The morning-of cheatsheet

One page. Read it out loud once. Do not open anything else.

Camera geometry

\[P_c = R P_w + t, \qquad C = -R^\top t, \qquad T^{-1} = \begin{bmatrix} R^\top & -R^\top t \\ 0 & 1\end{bmatrix}\]

\[K = \begin{bmatrix} f_x & s & c_x \\ 0 & f_y & c_y \\ 0 & 0 & 1\end{bmatrix}, \qquad u = f_x \frac{X_c}{Z_c} + c_x, \qquad v = f_y \frac{Y_c}{Z_c} + c_y\]

Order of operations: extrinsics, divide by \(z\), distortion, then \(K\).

\[\text{GSD} = \frac{h}{f_x}\ \text{m/px}. \qquad \text{90 m, } f = 2400 \Rightarrow 3.75\ \text{cm/px}; \text{ a 1 m target is 27 px.}\]

Subscript discipline: \(T_{w\leftarrow c} = T_{w\leftarrow b}\,T_{b\leftarrow c}\). Inner subscripts cancel or the expression is wrong.

Two-view geometry

\[x_2^\top E x_1 = 0,\quad E = [t]_\times R; \qquad p_2^\top F p_1 = 0,\quad F = K_2^{-\top} E K_1^{-1}\]

• \(E\): rank 2, singular values \((\sigma,\sigma,0)\), 5 dof. Project onto that manifold after computing it from a noisy \(F\).
• \(F\): rank 2, \(\det F = 0\), 7 dof. Epipoles are the null vectors.
• Eight-point: \(Af = 0\), smallest right singular vector, truncate to rank 2. Hartley normalisation or it does not work.
• Four poses from \(E\); cheirality (positive depth in both cameras) picks one.
• Degeneracies: pure rotation (\(t=0\), \(E=0\)) and planar scene. Fit \(H\) too and compare.

\[\boxed{k = \frac{\log(1-p)}{\log(1-w^s)}}\qquad w=0.5,\,p=0.99: \ s=4 \to 72,\ \ s=8 \to 1177,\ \ s=5 \to 146\]

\[\boxed{\sigma_Z \approx \frac{Z^2}{fB}\sigma_d}\qquad 90\ \text{m}, B=10\ \text{m}, f=2400, \sigma_d = 0.3\ \text{px} \Rightarrow \sigma_Z \approx 10\ \text{cm}\]

Depth error is quadratic in range, linear in baseline. Halve the baseline, double the error.

Triangulation is unreliable from: short baseline, large depth, pose error, calibration error, poor localisation, repeated texture, moving vegetation, water, occlusion.

Bundle adjustment

\[\min_{\{R_i,t_i\},\{X_j\}} \sum_{(i,j)} \rho\big(\|\pi(K_i(R_iX_j + t_i)) - p_{ij}\|\big)\]

Update on the manifold: \(R \leftarrow \exp([\delta\omega]_\times)R\), \(t \leftarrow \exp([\delta\omega]_\times)t + \delta t\).

\[\frac{\partial Y}{\partial\delta\omega} = -[Y]_\times,\quad \frac{\partial Y}{\partial\delta t} = I,\quad \frac{\partial Y}{\partial X} = R,\quad J_\pi = \begin{bmatrix}f_x & s\\0&f_y\end{bmatrix}\frac{1}{Z}\begin{bmatrix}1&0&-x\\0&1&-y\end{bmatrix}\]

Arrowhead Hessian. Schur complement:

\[(B - EC^{-1}E^\top)\delta_c = g_c - EC^{-1}g_p, \qquad \delta_p = C^{-1}(g_p - E^\top\delta_c)\]

500 cameras and 200k points: a \(3000\times3000\) solve instead of \(606{,}000\).

Gauge freedom is 7 dof (3 + 3 + 1 scale). Fix a camera, damp, or use ground control.

Huber: \(w(e) = \min(1, \delta/e)\). At \(e = 100\), \(\delta = 2\): weight 0.02.

Scale comes from GNSS baselines, altimeter, ground control points, known objects. Never from images.

Learned 3-D

Pointmap: \((3, H, W)\) per view, both in view 1's frame. Depth is the \(z\) channel; pose is weighted Umeyama; focal has a closed form.

Loss: scale-normalise both sides, then \(\sum (c_i e_i - \alpha\log c_i)\), optimum \(c^\star = \alpha/e\).

Splats: \((\mu, \Sigma, c, \alpha)\) with \(\Sigma = RSS^\top R^\top\) (PSD by construction). EWA: \(\Sigma_{2D} = JW\Sigma W^\top J^\top + \epsilon I\). Compositing: \(C = \sum_i c_i\alpha_i\prod_{j<i}(1-\alpha_j)\), same as NeRF.

Classical fails loudly (a hole, which downstream treats as unknown). Learned fails quietly (a confident wrong surface, which downstream accepts). That asymmetry is the answer to most "which would you use" questions.

Never take metric geometry off a splat without independent validation. Rendering quality is evidence about appearance.

The deliverability design

1. Start from what the vehicle consumes. A prior, with an explicit unknown.
2. Geometry (surface, bare earth, slope, relief, height above ground, occupancy, confidence).
3. Semantics (surfaces and hazards, thin structures separately).
4. Decision layer, separate from both: a conjunction of hard constraints, then ranking.

\[\text{safe}(x) = [\text{slope}<\tau_s]\wedge[\text{clearance}>\tau_c]\wedge[P(\text{hazard})<\tau_o]\wedge[\text{corridor clear}]\]

• Clearance is a radius. Largest inscribed disk = max of the distance transform. Area is the wrong statistic (the L-shaped strip).
• The corridor is vertical. A tethered droid descends from altitude. Wires dominate.
• Threshold \(p + n\sigma\), so uncertainty costs coverage instead of safety.
• Deliver iff \(P(\text{unsafe}) < C_{\text{decline}}/C_{\text{incident}}\). Four orders of magnitude gives \(10^{-4}\).
• Three actions: this site, an alternate site, decline and escalate.
• Priors expire. Validity horizon conditioned on what the site is made of.

Phases: rules and classical geometry, supervised semantics, learned multi-view geometry, richer priors, continuous feedback. Each with a promotion criterion set in advance.

Evaluation

A wire at 0.05% of pixels, missed entirely: pixel accuracy 99.95%, mIoU down 0.045, event recall 0.

Hierarchy: mIoU, per-class IoU, rare-hazard recall, resolution-conditioned recall, slices, end-to-end deliverability error, unsafe false-positive rate.

\[\boxed{p_{95\%} \approx 3/n}\qquad 10^{-4} \text{ needs } \approx 30{,}000 \text{ clean trials}\]

Split by property, never by tile. Calibrate, and check calibration in the low-probability regime, per slice.

Offline win that does not transfer: distribution mismatch, wrong metric, leakage, preprocessing difference, downstream trusting the new confidence differently, quiet-versus-loud failure character.

Scaling 10⁴ to 10⁷

200 images per property: 2 billion images, ~8 PB raw. Classical SfM at 30 core-min: 570 core-years. Feed-forward at 0.5 GPU-s: 1400 GPU-hours.

Changes: unit of work becomes a tile with a geospatial index; processing becomes incremental and change-driven; the expensive path becomes routed with a random audit; labelling becomes sampling plus mining; monitoring becomes regional and change-based; and the prior distribution system is a real system with staged rollout and rollback.

Complexity, for saying out loud

Operation	Time	Note
Connected components (flood fill)	\(O(HW)\)	\(O(HW)\) worst-case space; iterate, do not recurse
Exact Euclidean distance transform	\(O(HW)\)	Separable lower envelope, two passes
BFS on a grid	\(O(HW)\)	Optimal only for uniform cost; mark on enqueue
Dijkstra	\(O(HW\log HW)\)	Non-uniform cost; diagonals cost \(\sqrt2\)
A*	same, fewer expansions	Needs an admissible heuristic; octile for 8-connectivity
NMS	\(O(N^2)\), \(O(N\log N)\) sort	Sort dominates when suppression is heavy
Bundle adjustment step	\(O(C^3 + C^2P)\)	With Schur; \(O((6C+3P)^3)\) without
Triangulation (DLT)	\(O(V)\) per point	SVD of a \(2V\times4\) matrix

The five answers, compressed

1. 20 images, metric 3-D. Verify pairs by RANSAC on \(F\) with a homography check; select pairs by triangulation angle rather than match count; initialise from INS; triangulate; bundle adjust with GNSS position priors and ground control for metric scale; hold out check points to report honest accuracy; densify to a surface model.
2. Backyard safety. Vehicle contract first, then geometry, semantics, decision layer as a logged conjunction, clearance as a radius, corridor not patch, upper confidence bound, staged roadmap.
3. 97% and misses wires. Reject the metric, re-instrument at event level, diagnose resolution against imbalance against representation, then mitigate at the system level because detection recall alone will never be enough.
4. Offline win, production loss. Distribution, metric, leakage, integration, downstream calibration coupling, quiet-versus-loud failure character. Replay to isolate.
5. 10k to 10M. The arithmetic first, then tiles, incremental, routed verification, sampling plus mining, regional monitoring, and the distribution of the priors.

In the room

• Ask the two or three questions that change the design, then say how each one changed it.
• Say the complexity before you type.
• Write the tests unprompted.
• Name what is loud and what is quiet when you talk about failures.
• When you say "I would train X", say the promotion criterion in the next sentence.
• Four sentences, then stop talking.