Exercises

$\nabla f(\mathbf{p}) = \frac{\mathbf{p}}{\|\mathbf{p}\|}.$$

$\|\nabla f\| = \frac{\|\mathbf{p}\|}{\|\mathbf{p}\|} = 1 \quad \text{for all } \mathbf{p} \neq \mathbf{0}. \quad \blacksquare$$

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

Union and intersection preserve the correct zero level set but may violate the exact distance property ($\|\nabla f\| = 1$) away from the surface. The complement preserves the distance property exactly: $\|\nabla(-f)\| = \|\nabla f\| = 1$. $\blacksquare$

• $t_0 = 0$: $f = |5 - 0| - 1 = 4$, so $t_1 = 0 + 4 = 4$.
• $t_1 = 4$: $f = |5 - 4| - 1 = 0$. Surface hit!

In this case, sphere tracing converges in a single step because the ray passes through the centre of the sphere, where the SDF along the ray is exactly affine. $\blacksquare$

$f_{\text{cyl}}(\mathbf{p}) = \sqrt{p_x^2 + p_y^2} - R.$$

$\nabla f = \frac{1}{\sqrt{p_x^2 + p_y^2}}(p_x, p_y, 0)^T.KATEXPLACEHOLDER0END\|\nabla f\| = \frac{\sqrt{p_x^2 + p_y^2}}{\sqrt{p_x^2 + p_y^2}} = 1 \quad \text{for } (p_x, p_y) \neq (0, 0). \quad \blacksquare$$

• (a) $f = -1$, $o = \sigma(10) = 1/(1+e^{-10}) \approx 0.99995$ (inside).
• (b) $f = 0$, $o = \sigma(0) = 0.5$ (on the surface).
• (c) $f = 1$, $o = \sigma(-10) = 1/(1+e^{10}) \approx 0.00005$ (outside). $\blacksquare$

The positional encoding introduces frequencies $2^0, 2^1, \ldots, 2^{L-1}$ cycles per unit length. The maximum is $f_{\max} = 2^{L-1}$.

To resolve $\lambda = 5$ mm features in a $D = 1$ m scene: $f_{\max} \geq D/\lambda = 200$. Thus $2^{L-1} \geq 200$, giving $L \geq 9$ ($2^8 = 256 \geq 200$). In practice, $L = 10$ provides a safety margin. $\blacksquare$

$\frac{\partial \mathcal{L}}{\partial \theta} = \frac{2}{Q}\sum_{q=1}^{Q} \bigl(\hat{P}_{\text{MF}}(\mathbf{p}_q) - P_{\text{MF}}(\mathbf{p}_q)\bigr) \cdot \frac{\partial \hat{P}_{\text{MF}}(\mathbf{p}_q)}{\partial \theta}.$$

$\frac{\partial \hat{P}_{\text{MF}}}{\partial \theta} = \sum_{m,n} w_{mn} \Gamma_\phi \cdot \delta_\sigma'(f_\theta) \cdot \frac{\partial f_\theta}{\partial \theta},$$where$\delta_\sigma'(s) = -s/\sigma^2 \cdot \delta_\sigma(s)$is the derivative of the smoothed delta function. The gradient$\partial f_\theta / \partial \theta$is computed by standard backpropagation through the MLP.$\blacksquare$

The MF image has resolution limited by the PSF. However, the neural SDF is a continuous function, not a discrete pixel grid. The Eikonal constraint $\|\nabla f\| = 1$ enforces smoothness consistent with a valid distance function, acting as a strong prior that interpolates between the MF resolution elements.

Just as LASSO recovers sparse signals beyond the Nyquist limit by exploiting sparsity, the neural SDF recovers sub-resolution geometry by exploiting the SDF structure (smoothness + unit gradient). The neural network acts as an implicit regulariser, biasing the solution toward geometrically plausible shapes.

Super-resolution is not unlimited: the SNR and the number of views determine how much geometric detail can be reliably recovered. Features significantly smaller than the PSF may not be distinguishable from noise. $\blacksquare$

For TE polarisation: $$|R_\perp(\theta)|^2 = \left|\frac{\cos\theta - \sqrt{\epsilon_r - \sin^2\theta}}{\cos\theta + \sqrt{\epsilon_r - \sin^2\theta}}\right|^2.$$

At $\theta_1 = 30^\circ$: $\cos 30^\circ = \sqrt{3}/2$, $\sin^2 30^\circ = 1/4$. Setting $|R_\perp|^2 = 0.04$: $$\left|\frac{\sqrt{3}/2 - \sqrt{\epsilon_r - 1/4}}{\sqrt{3}/2 + \sqrt{\epsilon_r - 1/4}}\right|^2 = 0.04.$$ Taking the positive square root: $|R_\perp| = 0.2$, giving $\sqrt{3}/2 - \sqrt{\epsilon_r - 1/4} = 0.2(\sqrt{3}/2 + \sqrt{\epsilon_r - 1/4})$. Solving: $\epsilon_r \approx 4.0$.

Verification at $\theta_2 = 60^\circ$: $|R_\perp(60^\circ)|^2$ with $\epsilon_r = 4$ gives $\approx 0.16$, close to the measured $0.15$ (consistent within measurement noise). $\blacksquare$

p = torch.randn(N, 3, requires_grad=True)
sdf = model(p)  # shape (N, 1)
grad_sdf = torch.autograd.grad(
    outputs=sdf,
    inputs=p,
    grad_outputs=torch.ones_like(sdf),
    create_graph=True  # needed for Eikonal loss backprop
)[0]  # shape (N, 3)
eikonal_loss = ((grad_sdf.norm(dim=-1) - 1) ** 2).mean()

Setting create_graph=True builds the computational graph through the gradient computation, enabling backpropagation of the Eikonal loss through $\nabla f_\theta$ and into $\theta$. Without create_graph=True, the gradient $\nabla_\theta \mathcal{L}_{\text{eik}}$ would not be available. $\blacksquare$

For the sphere SDF $f(\mathbf{p}) = \|\mathbf{p}\| - R$: $\mathbf{n} = \nabla f / \|\nabla f\| = \mathbf{p}/\|\mathbf{p}\|$, which is the exact outward normal. The neural approximation $f_\theta$ has $\|\nabla f_\theta\| \approx 1$ near the surface (enforced by Eikonal loss), so the normals are accurate.

For $o(\mathbf{p}) = \sigma(-\alpha(\|\mathbf{p}\| - R))$: $\nabla o = -\alpha \sigma'(-\alpha f) \cdot \mathbf{p}/\|\mathbf{p}\|$. The direction is correct, but the magnitude depends on $\alpha$ and decays exponentially away from the surface. For a neural occupancy network, the gradient is noisy because the network learns a near-step function, whose gradient is poorly conditioned. $\blacksquare$

For an MLP with $L$ layers and skip connections, set the final layer bias to $-R_0$ and the final layer weights so that the output approximates $\|\mathbf{p}\| - R_0$. This can be achieved by: (1) using the geometric initialisation of SAL (Atzmon and Lipman, 2020), which sets the last layer to compute the distance from a sphere of radius $R_0$, or (2) pre-training the MLP on the analytical sphere SDF for a few hundred iterations.

Without geometric initialisation, the random MLP output has many disconnected zero level sets (small "bubbles"). The Eikonal loss and data loss must eliminate all spurious surfaces while growing the correct surface, which is slow and prone to local minima. With geometric initialisation, the SDF starts as a single large sphere that contracts toward the true surface --- a much smoother optimisation path. $\blacksquare$

NeRF's volume rendering equation integrates density and colour along a single ray per pixel. This works because the camera lens maps each pixel to a unique ray direction, providing angular selectivity. The loss compares rendered pixel colours to observed pixel colours, one ray at a time.

Each radar measurement $y_{m,n,k}$ (Tx $m$, Rx $n$, subcarrier $k$) integrates contributions from all scatterers in the scene, weighted by the Tx-Rx beamforming gain and propagation delay. There is no one-to-one mapping from measurements to spatial directions. Directly applying NeRF's per-ray rendering would require "focusing" the data first.

GeRaF uses the matched filter to produce the MF power image $P_{\text{MF}}(\mathbf{p}_q)$, which concentrates energy at scatterer locations. The MF power image serves as a pseudo-observation analogous to a camera image: each voxel $q$ corresponds to a spatial location, and the neural SDF renders the predicted MF power at that location. The L2 loss between observed and predicted MF power drives the training. $\blacksquare$

Sample $\mathbf{p} \sim \mathcal{U}(\Omega)$ uniformly in the bounding box. Advantage: simple, no bias, enforces the Eikonal equation globally. Disadvantage: most samples fall far from the surface, where the Eikonal constraint is automatically satisfied by a well-initialised network; these samples contribute little useful gradient.

Sample points near the current zero level set: $\mathbf{p} = \mathbf{p}_{\text{surface}} + \epsilon \cdot \mathbf{n}$ where $\epsilon \sim \mathcal{N}(0, \sigma_{\text{sample}}^2)$ and $\mathbf{n} = \nabla f_\theta / \|\nabla f_\theta\|$. Advantage: concentrates samples where the Eikonal constraint matters most. Disadvantage: biased toward the current surface estimate; may miss regions where the SDF is incorrect.

Use a mixture: 50% uniform, 50% near-surface. The uniform samples prevent global Eikonal violations; the near-surface samples ensure accurate surface geometry. $\blacksquare$

1. Evaluate $f_\theta(\mathbf{p})$ at all $N^3$ grid vertices.
2. For each of the $(N-1)^3$ voxels, classify each of 8 corners as inside ($f < 0$) or outside ($f > 0$).
3. Look up the triangle configuration from a precomputed table of $2^8 = 256$ cases.
4. For each edge where the sign changes, interpolate the vertex position using $\mathbf{p}_{\text{iso}} = \mathbf{p}_1 + f(\mathbf{p}_1)/(f(\mathbf{p}_1) - f(\mathbf{p}_2)) \cdot (\mathbf{p}_2 - \mathbf{p}_1)$.

The mesh resolution is determined by the grid spacing $\Delta = L/N$ where $L$ is the scene size. Features smaller than $\Delta$ are aliased. The computational cost is $O(N^3)$ for the MLP evaluations (the bottleneck) plus $O(N^3)$ for the marching cubes lookup (negligible). For $N = 256$, this requires $\sim 16.8$ million MLP evaluations. $\blacksquare$

The rendered MF power is $\hat{P}_{\text{MF}} \propto \Gamma_\phi(\mathbf{p}_q) \cdot \delta_\sigma(f_\theta(\mathbf{p}_q))$. A high observed power $P_{\text{MF}}$ can be explained by: (1) $\Gamma$ large and $f_\theta \approx 0$ (strong reflector on the surface), or (2) $\Gamma$ moderate and $f_\theta \approx 0$ with a different surface shape that concentrates the MF response. The product $\Gamma \cdot \delta_\sigma(f_\theta)$ admits infinitely many factorisations from a single observation.

Different views observe the same surface at different incidence angles. The geometry (SDF) is view-independent, but the effective reflectivity varies with angle (Fresnel coefficients). By requiring the SDF to explain all views simultaneously, the geometry is fixed, and the reflectivity is then determined by the residual. With $V \geq 2$ views at sufficiently different angles, the ambiguity is resolved. $\blacksquare$

Start with $\sigma_0 = \sigma_{\max}$ (e.g., several voxel widths) and decay exponentially: $$\sigma(t) = \sigma_{\max} \cdot \exp(-t / \tau),$$ with $\tau$ chosen so that $\sigma(T) \approx \sigma_{\min}$ (one voxel width or less).

• Large $\sigma$ (early training): The smoothed delta spreads the gradient signal over a wide region around the current surface, enabling the SDF to capture coarse geometry even when the initial surface is far from the truth.
• Small $\sigma$ (late training): The delta concentrates on the surface, forcing the SDF to precisely locate it. If $\sigma$ is small from the start, the gradient signal is too localised and the surface cannot move to the correct position.

This annealing is analogous to curriculum learning: coarse-to-fine optimisation avoids bad local minima. NeuS (Wang et al., 2021) uses a similar annealing for the logistic density function in SDF-based volume rendering. $\blacksquare$

A shared MLP backbone $h_\theta(\gamma(\mathbf{p})) \in \mathbb{R}^{256}$ feeds two heads:

• SDF head: $f_\theta(\mathbf{p}) = \text{Linear}(h_\theta) \in \mathbb{R}$.
• Material head: $\mathbf{m}_\theta(\mathbf{p}) = \text{Softmax}(\text{Linear}(h_\theta)) \in \Delta^{K-1}$, where $K = 4$ (concrete, glass, metal, drywall).

$\mathcal{L} = \mathcal{L}_{\text{data}} + \lambda_{\text{eik}} \mathcal{L}_{\text{eik}} + \lambda_{\text{mat}} \mathcal{L}_{\text{mat}},$$where$\mathcal{L}{\text{mat}} = -\sum_q \sum_k m_k^*(\mathbf{p}q) \log m{k,\theta}(\mathbf{p}q)$is the cross-entropy loss (if labels available), or$\mathcal{L}{\text{mat}} = \text{entropy}(\mathbf{m}\theta)$ to encourage low-entropy (decisive) classifications.

The Fresnel reflection coefficient depends on $\epsilon_r$, which is determined by the material class. The rendered MF power uses $\Gamma(\mathbf{p}) = \sum_k m_k \Gamma_k(\theta_i, \epsilon_{r,k})$, where $\Gamma_k$ is the Fresnel coefficient for material $k$ at incidence angle $\theta_i$. This makes the data fidelity loss depend on the material classification, providing a self-supervised signal for material segmentation. $\blacksquare$

• Exact Eikonal: No need for Eikonal regularisation; the analytical SDFs satisfy $\|\nabla f\| = 1$ exactly.
• Few parameters: An L-shaped room is described by $\sim 10$ parameters (wall positions, corner locations).
• Interpretability: Parameters have physical meaning (wall position = 3.5 m).
• Fast inference: No MLP evaluation; SDF computation is $O(1)$ per point.

• Expressiveness: Can represent arbitrary shapes (furniture, curved surfaces, irregular geometry).
• Data-driven: Learns from measurements without manual scene modelling.
• Joint optimisation: Geometry and materials are optimised end-to-end.

Use primitives when the scene is well-modelled by simple shapes (empty rooms, corridors, warehouses). Use neural SDFs when the scene contains complex or unknown geometry. A hybrid approach --- primitive backbone with neural residual --- combines the strengths of both: the primitives capture the dominant structure, and the neural network refines the details. $\blacksquare$

The first term forces $f_\theta(0) = 0$. The Eikonal term forces $|f_\theta'(x)| = 1$ for all $x \in [-1, 1]$. The only continuous functions satisfying both are $f(x) = x$ and $f(x) = -x$ (in the SDF convention, $f(x) = x$ with positive to the right). Both are valid SDFs for the surface $\{0\}$.

The true signed distance to the point $\{0\}$ on the real line is $f(x) = |x|$, which is the unsigned distance. However, $|x|$ is not differentiable at $x = 0$. The smooth neural approximation $f_\theta(x) \approx \sqrt{x^2 + \varepsilon^2} - \varepsilon$ for small $\varepsilon$ satisfies $|f'| \approx 1$ away from $0$ and transitions smoothly at $0$.

Under gradient descent with learning rate $\eta$, the Eikonal term converges at rate $O(1/t)$ (for smooth losses), while the surface term converges exponentially. The overall convergence is limited by the Eikonal term and depends on the MLP architecture (depth, width, activation function). $\blacksquare$