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Conditioning of the registration problem and observability of the six degrees of freedom.
§What is computed here, and what is not
What is computed is conditioning: how firmly the geometry of the
scene pins down each of the six motions. What is not computed is a
calibrated pose uncertainty — Censi’s closed form understates real
spread by orders of magnitude. That is why the field is called
conditioning and not covariance.
§The normalisation without which the answer is meaningless
The Jacobian’s columns carry different units: ∂e/∂ρ = n is
dimensionless while ∂e/∂φ = p × n is measured in metres. Singular
values of such a matrix are not comparable with one another, and
singular vectors that mix ρ and φ depend on the choice of units — a
verdict of “rotation about Z is degenerate” can flip when metres become
millimetres.
The cure is a change of variable: ξ' = [ρ; r_g·φ], where r_g is the
radius of gyration of the points about the centre of rotation. All six
coordinates are then in metres: one unit along a rotational axis means
one metre of displacement for a point at the characteristic distance.
The Jacobian in the new variables is J with its last three columns
divided by r_g, which is S⁻¹HS⁻¹ without ever forming H.
§Centre of rotation
The report is built about the centroid of the correspondences,
not about the coordinate origin. Otherwise the same scene, referred to
a distant origin, would get a different condition number: r_g would
measure how far away the scene is rather than how large it is. The
decomposition into degrees of freedom genuinely depends on the choice
of centre, so the centre is part of the report.
Structs§
- Analysis
- The full result of the analysis.
- Conditioning
- Conditioning of the problem.
- Correspondence
- A single point-to-plane correspondence.
- Observability
Criteria - What counts as reliable.
Enums§
- Observability
- How reliably a degree of freedom is determined.
Functions§
- analyse
- Analyses a set of correspondences.