PaperlayerView exact revision on arXiv

Source-linked AI summary

VeriCalib: Verifiable Target-Based LiDAR-Camera Calibration

Xiangcheng Hu, Xieyuanli Chen, Feiyi Chen, Shengli Ren, Shiyang Chen, Guoqiang Ye, Hao Wang, Ping Tan, Steven L. Waslander, Fuchun Sun, Jin Wu

arXiv:2610.28791v1cs.RO

TL;DR

Fitted-residual RMSE cannot indicate whether a calibration should be accepted or how to correct it, while independent-feature and nominal-noise assumptions can misstate uncertainty. VeriCalib combines structured covariance, directional precision, consistency checks, and fault-detection limits into a calibration state; on real-world data, the state also distinguishes precision from detectability requirements.

  • Problem

    Fitted-residual RMSE provides no criterion for accepting a calibration or choosing among re-extraction, recollection, and adding observations, while independent-feature and nominal-noise models can misstate uncertainty.

    Source: §1.1 · p. 2

  • Method

    VeriCalib combines structured covariance with weakest-direction pose guidance and consistency checks to report directional precision, detection limits, and a calibration state.

    Source: §5.3 · p. 9, §6.2 · p. 12, §9 · p. 26

  • Results

    A chassis-pair S1 calibration met the precision requirement but not the detectability requirement, whereas the S4 head-pair estimate from half the observations was verified by the other half.

    Source: §9 · p. 25

  • Takeaways & Limitations

    The calibration state indicates when to collect validation observations along the largest detection-limit direction rather than treating precision alone as verification.

    Source: §9 · p. 25

    Verification is internal to a batch: a bias common to all observations, including an extrinsic-equivalent intrinsic error, is indistinguishable from a different extrinsic.

    Source: §9 · p. 26

Abstract

from arXiv · show

Target-based extrinsic calibration estimates the rigid transform between sensors from a few observations of a known target, and its result is commonly assessed by the root-mean-square error of the fitted residuals. However, this scalar error is computed on the same observations and therefore cannot indicate whether to accept the result, re-extract, recollect or add observations. This paper presents VeriCalib, which formulates extrinsic calibration as a verifiable estimation problem and reports a calibration state. The key insight is a two-level extrinsic compensation mechanism, in which the extrinsic compensates the rigid part of each observation error at the residual level, and rotation and translation compensate each other at the parameter level. Consequently, the fitted residual can reveal neither an inconsistency between the observations and the model nor whether the estimated extrinsic is reliable in every direction. Accordingly, we assess the calibration through a structured covariance that jointly propagates the uncertainty of the camera and LiDAR board features with a known-geometry prior, and estimate its scale as a variance component from the residuals. On this covariance, we decouple rotation and translation to resolve the precision along each direction and guide data collection toward the weakest one. Since this precision presumes consistent observations, we further develop within-view and cross-view checks that examine the internal residual and the rigid discrepancy between observations, respectively, with thresholds that hold their significance level under an unknown noise level. We also extend the minimal detectable bias to the extrinsic to bound the smallest detectable fault along each direction. These criteria form a calibration state that indicates whether to accept the result or how to collect the data...

1 Introduction

Fitted RMSE cannot determine whether a calibration should be accepted or how to correct it, while uncertainty modeling and coupled rotation–translation constraints complicate reliability assessment. VeriCalib combines structured covariance, directional precision, consistency tests, and detection limits into a calibration state that guides subsequent action.

1.1 Motivation and Challenges

  • RMSE alone offers no criterion for choosing among accepting a result, re-extracting features, recollecting observations, or adding observations.The introduction illustrates this gap with two extraction errors in a 14-observation calibration.Source: §1.1 · p. 2, Figure 1 · p. 2
  • Independent features and nominal noise magnitudes can misstate uncertainty because camera features share a board pose, LiDAR features share a fitted plane, and noise varies.The resulting precision and test thresholds may lead to overconfident acceptance or rejection of consistent observations.Source: §1.1 · p. 2
  • Existing analyses can miss weak directions because rotation and translation compensate each other, while fixed robust thresholds lack a stated significance level and fault-detection bound.A faulty observation may pass undetected, and a consistent one may be discarded.Source: §1.1 · p. 2

1.2 Contributions

  • VeriCalib frames target-based extrinsic calibration as a verifiable estimation problem that reports a calibration state and the corresponding next procedure.Its two-level compensation explains why fitted residuals alone reveal neither observation inconsistency nor directional extrinsic reliability.Source: §1.2 · p. 2, §1.1 · p. 2
  • A structured covariance models shared camera board-pose and LiDAR board-plane uncertainty with a known-geometry prior, and its scale is estimated from absorbed residual components.The covariance underlies reported precision, thresholds, and detection limits.Source: §1.2 · pp. 2–3
  • Marginalized directional precision identifies the weakest extrinsic direction and guides the next target pose; within-view and cross-view checks assess consistency and directional detection limits.The checks use thresholds that retain their significance level under unknown noise magnitude, and the criteria combine into actionable calibration states.Source: §1.2 · p. 3

2 Related Work

Prior target-calibration work spans target types, uncertainty modeling, observability, and consistency checks, but the related-work discussion describes these analyses as separate. VeriCalib brings structured covariance, direction-resolved precision, and fault detection together on a common measurement factor.

2.1 Target-Based Calibration

  • Target-based methods use known patterns to provide cross-modal correspondences, with checkerboards, fiducials, spherical targets, and circular-hole boards among the reviewed targets.Four-hole boards have been used in several calibration methods, while targetless and learning-based methods do not exploit known-pattern geometric priors.Source: §2.1 · p. 3

2.2 Noise and Uncertainty Modeling

  • Uncertainty-aware methods model feature noise, but independent feature blocks can repeatedly count shared camera pose information and understate uncertainty along correlated directions.The paper assembles cross-modal residual covariance and estimates its scale as a global variance component.Source: §2.2 · pp. 3–4

2.3 Constraint Analysis

  • Observability and scalar information criteria do not reveal which physical directions remain weakly constrained; prior direction-resolved methods address registration rather than calibration.The paper applies Schur complements to structured covariance to obtain calibrated directional precisions.Source: §2.3 · p. 4

2.4 Consistency Assessment and Reliability

  • Geodetic reliability defines the smallest detectable fault at a specified significance level and power, but prior work does not report detection limits along extrinsic directions.Existing calibration checks include within-observation and cross-observation tests, often using thresholds without a noise-calibrated false-alarm rate.Source: §2.4 · p. 5

2 Related Work

  • The paper argues that prior methods address covariance, constraint analysis, and detectability in separate literatures, each relying on noise assumptions or another analysis’s output.VeriCalib formulates the three analyses on a single structured measurement factor.Source: §2.4 · p. 5

3 Problem Formulation

The problem is to estimate the rigid transform between a LiDAR and camera from observations of a known planar target. VeriCalib specifies a verifiable result: estimation on consistent observations, with checks for model consistency, directional precision, and detectable faults.

3.1 Target-Based Calibration

  • The unknown extrinsic maps LiDAR coordinates into camera coordinates, and the camera and LiDAR board poses must describe the same observed board pose.The target has four known circular-hole centers; the formulation also permits at least three non-collinear points with known board coordinates.Source: §3.1 · p. 5
  • The classical solver independently extracts camera and LiDAR hole centers, then jointly aligns their correspondences by a maximum-likelihood objective under correct correspondences and independent, isotropic, zero-mean Gaussian noise.The camera features share a PnP board pose, while LiDAR circles are fitted within a RANSAC-fitted plane.Source: §3.1 · p. 5
  • Orthogonal Procrustes gives a closed-form minimizer, and one observation can constrain all six degrees of freedom when its four hole centers are correctly associated and non-collinear.Real observations can violate the classical assumptions through shared feature estimates and gross extraction errors; the paper models the former and detects the latter.Source: §3.1 · p. 5

3.2 Observation Model

  • The observation model stacks correspondence residuals and distinguishes measurement noise from faults, including local errors with components that no rigid perturbation explains.A common-mode fault equivalent to an extrinsic change is indistinguishable from a different extrinsic; differential faults displace groups of observations relative to one another.Source: §3.2 · pp. 5–6

3.3 The Verifiable Calibration Problem

  • The verifiable-calibration problem requires estimating the extrinsic and covariance from observations consistent with the model while meeting precision and detectable-fault requirements.It also requires consistency tests at level α and validation on held-out observations with power 1 − β.Source: §3.3 · p. 6
  • The calibration state reports the first unmet requirement and pairs it with a recommended procedure, but verification is internal to observations within the same batch.Therefore, common-mode faults can remain unobservable even when validation information has full rank.Source: §3.3 · p. 6
  • The local factor model assumes conditionally Gaussian, locally linear residuals, covariance shapes correct up to a global scale, zero-mean accepted residuals, and independence across observations.The covariance is frozen at the current estimate, making the likelihood a quasi-likelihood rather than differentiating its rotation dependence and log-determinant.Source: §3.3 · p. 6

4 Two-Level Extrinsic Compensation

VeriCalib analyzes two forms of compensation: the extrinsic absorbs rigid components of observation errors, and rotation and translation can compensate each other. These effects explain why fitted residuals or aligned projections alone cannot establish consistency or directional reliability.

4.1 Residual-Level Compensation

  • At the residual level, each observation’s whitened residual separates into a component an extrinsic perturbation can absorb and an invariant internal component that remains testable.For the four-hole target, the internal residual has five dimensions corresponding to non-rigid in-plane deformations.Source: §4.1 · pp. 6–7
  • A rigid component of a fault shifts the common estimate and is distributed across residuals, so fitted residuals neither reveal the extrinsic error nor locate corrupted observations.For a four-hole observation, the internal residual isolates in-plane LiDAR extraction error because camera and fitted-plane errors move centers rigidly.Source: §4.1 · p. 7

4.2 Parameter-Level Compensation

  • At the parameter level, rotation and translation compensate: the Schur-complement information matrices account for information each parameter block can absorb from the other.A pitch and a compensating translation can yield nearly the same target observation; fixing translation therefore overstates rotation precision relative to marginalizing it.Source: §4.2 · p. 7

5 Structured Covariance Modeling

VeriCalib builds calibration precision and consistency thresholds from a structured covariance whose sensor-specific shape is derived from the measurement process and whose scale is estimated from residuals. The model accounts for shared camera-pose and LiDAR-plane errors, then adapts the factor construction for checkerboards.

5.1 Covariance Weighting Theorem

  • For ω = 2, covariance-shape error loses at most 12.5% of variance; correctly weighted estimates also do not lose precision when a consistent observation is added.The theorem separates covariance shape, derived from each sensor's measurement process, from batch scale, estimated from residuals.Source: §5.1 · p. 8
  • A fault on a precise observation can make the weighted estimate less accurate than the unweighted estimate, so weighting and consistency assessment are applied together.Source: §5.1 · p. 8

5.2 Shared-Pose Camera Covariance

  • Propagating one PnP board-pose covariance creates cross-hole camera correlations; rigid pose errors can shift the estimate without appearing in the internal residual.Keeping only diagonal hole blocks counts the PnP information once per hole.Source: §5.2 · pp. 8–9

5.3 Shared-Plane LiDAR Covariance

  • The LiDAR model shares a fitted board plane across hole centers and regularizes partial-arc circle fits with a known-radius prior.A saddle mode has no measurement and is removed rather than assigned a variance; the prior can contract radial uncertainty on a synthetic short arc but yield a larger covariance on the observed partial arc when it counts only Neff points.Source: §5.3 · pp. 9–10, Figure 5 · p. 10

5.4 Joint Residual Covariance Estimation

  • Combining camera and LiDAR blocks yields a joint residual covariance of rank 11 after projecting out the unmeasured saddle mode.Source: §5.4 · p. 10, §5.3 · p. 10

5.5 Two-Stage Scale Estimation

  • The method estimates scale from absorbed residual components, then re-estimates it on the accepted observations so rejected observations do not inflate reported precision or detection limits.Under the stated model, the estimate is unbiased to first order.Source: §5.5 · pp. 10–11

5.6 Checkerboard-Target Instantiation

  • For checkerboards, a known-geometry prior regularizes the LiDAR rectangle fit, and the residual covariance has rank 6 with no within-observation degrees of freedom.Batch consistency therefore relies on the cross-observation test.Source: §5.6 · p. 11

6 Directional Information and Acquisition Guidance

VeriCalib decouples rotation and translation to assess precision by physical direction, then uses the weakest direction to guide the next target pose. Its analysis also shows that rotation precision depends on orientation diversity and relative board-center spread, not common standoff alone.

6 Directional Information and Acquisition Guidance

  • Marginal information resolves rotation and translation precision separately, revealing the weakest direction that coupled observability analysis can mask.Source: §6 · p. 11

6.1 Closed-Form Directional Information

  • Rotation information is invariant to a common board-position offset when residual covariance is position-independent; orientation diversity and relative center spread determine rotation precision.For a frontal board under the stated isotropic-noise conditions, rotation precision is independent of depth, while lateral translation precision decreases with depth.Source: §6.1 · pp. 11–12

6.2 Weakest-Direction Data-Collection Guidance

  • The weakest-direction score ranks feasible target poses by their first-order increase in the failing block's smallest information eigenvalue.The selected orientation and range strengthen the failing direction, subject to radius and consistency diagnostics.Source: §6.2 · p. 12

7 Consistency Assessment

VeriCalib combines calibrated observation checks, directional precision and fault-detection limits into a calibration state that recommends whether to accept the result or what action to take. The checks distinguish internal observation errors from rigid discrepancies across observations and assess the accepted set before reporting precision and detectability.

7.1 Calibrated Test Thresholds

  • Fitted residuals are deflated, especially with few observations, so VeriCalib refers residual tests to an F distribution using an independent scale estimate.A scale estimated from the same residuals does not remove the bias.Source: §7.1 · p. 12

7.2 Observation Diagnosis

  • Within-view checks detect errors not explained by a rigid extrinsic perturbation, while cross-view checks test each observation against an extrinsic estimated from the others.A consensus search handles correlated contamination; if comparable inconsistent consensus sets remain, the procedure recommends recollection.Source: §7.2 · p. 13

7.3 Batch Consistency Test

  • The batch test uses upper and lower tails to distinguish excess internal error from rigid discrepancy, but independent rigid errors can be indistinguishable from underestimated camera covariance.Both can leave the estimate unbiased, and their mean is included in the reported covariance; a held-out validation set resolves this ambiguity.Source: §7.3 · p. 14

7.4 Directional Minimal Detectable Bias

  • Directional minimal detectable bias bounds the smallest differential fault a focused validation test detects along a rotation or translation direction.Random-split limits refer to a fit on half the accepted set; with equal halves and a detectability requirement three times the precision requirement, about 3.5 times as many observations are called for.Source: §7.4 · p. 15

7.5 Calibration State and Recommended Procedure

  • The calibration state reports the first unmet consistency, precision or detectability requirement, with a corresponding recommended procedure.A precision-limited state calls for an observation along the weakest Schur direction; a validation-limited state calls for validation observations along the largest detection-limit direction, or external checks or a modified design if no target pose constrains it.Source: §7.5 · pp. 15–16

8 Experiments

The experiments evaluate VeriCalib in simulation and on real-world data from multiple sensor platforms. They assess covariance prediction and weighting accuracy, including the effect of model mismatch.

8.1 Experimental Setup

  • The evaluation includes simulation with a known extrinsic and real-world recordings on four platforms, using the same observations and board configuration for compared methods.The real-world sets include a mobile manipulator, handheld stereo-camera platform and flash-LiDAR platform.Source: §8 · p. 16, §8.1 · p. 16

8.2 Covariance Prediction and Weighting Accuracy

  • 1.89× under-prediction by the independent-hole model versus a 0.90 variance ratio for the structured model in simulation; on real PnP output, median VRs were 1.05 and 1.07 for head and chassis pairs.The independent-hole model under-predicted real-data scatter by 3.6 to 3.7×; in a noise sweep, structured-model VRs were 0.99 to 1.03.Source: §8.2 · p. 18
  • 0.5 to 2.4 mm error reduction from weighting over Ours-U as observation heterogeneity increased in synthetic batches; predicted and empirical RMS errors differed by less than 0.05 mm.An unmodeled 6 mm offset reduced the error reduction to 1.3 mm, or 0.5 mm when it affected only the nearest observations.Source: §8.2 · p. 18
  • The weighted estimate was more repeatable than Ours-U on six of seven robot and FS sets, with gains of 0.01° to 0.24° and 0.2 to 11.5 mm.The shared-plane covariance had split-half VRs of 0.96 to 1.57 and was closer to one on eight of nine sets than the independent-hole covariance.Source: §8.2 · pp. 18–19
  • Under range-dependent shape error, reported σt matched the 7.5 mm error after scale estimation, while the trend test rejected 14% of trials versus 8% under the nominal model.A common bias remained statistically nominal.Source: §8.2 · p. 19

8.3 Directional Precision and Acquisition Efficiency

The experiments examine whether marginal directional precision guides acquisition and whether the calibration state detects faults under unknown noise. The weakest-direction score reduces observation needs in supported settings, while scale-aware checks identify faults and flag batches that cannot be reliably cleaned.

8.3 Directional Precision and Acquisition Efficiency

  • A diagonal-block analysis understates worst-direction standard deviation by factors of 1.16–1.74 for rotation and 1.83–2.56 for translation, often identifying the wrong weakest axes.The weakest axes differ by 42°–86° in rotation and 62°–89° in translation; on synthetic batches, the translation factor remains about three.Source: §8.3 · p. 19
  • On S4, the weakest-direction score reaches the 1.5× full-set precision target with fewer observations than random or far-board-first ordering.Under a strict rotation requirement, it uses up to 30% fewer observations than design criteria; under the default requirement, it requires comparable counts to those criteria and fewer than random or far-first ordering.Source: §8.3 · p. 19
  • On three recorded sets, state-guided additions meet both precision and detectability requirements with the fewest observations among the tested collection loops.On chassis S2, the unselected observations confirmed the verified state in all 10 replays.Source: §8.3 · p. 19

8.4 Fault Detection and Detection-Limit Validation

  • At nominal noise with one 30 mm fault, VeriCalib detects 91.2% of faults versus 62.3% for LOO influence, while removing 1.5% of fault-free observations.At twice nominal noise, VeriCalib removes 1.4–1.5% of fault-free observations as its thresholds follow the batch-estimated scale; the fixed-threshold baselines remove 76–90%.Source: §8.4 · pp. 20–21
  • With three of eight observations independently displaced, VeriCalib reports 67% of batches inconsistent or requiring validation instead of returning a corrupted estimate.For shared offsets, it rejects 67–87% of multi-fault batches at the batch level, while none of the baselines reports such a decision.Source: §8.4 · p. 21
  • On real residual splits, injected faults at the predicted MDBeff are detected in 76–85% of rotation trials and 73–89% of translation trials on S1, S2, and FS-B.Without a fault, the test rejects 0–9% of splits, against the nominal 5% level.Source: §8.4 · p. 21

8.5 Ablation Study and Runtime Analysis

The ablations measure the contributions of covariance and diagnosis, then repeated-session experiments test calibration states and independent validation. The results show both how inconsistent batches are identified and how estimates are confirmed across recording days.

8.5 Ablation Study and Runtime Analysis

  • On fault-free synthetic data, structured covariance lowers median error from 4.84 to 4.48 mm; diagnosis removes 1.9% of observations and changes median error by 0.04 mm.Under independent offsets, covariance alone cannot recover the estimate because a mis-extracted hole is biased rather than noisy; diagnosis raises the detection fraction from 72% to 91%.Source: §8.5 · p. 22
  • The complete procedure processes 35 observations in less than 13 s on a single thread, including consensus search, diagnosis, state assessment, and 200 detection-limit splits.The radius-prior fits, structured covariance, and weighted solution take 29–216 ms for 5–35 observations, within about 1.3× FAST-Calib's back end.Source: §8.5 · p. 22

8.6 Calibration States across Repeated Sessions

  • The batch test accepts both S1 pairs but rejects both S2 pairs and both combined sets in the lower tail; the combined sets are also inconsistent under the range-trend test.The reported ratios are 0.61 and 0.39 on S2 and 0.27 and 0.24 on combined sets.Source: §8.6 · pp. 22–23
  • On S1, the head pair is verified, while the chassis pair is validation-limited; both S2 pairs meet precision requirements but are directed to held-out validation.The S1 head precision is 0.066° and 1.8 mm; the combined sets are inconsistent.Source: §8.6 · p. 23
  • A large SOD can still be acceptable when its 44 mm held-out error falls within an 83 mm depth-axis PnP standard deviation, whereas a 6.6 mm SOD is rejected by the within-view test.Thus, deviation magnitude alone separates neither corrupted from informative observations nor precise from imprecise ones.Source: §8.6 · p. 23
  • On the cross-day chassis hold-out, both estimates meet precision and detectable-fault requirements, and each is confirmed by the independent recording day.The held-out deviations are 0.099° and 5.0 mm, and 0.070° and 3.8 mm, below the hold-out detection limits.Source: §8.6 · p. 24

8.7 Generalization to a Checkerboard Target

The checkerboard experiment applies the framework with a target-specific factor, while a stereo reference tests accuracy and dynamic-observation handling. The results show checkerboard transfer and improved stereo agreement after filtering observations with predicted motion.

8.7 Generalization to a Checkerboard Target

  • On the checkerboard, weakest-direction ordering reaches 1.5× full-set precision in 12 head and 6 chassis boards, versus 19 and 11 for random ordering and 27 and 18 for far-first ordering.The procedure retains 30 head and 17 chassis boards, reports the head batch inconsistent, and finds the chassis batch precision-limited.Source: §8.7 · p. 24

8.8 Accuracy and Fault Detection against a Stereo Reference

  • On accepted observations, VeriCalib differs from the stereo reference by 0.23° and 3.0 mm, versus 0.35° and 6.5 mm for FAST-Calib.The VeriCalib rotation discrepancy is about 1.2× its predicted worst-direction standard deviation of 0.19°.Source: §8.8 · p. 24
  • Predicted board motion between image and LiDAR sampling explains 81% and 84% of residual energy, and the cross-view statistic rises with displacement while the within-view statistic rises only weakly.The batch test rejects both pairs in the lower tail, consistent with rigid error in the absorbed subspace.Source: §8.8 · p. 25
  • At a 5 mm displacement threshold, the first pair's stereo discrepancy falls to 0.04° and 1.9 mm, within reference uncertainty, but its detection limits exceed the requirement.The threshold leaves 20 and 18 observations; at 8 mm, consistency assessment excludes seven observations and reduces the discrepancy to 0.06°.Source: §8.8 · p. 25

9 Discussion

The discussion interprets VeriCalib’s calibration states as both reliability checks and guides for data collection. It also identifies limits on weighting, verification scope, and transfer to other pose-estimation problems.

9 Discussion

  • 3.5 times the observations are needed to meet detectability after just meeting precision under (δR, δt) = 3(ϵR, ϵt).The chassis pair of S1 meets precision but not detectability; S4’s 79 and 89 accepted observations reach detection limits below the requirement, and half the head-pair observations are verified by the other half.Source: §9 · p. 25
  • Target placement should vary board orientation and cover the field of view before increasing distance, while the weakest-direction score identifies what the next observation strengthens.When detectable-fault requirements apply, data collection should also reserve a validation set whose information contributes to the detection limit.Source: §9 · p. 26
  • Weighting does not improve the estimate when the predicted covariance reduction vanishes, and its benefit grows with covariance heterogeneity when the covariance shape is close to true.On S5 predicted differences are at most 0.04° and 1 mm; a fault on a precise observation can displace the weighted estimate more than the unweighted one, motivating consistency assessment alongside weighting.Source: §9 · p. 26
  • Residual variance on real-world sets exceeds the nominal model by an estimated scale of 2.8 to 29.5.In simulation, nominally fixed thresholds remove most fault-free observations at twice nominal noise, whereas thresholds using an independent scale estimate retain their level.Source: §9 · p. 26
  • The framework requires an observation residual, supported covariance, and local extrinsic Jacobian; checkerboards provide these for two shared-pose modality blocks.Transfer also requires checking whether consistency statistics retain degrees of freedom.Source: §9 · p. 26
  • Verification is internal to the batch: a bias common to all observations is indistinguishable from a different extrinsic under Assumption 1.If the batch cannot attribute a rigid discrepancy, the procedure requests a held-out validation set rather than accepting it.Source: §9 · p. 26
  • The authors expect two-level compensation wherever a rigid pose is estimated from a few structured observations, including hand-eye, multi-LiDAR, and camera-IMU calibration.They identify deriving structured covariance, directional precision, and consistency checks for these problems as future work.Source: §9 · p. 26

10 Conclusion

The conclusion frames target-based LiDAR-camera calibration as verifiable estimation rather than residual inspection. VeriCalib reports consistency, directional precision, and detection limits, and directs data collection toward the weakest direction.

10 Conclusion

  • Two-level extrinsic compensation explains why fitted residuals reveal neither observation inconsistency nor directional reliability, motivating a calibration state that reports consistency, directional precision, and detection limits at a stated significance level.The state also directs data collection toward the weakest direction.Source: §10 · p. 26

Declaration of conflicting interests

The paper reports no declared conflicts of interest and no funding support.

Declaration of conflicting interests

Funding

  • The authors received no funding support for the research.Source: Funding · p. 26

A Proofs and Derivations

The appendix proves results from the main text and derives covariance and information expressions, including for the checkerboard target. Its derivations address estimator weighting, bias invariance, directional information, and target-feature uncertainty.

A Proofs and Derivations

  • The appendix derives results from Sections 3–6, stereo-composition invariance, and main-text covariance and information expressions, including checkerboard-target expressions.Source: A Proofs and Derivations · p. 29
  • A common extrinsic-equivalent bias shifts the estimated extrinsic but leaves fitted residuals, cross-validated residuals, and cross-subset differences invariant.The stereo composition is likewise insensitive to common extrinsic-equivalent LiDAR bias and examines differential camera errors.Source: A Proofs and Derivations · p. 29
  • For the linearized model, the weighted estimator’s covariance is bounded below by the correctly weighted estimator’s covariance, with equality when the estimates coincide.The proof also gives an upper bound using the generalized eigenvalues of the observation and assumed covariance pairs.Source: A Proofs and Derivations · p. 30
  • A radius prior regularizes circle-center covariance along the radial axis of a partial arc, while the arc-tangential direction remains weak for short arcs.The prior’s effect is direction-dependent and is not removed by the global scale.Source: A Proofs and Derivations · p. 30
  • A variational minimization defines decoupled directional precision by optimally co-adjusting nuisance parameters, yielding a conservative reading even under rank deficiency.Source: A Proofs and Derivations · p. 31
  • Rotation information decomposes into centered point-cloud inertia, with board orientation diversity and spread determining the contribution; common standoff adds no rotation information once translation absorbs it.Source: A Proofs and Derivations · p. 31
  • The checkerboard LiDAR fit uses a rigid rectangle of known outer size fitted to boundary points, with its fixed half-extents removing weak size directions.The four corners share one fitted pose and have correlated covariance rather than independent corner errors.Source: A Proofs and Derivations · p. 31
  • A non-degenerate rigid checkerboard has no within-observation degrees of freedom, so its within-view check does not apply and consistency relies on a cross-observation test.A free-rectangle fit monitors the known board dimension, with a systematic size gap determined once and fixed for both sensors.Source: A Proofs and Derivations · p. 31

B Consistency Statistics

The appendix develops residual, subset, and held-out checks for model consistency, alongside diagnostics for scale, directional fault detection, and observation influence. Their references depend on stated Gaussian, rank, and selection conditions, and some thresholds are approximate.

Expected influence of a fault-free observation

  • Influence reflects observation geometry, not correctness; at the fit, the expected displacement of a fault-free observation is determined by its leverage.The camera-feature error described in the proposition changes the residual covariance but, as a zero-mean error, does not bias the weighted estimate to first order.Source: B Consistency Statistics · p. 32

Radius diagnostics

  • The radius diagnostic compares free-fit radii with the prior: its reference is exact for a radius drawn from the prior and conservative for a fixed radius equal to r.For a fully sampled boundary, the prior is inert; for the most partial arc, the induced center displacement is reported as a bias budget.Source: B Consistency Statistics · p. 32

Observation-level and subset statistics

  • Internal and cross-fitted observation statistics use different residual scales; for a fixed full-rank four-hole Gaussian model before selection, the cross-fitted scale is independent of the estimate and residual.The cross-fitted references are mean-matched but not exactly calibrated under scale mismatch between the subspaces.Source: B Consistency Statistics · p. 32

Cross-subset statistics

  • A cross-subset Mahalanobis distance tests whether two sufficiently large disjoint subsets converge to different solutions, but cannot detect a bias shared by both.The stated distribution is exact for a fixed linear Gaussian design; each subset must have at least two observations and full-rank information.Source: B Consistency Statistics · p. 32

Residual-trend test

  • The range-trend test regresses normalized residuals on LiDAR range; neglecting the leverage term makes its threshold approximate, with false-alarm behavior checked empirically.A large scale with a large discrepancy indicates inconsistent observations, whereas a large scale with a small discrepancy indicates an undersized but approximately proportional noise model.Source: B Consistency Statistics · pp. 32–33

Scale references

  • The reported accepted-set scale comes from absorbed residual components that the common extrinsic cannot remove because observations require different corrections.The working scale is for minimal-consensus evaluation, independent references calibrate tests, and the accepted-set scale determines precision and detection limits.Source: B Consistency Statistics · p. 33

Hold-out confirmation

  • The held-out validation test compares a validation-only correction with the fixed estimate, while its focused worst-direction statistic defines a directional detection limit.The six-degree-of-freedom test and focused statistic have different non-centralities, so their detection limits are not in a fixed ratio.Source: B Consistency Statistics · p. 33

Focused validation statistic and equal-half splits

  • A focused validation limit is about 5.6 times the full-set worst-direction standard deviation, requiring about 3.5 times as many observations as the precision requirement under the stated equal-half assumptions.With the default requirement δ = 3ϵ, the worst-direction standard deviation must be below 0.54 ϵ.Source: B Consistency Statistics · p. 33

C Additional Experimental Results

Additional experiments evaluate VeriCalib’s state-guided collection, selected subsets, held-out alignment, and sensitivity to modeling and decision parameters. The results show better data efficiency and held-out alignment in the reported settings, while several sensitivity changes leave the state largely stable or reveal persistent inconsistency.

Projection consistency on held-out scenes

  • 19 of 26 held-out score–extrinsic pairs at least 0.7° from VeriCalib were significantly worse, none was better, and VeriCalib had the smallest absolute edge offset in every shown image.The seven non-significant cases all concerned Dom2019.Source: C Additional Experimental Results · p. 34, Figure 20 · p. 34

Accuracy of selected subsets

  • 0.09°–0.15° and 4.2–5.0 mm were the median rotation and translation deviations for weakest-direction subsets of three to five observations on head S2, versus 0.16°–0.19° and 6.0–8.2 mm for random subsets.Weakest-direction subsets also had smaller reported deviations than random subsets on FS-B and the S4 chassis pair; on the S4 head pair the rotation advantage was smaller, with comparable translation.Source: C Additional Experimental Results · pp. 34–35

State-guided collection

  • 10, 10, and 12.5 observations were needed for state-guided collection to meet both requirements, versus 11, 11.5, and 19 for random ordering on the three sets.On chassis S2, all 10 replays requested validation, and unselected observations confirmed the verified state in all 10.Source: C Additional Experimental Results · p. 35

LiDAR edge-noise model

  • 12.99 mm against 13.27 mm on average was the held-out discrepancy for the range-incidence model versus the beam model, and its residual scale was smaller on all three raw-support sets.The comparison used otherwise identical structured factors on sets with K ≥ 14.Source: C Additional Experimental Results · p. 35, Table 12 · p. 35

Parameter sensitivity

  • 19.5, 21.9, and 25.4 mm were the median MDBt values as required power rose from 0.70 through 0.80 to 0.90; lowering α from 0.10 through 0.05 to 0.01 raised MDBt from 19.5 through 21.9 to 26.8 mm.Across αbias = 0.005 to 0.05, the diagnosis detected at least 0.99 of injected faults, while false-removal rates ranged from 0.007 to 0.051.Source: C Additional Experimental Results · p. 35, Table 13 · p. 35
  • 12.7 to 0.8 on the head pair and 12.8 to 2.0 on the chassis pair were the scale changes when boundary-point equicorrelation rose from 0 to 0.3, while accepted sets mostly stayed unchanged.Precision and detection limits changed by less than 15% on the head and 13% on the chassis relative to default correlation, although the chassis batch decision differed for ρ̄ ≥ 0.1.Source: C Additional Experimental Results · p. 35

Related papers

Loading 2610.28791v1…