LIDAR GS-SLAM · COMPUTED TEACHING EXPERIMENTS
Inspect a covariance. Weight a registration. Spend a finite Gaussian budget.
The smallest eigenvector is a normal only when the local surface supports one.
Press Step or Solve to reduce the actual weighted Mahalanobis objective.
Click a Gaussian to inspect its normalized geometry and residual score.
Synthetic teaching model. The browser computes geometry and metrics locally. This is not a reproduction of the paper’s experiments.
A seeded set of 180 three-dimensional samples defines the empirical covariance C. A symmetric eigendecomposition supplies ordered eigenvalues λ₀ ≥ λ₁ ≥ λ₂ and orthonormal axes. A Gaussian disk uses the two leading axes, while the smallest-eigenvalue axis is a candidate normal. The left plot shows a projection of the actual samples and eigenvectors. The right plot uses the same orientation with range-adaptive scale.
The neighborhood uses normalized local coordinates. Here σ̂ⱼ = √(λⱼ/λ₀), so its largest standard deviation is one. The paper reuses the tracker's normalized covariance geometry and scales the initial surface footprint with range. This experiment instantiates its Eq. (6) as s⁽⁰⁾ = clip(0.04 r σ̂ᵗᵃⁿ, 0, 2 m). The factor 0.04 and the cap are teaching constants, not reported optimal parameters.
A plane typically has two substantial tangent eigenvalues and a much smaller normal eigenvalue. A line has two similar small eigenvalues, so many normal directions fit. An approximately isotropic cloud has no preferred plane. Sampling noise can still separate numerical eigenvalues. The displayed warning uses relative eigengap and curvature as a teaching heuristic, not the authors' trackability rule. Normal sign is also ambiguous.
The ellipsoid describes the spatial spread of a local point neighborhood. It does not by itself estimate the uncertainty of the complete sensor pose. The raw shape proxies are L = (λ₀ − λ₁)/(λ₀ + ε) and C = λ₂/(λ₀ + λ₁ + λ₂ + ε). The map-control experiment adds population normalization and registration residuals before combining them.
The displayed source points align to fixed targets through a planar rigid transform T = (tₓ, tᵧ, θ). The code minimizes Σᵢ wᵢ dᵢᵀ(Cᵢᵗ + R Cᵢˢ Rᵀ)⁻¹dᵢ, with dᵢ = R pᵢ + t − qᵢ. This toy chooses isotropic source covariances, making R Cᵢˢ Rᵀ = Cᵢˢ exactly. Gauss–Newton computes each pose update, and a line search accepts a step only when the displayed objective decreases.
The paper's Eqs. (2)–(5) derive κ from the two in-plane Gaussian scales, normalize κ across the target population, and cap its reciprocal weight at 10. This experiment follows that min–max normalization and uses ε = 0.001 in the final reciprocal. We assign the illustrative two-dimensional residual covariances separately from the in-plane scale-ratio metadata. The plotted ellipses show those residual covariances. This is a reduced weighting demonstration, not a shared three-dimensional Gaussian reconstruction.
Two stable surfaces constrain the pose. A separate group has more elongated Gaussian geometry and an adjustable systematic offset. The association between elongation and bias is constructed so the effect of weighting can be inspected. It is not evidence that this weighting always improves tracking. Sliders and the weight toggle restart the optimizer. Ground truth only produces the displayed translation and heading errors.
This experiment omits correspondence search, deskewing, three-dimensional motion, map optimization, observation masks, keyframes, and loop closure. The released implementation uses a fixed normalization clip((κ − 1)/3, 0, 1), while the paper describes population min–max normalization. Both apply an inverse weight capped at 10. The live experiment uses the paper formulation consistently.
Each initial primitive has a sample covariance from 48 seeded local three-dimensional points and an assigned synthetic registration residual. The code normalizes linearity and curvature between their population 5th and 95th percentiles, then clips them to [0, 1]. Residuals divide by their 95th percentile and clip. The score is cᵢ = clip(0.55 L̃ᵢ + 0.30 C̃ᵢ + 0.15 r̃ᵢ, 0, 1), following Eq. (9) and the released feature computation.
The lower-score candidates form groups on this two-dimensional patch. Each group keeps its highest-opacity primitive and expands that representative to conservatively enclose the removed members' original one-standard-deviation support ellipses. This is a deliberate toy simplification: the paper uses multiple orthogonal projection passes and a coverage quantile. The toy caps pruning at 35% of the initial count and rejects expanded support radii above 1.5 m.
The selected high-score candidates each become two children, offset symmetrically by ±0.45 times the major scale along their major axis. Each child scale is 0.72 times the parent's scale. These offsets and the 12% split cap are illustrative choices. The new children would need later mapping gradients to specialize. No reconstruction improvement follows automatically from the initialization shown here.
The coverage readout is the fraction of a fixed, dense set of synthetic surface samples that lies within at least one displayed one-standard-deviation support ellipse. It is not the paper's mesh F-score, nor a measure of map correctness or free-space consistency. The flat patch and polyline detail are an unrolled two-dimensional schematic. Every control recomputes one pass from the same original map, so comparisons remain reproducible.