:
Sphere Tracing

Sphere tracing finds the intersection of a ray $\mathbf{r}(t) = \mathbf{o} + t\mathbf{d}$ with the zero level set of an SDF $f$ . The algorithm iterates:

$t_{k+1} = t_k + f(\mathbf{r}(t_k)), \qquad t_0 = t_{\min}.$

At each step, the SDF value $f(\mathbf{r}(t_k))$ gives the distance to the nearest surface. Since $\|\nabla f\| = 1$ , the ball of radius $f(\mathbf{r}(t_k))$ is guaranteed to be surface-free, so the step never overshoots the surface.

Termination: When $|f(\mathbf{r}(t_k))| < \varepsilon$ (surface hit) or $t_k > t_{\max}$ (ray missed).

Sphere tracing is much more efficient than dense ray marching because it takes large steps in empty regions and small steps near surfaces. For RF imaging, sphere tracing enables efficient computation of ray-surface intersections needed for differentiable rendering (Section 25.2).

Sphere Tracing Algorithm

Complexity: Convergence is linear: near the surface,

f(\mathbf{r}(t_k)) \to 0

at rate

O(1/k)

for convex surfaces.

Input: Ray origin

\mathbf{o}

, direction

\mathbf{d}

,

SDF

f

, tolerance

\varepsilon

, maximum distance

t_{\max}

Output: Intersection distance

t^*

or NO_HIT

1.

t \leftarrow 0

2. while

t < t_{\max}

do

3.

\quad \mathbf{p} \leftarrow \mathbf{o} + t \cdot \mathbf{d}

4.

\quad d_{\text{sdf}} \leftarrow f(\mathbf{p})

5.

\quad

if

|d_{\text{sdf}}| < \varepsilon

then

6.

\quad\quad

return

t

7.

\quad

end if

8.

\quad t \leftarrow t + d_{\text{sdf}}

9. end while

10. return NO_HIT

For neural SDFs where $\|\nabla f_\theta\| \neq 1$ exactly, a safety factor $\alpha \in (0, 1)$ can be used: $t_{k+1} = t_k + \alpha \cdot f_\theta(\mathbf{r}(t_k))$ .

2D Signed Distance Field Visualization

Visualise the signed distance field for different 2D shapes. Blue regions are outside (positive SDF), red regions are inside (negative SDF), and the white contour marks the zero level set (the surface). Observe how $\|\nabla f\| = 1$ everywhere for exact SDFs.

Parameters

Shape

Grid resolution100

Sphere Tracing in 2D

Watch sphere tracing find the ray-surface intersection for a 2D SDF. Each step advances by the SDF value at the current position (shown as a circle). In empty regions the steps are large; near the surface they shrink to zero.

Parameters

Scene shape

Ray angle (degrees)0

Example: Analytical SDFs for RF-Relevant Primitives

Write the SDF for (a) a sphere of radius $R$ centred at the origin, (b) an infinite half-space (modelling a wall), and (c) a cylinder of radius $R$ along the $z$ -axis (modelling a pipe or column). Verify the Eikonal equation for each.

Solution

Sphere

$f_{\text{sphere}}(\mathbf{p}) = \|\mathbf{p}\| - R.$ $Gradient:$ \nabla f = \mathbf{p}/|\mathbf{p}| $, so$ |\nabla f| = 1 $for$ \mathbf{p} \neq \mathbf{0}$.

Half-space (wall at $x = 0$)

$f_{\text{wall}}(\mathbf{p}) = p_x.$ $Gradient:$ \nabla f = (1, 0, 0)^T $, so$ |\nabla f| = 1 $everywhere. The zero level set is the plane$ x = 0$.

Cylinder (radius $R$, axis along $z$)

$f_{\text{cyl}}(\mathbf{p}) = \sqrt{p_x^2 + p_y^2} - R.$ $Gradient:$ \nabla f = (p_x, p_y, 0)^T / \sqrt{p_x^2 + p_y^2} $, giving$ |\nabla f| = 1 $away from the$ z$-axis.

Example: Constructive Solid Geometry via SDF Operations

Given SDFs $f_A$ and $f_B$ for two objects, construct the SDF for (a) the union $A \cup B$ and (b) the intersection $A \cap B$ . Explain why these operations do not preserve the exact distance property.

Solution

Union and intersection

Union: $f_{A \cup B}(\mathbf{p}) = \min(f_A(\mathbf{p}), f_B(\mathbf{p}))$ .
Intersection: $f_{A \cap B}(\mathbf{p}) = \max(f_A(\mathbf{p}), f_B(\mathbf{p}))$ .

The zero level sets are correct: a point is inside $A \cup B$ iff $\min(f_A, f_B) < 0$ .

Distance property violation

Consider two overlapping spheres. At a point equidistant from both surfaces but closer to a third point on the union boundary, $\min(f_A, f_B)$ overestimates the true distance to the union surface. Formally, $\|\nabla \min(f_A, f_B)\| \leq 1$ but may be strictly less than $1$ near the intersection seam. For neural SDFs, the Eikonal regulariser (Section 25.4) corrects this during training.

SDF vs. Voxel Grid: Memory and Resolution

A voxel grid at resolution $N$ requires $O(N^3)$ storage. For $N = 512$ , this is $\sim 134$ million voxels --- prohibitive for many real-time applications. A neural SDF with a modest MLP ( $\sim 10^5$ parameters) can represent the same geometry continuously, with the surface extractable at arbitrary resolution via marching cubes.

For RF imaging, the practical benefit is even more pronounced: the wavelength $\lambda$ at $60$ GHz is $\sim 5$ mm, so $\lambda$ -scale voxels over a $5 \times 5 \times 3$ m room require $\sim 10^9$ voxels. A neural SDF bypasses this entirely.

Historical Note: The Eikonal Equation: From Optics to Neural Geometry

1828--2020

The Eikonal equation $\|\nabla f\| = 1$ originates in geometrical optics, where it describes the propagation of wavefronts in media with unit refractive index. Hamilton (1828) and later Bruns (1895) developed the theory in the context of ray optics. The equation reappeared in computational geometry when Sethian (1996) used it as the foundation of the fast marching method for computing distance functions on grids. Park et al. (2019) brought the Eikonal equation into deep learning by using it as a regulariser for neural SDFs, and Gropp et al. (2020) showed that Eikonal regularization alone --- without ground-truth distance supervision --- suffices to learn valid SDFs from raw point clouds.

Historical Note: DeepSDF and the Rise of Neural Implicit Representations

2019--2020

Park et al. introduced DeepSDF at CVPR 2019, demonstrating that a single MLP can represent hundreds of 3D shapes via a learned latent code. The auto-decoder architecture (no encoder; optimise the latent code directly) proved both simple and powerful. Combined with Mildenhall et al.'s NeRF (2020), DeepSDF sparked the "neural implicit revolution" in computer vision --- a shift from explicit discrete representations (meshes, voxels) to continuous, differentiable function approximations. For RF imaging, this shift is particularly natural: electromagnetic fields are themselves continuous functions of space, and neural implicits provide a native representation.

Quick Check

The Eikonal equation $\|\nabla f(\mathbf{p})\| = 1$ implies which of the following?

The SDF is a convex function.

The SDF is Lipschitz continuous with constant $1$ .

The surface normal is always vertical.

The SDF has no local minima.

Correction:

The SDF is Lipschitz continuous with constant

1

.

$\|\nabla f\| = 1$ everywhere implies $|f(\mathbf{p}) - f(\mathbf{q})| \leq \|\mathbf{p} - \mathbf{q}\|$ , which is exactly Lipschitz continuity with constant $1$ .

Quick Check

In sphere tracing, what determines the step size at iteration $k$ ?

A fixed step size chosen before tracing.

The SDF value $f(\mathbf{r}(t_k))$ at the current position.

The gradient magnitude $\|\nabla f\|$ .

The dot product of the ray direction and the surface normal.

Correction:

The SDF value

f(\mathbf{r}(t_k))

at the current position.

The SDF value gives the distance to the nearest surface, so stepping by $f(\mathbf{r}(t_k))$ is the largest step that guarantees no overshoot.

Common Mistake: Approximate SDFs Break Sphere Tracing

Mistake:

Treating a neural network output as an exact SDF and using full-step sphere tracing: $t_{k+1} = t_k + f_\theta(\mathbf{r}(t_k))$ .

Correction:

Neural SDFs satisfy $\|\nabla f_\theta\| \approx 1$ but not exactly. If $\|\nabla f_\theta\| < 1$ locally, the network overestimates the distance and the ray may overshoot the surface. Use a safety factor $\alpha \in (0.5, 0.9)$ or clip the step size: $t_{k+1} = t_k + \min(f_\theta(\mathbf{r}(t_k)),\, \delta_{\max})$ .

Common Mistake: Spectral Bias Without Positional Encoding

Mistake:

Training an MLP directly on 3D coordinates $\mathbf{p}$ without positional encoding, expecting it to learn sharp features at the wavelength scale.

Correction:

Standard MLPs are biased toward low-frequency functions. The positional encoding $\gamma(\mathbf{p})$ lifts the input into a high-dimensional space where high-frequency targets become smooth, enabling the MLP to represent sub-wavelength detail. For RF imaging at $f_0 = 60$ GHz ( $\lambda \approx 5$ mm), features at the wavelength scale require $L \geq 10$ encoding levels.

Signed Distance Function (SDF)

A scalar field $f(\mathbf{p})$ that returns the signed distance from a point $\mathbf{p}$ to the nearest surface: negative inside, zero on the surface, positive outside.

Eikonal Equation

The constraint $\|\nabla f(\mathbf{p})\| = 1$ that characterises valid signed distance functions. Used as a regularisation loss during neural SDF training.

Related: Signed Distance Function (SDF)

Zero Level Set

The surface $\mathcal{S} = \{\mathbf{p} : f(\mathbf{p}) = 0\}$ implicitly defined by a signed distance function.

Sphere Tracing

A ray-marching algorithm that exploits the distance property of SDFs to take variable-size steps along a ray, reaching the surface in $O(\log(1/\varepsilon))$ steps for smooth geometry.

Related: Signed Distance Function (SDF)

Key Takeaway

Signed distance functions provide a continuous, memory-efficient representation of 3D geometry. The Eikonal equation $\|\nabla f\| = 1$ characterises valid SDFs and serves as a regulariser for neural parameterisations. Sphere tracing enables efficient ray-surface intersection. For RF imaging, neural SDFs bypass the $O(N^3)$ voxel grid bottleneck while providing surface normals essential for physically-grounded scattering models.

Signed Distance Functions for RF

Why Signed Distance Functions for RF Imaging?

Definition: Signed Distance Function

Theorem: The Eikonal Equation

Lipschitz property

Lower bound on the gradient

Combining

Definition: Neural SDF (DeepSDF)

Definition: Sphere Tracing

Sphere Tracing Algorithm

2D Signed Distance Field Visualization

Parameters

Sphere Tracing in 2D

Parameters

Example: Analytical SDFs for RF-Relevant Primitives

Sphere

Half-space (wall at $x = 0$)

Cylinder (radius $R$, axis along $z$)

Example: Constructive Solid Geometry via SDF Operations

Union and intersection

Distance property violation

SDF vs. Voxel Grid: Memory and Resolution

Historical Note: The Eikonal Equation: From Optics to Neural Geometry

Historical Note: DeepSDF and the Rise of Neural Implicit Representations

Quick Check

Quick Check

Common Mistake: Approximate SDFs Break Sphere Tracing

Common Mistake: Spectral Bias Without Positional Encoding

Signed Distance Function (SDF)

Eikonal Equation

Zero Level Set

Sphere Tracing

Key Takeaway

Definition:
Signed Distance Function

Definition:
Neural SDF (DeepSDF)

Definition:
Sphere Tracing