Exercises

$\mathbf{x}^\mathsf{T}\boldsymbol{\Sigma}\mathbf{x} = \mathbf{x}^\mathsf{T}\mathbf{R}\mathbf{S}\mathbf{S}^\mathsf{T}\mathbf{R}^\mathsf{T}\mathbf{x} = \|\mathbf{S}^\mathsf{T}\mathbf{R}^\mathsf{T}\mathbf{x}\|^2 \geq 0.$$

Since $\|\mathbf{y}\|^2 \geq 0$ for any vector $\mathbf{y}$, the matrix $\boldsymbol{\Sigma}$ is PSD for any rotation $\mathbf{R}$ and any scale $\mathbf{S}$. It is strictly positive definite when all diagonal entries of $\mathbf{S}$ are nonzero. $\blacksquare$

• $T_1 = 1$ (no preceding Gaussians)
• $T_2 = 1 - \alpha_1 \cdot 1 = 0.2$
• $T_3 = (1 - \alpha_1)(1 - \alpha_2) = 0.2 \times 0.4 = 0.08$

$\hat{C} = f_1 \alpha_1 T_1 + f_2 \alpha_2 T_2 + f_3 \alpha_3 T_3KATEXPLACEHOLDER0END= 1.0 \times 0.8 \times 1 + 0.5 \times 0.6 \times 0.2 + 0.2 \times 0.9 \times 0.08KATEXPLACEHOLDER1END= 0.8 + 0.06 + 0.0144 = 0.8744.$$The first Gaussian dominates because it is closest to the camera and has high opacity.$\blacksquare$

The quaternion $\mathbf{q} = (w, x, y, z) = (1, 0, 0, 0)$ gives $\mathbf{R} = \mathbf{I}_3$ (the identity matrix), since:

$$\mathbf{R} = \begin{pmatrix} 1 - 2(y^2 + z^2) & 2(xy - wz) & 2(xz + wy) \\ 2(xy + wz) & 1 - 2(x^2 + z^2) & 2(yz - wx) \\ 2(xz - wy) & 2(yz + wx) & 1 - 2(x^2 + y^2) \end{pmatrix} = \mathbf{I}_3.$$

With $\mathbf{R} = \mathbf{I}$, the covariance becomes $\boldsymbol{\Sigma} = \mathbf{S}\mathbf{S}^\mathsf{T} = \text{diag}(s_x^2, s_y^2, s_z^2)$. The Gaussian is axis-aligned with semi-axes along the coordinate axes. No rotation is applied. $\blacksquare$

• Model 1: $(\hat{P}_1 - P)^2 = (-77 - (-80))^2 = 9$ dB$^2$
• Model 2: $(\hat{P}_2 - P)^2 = (-83 - (-80))^2 = 9$ dB$^2$

Both predictions have equal error in dB scale.

Converting: $P = 10^{-8}$ mW, $\hat{P}_1 = 10^{-7.7} \approx 2 \times 10^{-8}$ mW, $\hat{P}_2 = 10^{-8.3} \approx 5 \times 10^{-9}$ mW.

• Model 1: $(\hat{P}_1 - P)^2 = (2 \times 10^{-8} - 10^{-8})^2 = 10^{-16}$
• Model 2: $(\hat{P}_2 - P)^2 = (5 \times 10^{-9} - 10^{-8})^2 = 2.5 \times 10^{-17}$

Model 2 has $4\times$ smaller loss in linear scale, despite having the same absolute error in dB. Linear-scale loss is dominated by high-power regions, ignoring errors at low power. $\blacksquare$

Let $\pi: \mathbb{R}^3 \to \mathbb{R}^2$ be the perspective projection. The Jacobian at $\boldsymbol{\mu}$ is $\mathbf{J} = \frac{\partial \pi}{\partial \mathbf{p}}\big|_{\boldsymbol{\mu}} \in \mathbb{R}^{2 \times 3}$.

Under the linear approximation (EWA splatting), the projected covariance is:

$$\boldsymbol{\Sigma}' = \mathbf{J}\mathbf{W}\boldsymbol{\Sigma}\mathbf{W}^\mathsf{T}\mathbf{J}^\mathsf{T} \in \mathbb{R}^{2 \times 2},$$

where $\mathbf{W}$ is the world-to-camera rotation.

Since $\boldsymbol{\Sigma}$ is PSD and $\mathbf{J}\mathbf{W}$ is a $2 \times 3$ matrix with rank 2 (for non-degenerate cameras), $\boldsymbol{\Sigma}'$ is a $2 \times 2$ PSD matrix. The projected density is therefore a 2D Gaussian with elliptical level sets, centred at $\pi(\boldsymbol{\mu})$. $\blacksquare$

For $L = 0$: $\hat{P}(\hat{\mathbf{d}}) = a_{0,0} Y_0^0 = a_{0,0}/(2\sqrt{\pi})$, which is constant over all directions. This is an omnidirectional (isotropic) scatterer.

The angular resolution of spherical harmonics of degree $L$ is approximately $\Delta\theta_{\text{SH}} \sim \pi/L$. To represent a specular reflection with beamwidth $\Delta\theta$, we need:

$$L \geq \frac{\pi}{\Delta\theta}.$$

For example, a specular reflector with $10^\circ$ beamwidth requires $L \geq \pi/(10\pi/180) = 18$. In practice, $L = 3$--$4$ suffices for typical RF scatterers because RF scattering patterns are broader than optical ones. $\blacksquare$

The positional gradient $\partial\mathcal{L}/\partial\boldsymbol{\mu}_k$ indicates the direction in which moving the Gaussian centre would most reduce the rendering loss. A large magnitude means the Gaussian is being "pulled" strongly --- it needs to cover more area than its current position and shape allow.

If a single Gaussian cannot adequately represent the local scene structure (e.g., a corner, an edge, or a transition between materials), duplicating it allows two Gaussians to share the responsibility. One stays near the current position; the clone shifts toward the gradient direction. After further optimisation, the two Gaussians settle into positions that jointly cover the under-reconstructed region.

Cloning is used for small Gaussians in under-represented regions; splitting is used for large Gaussians covering too much area. The distinction is controlled by the scale threshold $\tau_s$. $\blacksquare$

$\lambda = \frac{c}{f_0} = \frac{3 \times 10^8}{28 \times 10^9} \approx 1.07 \text{ cm}.$$

Recommended spacing: $2\lambda \approx 2.1$ cm.

For a $10 \times 10$ m area at 2.1 cm spacing: $(10/0.021) \times (10/0.021) \approx 476 \times 476 \approx 226{,}000$ measurements.

Over 200,000 measurements is impractical for manual or even robotic data collection. This highlights why visual priors (RFCanvas) or coarser sampling with interpolation are essential at mmWave frequencies. Alternatively, phased-array beam scanning can acquire directional measurements more efficiently. $\blacksquare$

• Geometric per Gaussian: $3 + 4 + 3 = 10$
• RF per Gaussian: $1 + (2+1)^2 = 1 + 9 = 10$
• Total per Gaussian: $20$
• Total scene: $1000 \times 20 = 20{,}000$
• Frozen (geometric): $1000 \times 10 = 10{,}000$

Frozen/total $= 10{,}000/20{,}000 = 50\%$.

Half the parameters are frozen, meaning the RF optimisation operates in a 10,000-dimensional subspace. With 20 measurements, this is still severely under-determined ($20 \ll 10{,}000$), which is why additional regularisation (smoothness, material-based priors) is important. $\blacksquare$

The $k$-th Gaussian contributes $f_k \alpha_k G_k T_k$ to $\hat{C}$. The direct gradient is:

$$\frac{\partial \hat{C}}{\partial \alpha_k}\bigg|_{\text{direct}} = f_k G_k T_k.$$

For $i > k$, the transmittance $T_i$ depends on $\alpha_k$: $\partial T_i / \partial \alpha_k = -G_k \prod_{j<i, j\neq k}(1-\alpha_j G_j)$. This gives:

$$\frac{\partial \hat{C}}{\partial \alpha_k}\bigg|_{\text{indirect}} = -G_k \sum_{i>k} f_i \alpha_i G_i \frac{T_i}{1 - \alpha_k G_k}.$$

$\frac{\partial \hat{C}}{\partial \alpha_k} = f_k G_k T_k - G_k \sum_{i>k} f_i \alpha_i G_i \frac{T_i}{1 - \alpha_k G_k}.$$The gradient of the loss follows by the chain rule:$\partial\mathcal{L}/\partial\alpha_k = 2(\hat{C} - C_{\text{target}}) \cdot \partial\hat{C}/\partial\alpha_k$. The dependence on$T_k$(and hence on all preceding opacities) means that gradients flow through the depth ordering, coupling all Gaussians along each ray.$\blacksquare$

$|s_1 + s_2|^2 = |\sigma_1 e^{j\phi_1} + \sigma_2 e^{j\phi_2}|^2KATEXPLACEHOLDER0END= \sigma_1^2 + \sigma_2^2 + 2\sigma_1\sigma_2\cos(\phi_1 - \phi_2).$$

• Constructive: $\cos(\Delta\phi) = 1 \Rightarrow |s|^2 = (\sigma_1 + \sigma_2)^2$.
• Destructive: $\cos(\Delta\phi) = -1 \Rightarrow |s|^2 = (\sigma_1 - \sigma_2)^2$.
• Incoherent: $|s|^2 = \sigma_1^2 + \sigma_2^2$.

For $\sigma_1 = \sigma_2 = \sigma$: constructive gives $4\sigma^2$, incoherent gives $2\sigma^2$, ratio $= 2$ (3 dB). For the full range: constructive/destructive ratio is $(2\sigma)^2/0 = \infty$ (cancelled completely). $\blacksquare$

The measurement at subcarrier $k$ and spatial position $q$ is:

$$y_{k,q} = \sum_{n=1}^N c_n [\mathbf{A}_{\text{freq}}]_{k,n} [\mathbf{A}_{\text{space}}]_{q,n} + w_{k,q}.$$

Representing $c_n$ via Gaussians: $c_n = \sum_{i=1}^{N_G} p_i G_i(\mathbf{p}_{n})$, the measurement becomes:

$$y_{k,q} = \sum_{i=1}^{N_G} p_i \underbrace{\left(\sum_n G_i(\mathbf{p}_{n}) [\mathbf{A}_{\text{freq}}]_{k,n}\right)}_{\text{freq. splatting}} \underbrace{[\mathbf{A}_{\text{space}}]_{q,\cdot}}_{\text{spatial splatting}}.$$

The frequency splatting requires $O(N_G \cdot K)$ operations and the spatial splatting $O(N_G \cdot Q)$, compared to $O(N_G \cdot K \cdot Q)$ for the joint operation. The total cost is $O(N_G(K + Q))$ rather than $O(N_G K Q)$. $\blacksquare$

A pruned Gaussian $k$ with $\alpha_k < \epsilon_\alpha$ contributes at most $\|f_k\| \cdot \alpha_k \cdot G_k(\mathbf{u}) \cdot T_k \leq f_{\max} \cdot \epsilon_\alpha$ to any pixel $\mathbf{u}$ (since $G_k \leq 1$ and $T_k \leq 1$).

If $N_p$ Gaussians are pruned, the worst-case pixel error is:

$$|\hat{C}_{\text{pruned}}(\mathbf{u}) - \hat{C}(\mathbf{u})| \leq N_p \cdot f_{\max} \cdot \epsilon_\alpha.$$

For $N_p = 5000$, $f_{\max} = 1$, $\epsilon_\alpha = 0.01$: error $\leq 50$, which is a loose bound. In practice, the transmittance $T_k$ ensures that deeply occluded pruned Gaussians contribute negligibly, making the actual error much smaller. $\blacksquare$

The alpha-compositing depends on the depth order of Gaussians. Swapping the order of two Gaussians with similar depth creates a discontinuity in the rendering function. This makes the loss landscape non-smooth (not just non-convex). In practice, the sorting is treated as fixed during each gradient step and updated periodically.

The transmittance $T_k = \prod_{j<k}(1 - \alpha_j G_j)$ is a product of terms, each depending on different parameters. This creates coupled non-linear interactions between Gaussians --- the gradient of one Gaussian's opacity depends on all preceding Gaussians' opacities.

The quaternion-to-rotation conversion $\mathbf{q} \to \mathbf{R}$ is non-linear, and the rotation group SO(3) is non-convex. Additionally, antipodal quaternions $\mathbf{q}$ and $-\mathbf{q}$ represent the same rotation, creating symmetries in the loss landscape.

Despite non-convexity, 3DGS optimisation works well because: (1) The initialisation from SfM or a grid provides a warm start near a good basin. (2) Adaptive density control adds/removes parameters, exploring the loss landscape more broadly than fixed-parameter optimisation. (3) The per-Gaussian parameters are largely decoupled in non-overlapping regions. $\blacksquare$

The Gaussian scene has $\Theta \in \mathbb{R}^{Nd}$ parameters. Each measurement $P_{\text{dB}}(\ntn{rx_pos}_m)$ provides one scalar equation. For the system to be determined, we need $M \geq Nd$ (parameter counting).

However, the rendering equation is non-linear in $\Theta$, so the $M$ equations are not independent linear constraints. The effective number of independent constraints depends on the Jacobian rank $\text{rank}(\partial \hat{\mathbf{P}}/\partial \Theta)$. For well-separated Gaussians, the Jacobian has full rank and $M \geq Nd$ suffices. For overlapping Gaussians, degeneracies reduce the rank.

In compressed sensing with a linear model $\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}$, recovery of an $S$-sparse signal in $\mathbb{C}^Q$ requires $M \geq O(S \log(Q/S))$ measurements under RIP. For Gaussian splatting with $S$ scatterers (each modelled as one Gaussian with $d$ parameters), the analogous requirement is $M \geq O(Sd \log(Q/S))$ --- the per-scatterer cost increases by the factor $d$ due to the additional shape parameters.

For $S = 100$ scatterers in a $Q = 10{,}000$-voxel space with $d = 20$: compressed sensing needs $M \geq O(200\,\log\,100) \approx 1{,}000$ measurements; Gaussian splatting needs $M \geq O(2{,}000\,\log\,100) \approx 10{,}000$ in the worst case. The visual prior (RFCanvas) dramatically reduces this by fixing the geometric parameters. $\blacksquare$