Limits and Unavailable Results

A covariance is an honest statement about noise, not about everything that can be wrong with a calibration. This chapter lists the cases where none can be given, and where one is given but answers a narrower question than it seems to.

When It Is Unavailable

rank_deficient is a statement about the data, not a failure of the calibration: the solve still returns its best parameters. A Scheimpflug camera whose views do not separate the sensor tilt from a principal-point shift, for example, reports tilt_y, cx and the board translations as unobservable together.

Scope

Every workflow reports the uncertainty of its final solve. When a hand-eye solve refines the robot poses (the default), it also estimates the robot's orientation and position error from the corrections, weighed against the corner noise the residuals show (robot_poses.estimate_sigma). The estimate is then the maximum-likelihood one and its covariance holds: on synthetic eye-in-hand scenes the normalized errors had an RMS of 1.02–1.13, from priors that matched the robot's error and from priors ten times too tight or too loose. With the estimate off, the configured robot_poses.rot_sigma and trans_sigma are taken as the robot's true accuracy: too loose, and the hand-eye and target standard deviations are pessimistic; too tight, and they are too small, with nothing in the residuals to say so.

Without refinement the robot poses are taken as exact. Their errors then show up in the corner residuals, but as an error shared by every corner of a view rather than independent noise, and the covariance treats it as independent: with 476 corners a view, it was 16–17 times too small on the same scenes. Refine the robot poses whenever their error is not well below the corner noise.

What a Covariance Does Not Cover

• Systematic error. The covariance measures the scatter caused by noise. An error that differs from view to view (a lens model or target that does not fit exactly) is in it, through the view scatter that scales the covariance (Parameter Uncertainty). An error common to every view, such as a wrong board cell size, leaves it small and the estimate biased; watch the per-feature residuals for structure. Calibrating kuka_1, the three ds8 cameras and the two stereo cameras from two disjoint halves of their views, the two projections differ by 0.8 to 1.2 times their predicted σ, and by 1.0 on average (2.2 to 5 times without the view scatter). One camera is a noisy check: with an exactly known σ, a camera's ratio still spans about 0.7 to 1.2.
• Second-order error. The covariance is first order. Weak views (a rig whose boards stay centred and barely tilted) can leave each parameter's σ right while a combination of them errs several times its σ; second_order predicts it and the quality report's covariance_linearity check warns (Reading the Covariance, Capturing Calibration Images). The covariance itself is not corrected for it.
• Fixed parameters. It is conditional on the parameters the final solve held fixed (fixed in the result). Fixing a parameter the data cannot pin down makes the others look better determined than they are: a rig with rig.refine_intrinsics: false holds each camera's intrinsics, and its extrinsics' σ leaves their uncertainty out.
• Robot poses. They are taken as exact unless the hand-eye solve refines them, as described in the Scope section above.
• The shape of stated point covariances. Per-point covariances are trusted up to one common factor, which σ̂₀ estimates. Covariances that are right for some points and wrong for others (one image overstated, say) are not detected by σ̂₀ and weigh the points wrongly.
• The camera's own pose. The projection map varies only the camera's parameters (Projection Uncertainty).