VeriCalib: Verifiable Target-Based LiDAR-Camera Calibration

Xiangcheng Hu; Xieyuanli Chen; Feiyi Chen; Shengli Ren; Shiyang Chen; Guoqiang Ye; Hao Wang; Ping Tan

cs.ROOct 20, 2026

VeriCalib: Verifiable Target-Based LiDAR-Camera Calibration

Authors: Xiangcheng Hu, Xieyuanli Chen, Feiyi Chen, Shengli Ren, Shiyang Chen, Guoqiang Ye, Hao Wang, Ping Tan,

Organizations: Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology, Hong Kong, China · X Square Robot Technology (Shenzhen) Co., Ltd., Shenzhen, China · College of Intelligence Science and Technology at National University of Defense Technology, Changsha, Hunan, China · Independent Researcher ·

Abstract

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...

Figures & tables

Figure 1: Head-pair calibration from $14$ observations, two with hole centers misplaced by automatic rim extraction ( C : true rim and center in black and white, extraction in red). A : LiDAR scans projected with the extrinsics of FAST-Calib and VeriCalib (tags: translation deviation from the reference). The RMSE of $34.0$ mm neither reveals nor locates the errors, whereas VeriCalib removes both and one further observation ( B : accepted in teal, removed in red) and reports the state, validation-limited (V), with its translation precision $\sigma_{t}$ .

Figure 2: VeriCalib overview on the head pair. Left : observations with projected LiDAR points (blue; red box: head camera; cyan boxes: chassis LiDAR and camera) and their fitted RMSE. Center : structured factor $\mathcal{F}_{k}$ of one observation with the $1\sigma$ hole-center residual covariance (navy ellipses), the LiDAR boundary points (blue dots), and the absorbed (green) and internal (red) residual parts; the boxes derive the scale $\hat{s}$ , the directional information $\bm{\Lambda}$ , and the MDB, which depend on one another. Right : calibration state (X/P/V/OK: inconsistent, precision-limited, validation-limited, verified), accepted (navy), $p<0.05$ (salmon), and removed (red) board poses, detection-limit ellipse with its weakest direction $\bm{v}_{\max}$ (green), and recommended next pose (green dashed).

[][]			Uncertainty modeling			Constraint analysis		Result assessment
[][]									Fault test
[][]	Method	Venue	LiDAR	Camera	Scale	Analysis	Guidance	Metric	Within	Cross	Detect. limit
	Zhou et al. (2018a)	IROS	$\times$	$\times$	$\times$	Coupled	Empirical	Ref.	$\checkmark$	$\times$	$\times$
	Domhof et al. (2019)	ICRA	Indep. †	Indep. †	Est. †	$\times$	$\times$	RMSE	$\checkmark$	$\times$	$\times$
	velo2cam ( Beltrán et al., 2022 )	T-ITS	$\times$	$\times$	$\times$	$\times$	Empirical	Repeat.	$\checkmark$	$\times$	$\times$
	LCE-Calib ( Jiao et al., 2023 )	T-MECH	$\times$	Indep.	$\times$	$\times$	$\times$	Ref.	$\checkmark$	$\times$	$\times$

Table 1: Comparison of extrinsic calibration methods.

Figure 3: Two-level extrinsic compensation ( A – C : principle; D – F : chassis-pair data). A , D : a hole-center error split into absorbed and internal parts. B , E : a pitch and the compensating translation yield nearly the same target observation. C , F : rotation precision with the translation fixed ( $\sigma_{\mathrm{f}}$ ) and marginalized ( $\sigma_{R}$ ); C: $\sigma=5$ mm, $b=0.2$ m.

Figure 4: Shared-pose camera covariance. Left :

3\sigma

hole displacements and correlations of the shared-pose (Joint) and independent-hole (Indep.) models. Right : hole centers of PnP solutions from noise-perturbed marker corners on a head-pair observation (

\times 60

), with

3\sigma

ellipses of each hole and of their centroid (dashed: Indep.).

Figure 5: Short-arc LiDAR center covariance ( $3\sigma$ ) of the free-radius fit (Free) and of the radius prior with $N_{\mathrm{eff}}$ points (Prior); Rim: boundary points. Left : synthetic $\pm 25^{\circ}$ arc. Right : partial-arc hole of a head-pair observation ( $\times 12$ ).

Figure 6: Covariance shape and scale. Left : nominal (dashed) and

\hat{s}

-scaled (solid) covariances of the joint and independent-hole models against the ground truth (GT). Right : ECDF of

\mathrm{NIS}_{k}/d_{k}

before (Nom.) and after (

/\hat{s}

) scale estimation (solid: head; dashed: chassis).

Panel of Figure 7 — Figure 7: Verification pipeline ( Algorithm 1 ).

Figure 8: Consistency tests. Left : null and fitted NIS densities with threshold

\tau

, and rigid and local faults. Right : within-view and cross-view statistics of the head (circles) and chassis (triangles) observations, normalized by their thresholds.

Figure 9: Observation diagnosis in the influence-inconsistency plane. Left : removal criterion against leave-one-out (LOO) selection. Right : head (circles) and chassis (triangles); blue: accepted; salmon: $p<0.05$ ; red: removed; X: inconsistent.

Figure 10: Directional detection limits. Left : two-dimensional section (shaded: validation-limited). Right : translation detection limits of the head and chassis pairs along the principal axes, with the

1\sigma

precision core and weakest direction.

Figure 11: Platforms and targets (navy: cameras; red: LiDARs). A – C : mobile manipulator with the head and chassis cameras. D , E : four-hole target and checkerboard. F : handheld platform with stereo cameras and an Ouster LiDAR. G : FS platform with a flash LiDAR.

				Camera 1		Camera 2
Platform	Set	LiDAR	Target	FOV [ ∘ ]	$K$	FOV [ ∘ ]	$K$
Robot-A	S1	Mid-360	Four-hole	$83{\times}53$	$14$	$107{\times}91$	$14$
	S2		Four-hole		$21$		$26$
Robot-B	S3	Mid-360	Checkerboard	$83{\times}53$	$37$	$107{\times}91$	$20$
	S4		Four-hole		$94$		$101$
Handheld	S5	Ouster	Four-hole	$80{\times}70$	$147$	$80{\times}70$	$92$

Table 2: Sensors and observation counts of the real-world sets.

		Rot. [ ∘ ] $\downarrow$			Trans. [mm] $\downarrow$			VR
Set	$\|\mathcal{S}\|$	Ours-U	Indep.	Joint	Ours-U	Indep.	Joint	Indep.	Joint
S1-H	$14$	$0.237$	$0.217$	$0.211$	$5.5$	$4.6$	$3.7$	$2.03$	$1.47$
S1-C	$13$	$0.444$	$0.342$	$0.297$	$12.0$	$11.1$	$8.5$	$2.09$	$1.37$
S2-H	$18$	$0.141$	$0.136$	$0.133$	$6.1$	$5.9$	$5.9$	$1.51$	$1.41$
S2-C	$23$	$0.371$	$0.138$	$0.126$	$16.7$	$5.7$	$5.1$	$1.24$	$0.96$
S4-H	$79$	$0.080$	$0.093$	$0.085$	$2.7$	$3.6$	$2.8$	$2.03$	$1.57$

Table 3: Split-half repeatability of the extrinsic under three weightings on the real-world sets.

Figure 12: Real-data covariance and acquisition. A : variance ratio of the joint and independent-hole (Indep.) camera covariances (dashed: unity). B , C : worst-direction

\sigma_{t}

of six orderings from one initial pair on the head and chassis sets (dotted:

8

mm target; markers: first

K

reaching it).

	Ours-U		Ours-W
PnP noise	$e_{R}$ [ ∘ ]	$e_{t}$ [mm]	$e_{R}$ [ ∘ ]	$e_{t}$ [mm]
Constant	$0.111$	$5.60$	$0.103$	$5.09$
Depth-dependent	$0.165$	$7.36$	$0.115$	$4.97$
Depth-dep. + offset (all)	$0.216$	$10.32$	$0.189$	$9.02$
Depth-dep. + offset (near)	$0.187$	$9.27$	$0.177$	$8.75$

Table 4: Weighting efficiency on synthetic batches.

Generating model	$\hat{s}$	Rej. [%]	$\sigma_{t}$ [mm]	$e_{t}$ [mm]
Nominal	$1.00$	$8$	$4.4$	$5.3$
Camera noise $\times 2$	$3.21$	$7$	$7.9$	$10.1$
LiDAR noise $\times 2$	$1.80$	$8$	$5.9$	$6.5$
Range-dependent shape	$2.92$	$14$	$7.5$	$7.5$
Common-mode bias	$0.99$	$6$	$4.4$	$9.3$

Table 5: Covariance-model perturbations on synthetic batches.

Figure 13: Contribution of each observation to the head-pair extrinsic, with the observations sorted by depth. A : board images shaded by the resulting difference $\Delta\sigma_{t}$ of the worst-direction translation precision (salmon frames: accepted with $p<0.05$ ; red frames: removed). B : $\Delta\sigma_{t}$ when the observation is removed from the accepted-set estimate, or added if excluded. C : resulting translation shift $\|\Delta\bm{t}\|$ .

$(\epsilon_{R},\epsilon_{t})=(0.15^{\circ},8\,\mathrm{mm})$
Set	Ours	D/A/E	Rand.	Far
Synthetic	$4$	$4$	$6$	$7$
S1+S2 H, C	$3$ – $4$		$5$ – $6$	$5$ – $12$
$1.5\times$ full-set precision
S4 H	$24$	$23$ – $29$	$41.5$	$72$
S4 C	$22.5$		$47.5$	$79$

Table 6: Median number of observations

\downarrow

of the selection rules on synthetic and real-world sets.

Noise at the nominal magnitude
		Detected [%]				Falsely removed [%]				Rej. [%]
Faulty	Fault	LOO	RANSAC	Huber	Ours	LOO	RANSAC	Huber	Ours	Ours
$0/8$	None	–	–	–	–	$12.5$	$2.2$	$2.8$	$1.7$	$10.0$
$1/8$	Offset $30$ mm	$62.3$	$86.0$	$96.0$	$91.2$	$5.4$	$5.4$	$4.6$	$1.5$	$12.8$
$1/8$	Hole swap	$11.0$	$100.0$	$100.0$	$100.0$	$12.7$	$2.2$	$3.0$	$1.9$	$10.5$
$3/8$	Indep. $30$ mm	$27.7$	$83.3$	$94.3$	$50.4$	$3.4$	$13.3$	$6.2$	$1.1$	$66.8$

Table 7: Outlier removal on synthetic batches.

Figure 14: Rotation (left) and translation (right) deviations of head S1 extrinsics from the reference (dashed): fault-free single-observation solutions (grey), the two corrupted observations (red crosses), and the joint solutions.

Figure 15: Detection limit and calibration state on fault-free synthetic batches. A : detection rate of the focused test against a translation bias (dots: empirical; curve: predicted; dashed:

80\%

power). B – D :

\sigma_{t}

\mathrm{MDB}_{R}

, and

\mathrm{MDB}_{t}

against

K

(dashed: requirements). E : hold-out against random-split

\mathrm{MDB}_{t}

. Boxes: quartiles; whiskers:

5

th to

95

th percentiles; logarithmic axes in C–E.

Diagnosis
Method	Fault-free	Indep.	Shared	Head S1
Ours-U	$4.84$	$9.56$	$14.09$	$13.7$
Ours-W	$4.48$	$8.85$	$13.41$	$16.2$
Ours-WC	$4.48$	$6.93$	$13.64$	$0.37$
Ours (full)	$4.52$	$5.85$	$13.18$	$0.32$
Retained, cons.	$16$	$13$	$13$	$12$

Table 8: Translation error [mm]

\downarrow

of the cumulative ablation.

Figure 16: Observation decisions on the combined robot-platform batches. Rows: joint extrinsic of each method; columns: an accepted observation, observations removed by the within-view and the cross-view check, and a far three-marker board explained by its covariance. Cyan: projected rim points; white and blue: camera and LiDAR hole centers; labels: held-out alignment error; bottom: $3\sigma$ residual ellipses ( $\times 5$ ) and smallest $p$ -value; frames: teal, accepted; salmon, $p<0.05$ ; red, removed.

Estimation
Stage	$K{=}5$	$K{=}14$	$K{=}35$
FAST-Calib	$22.0$	$62.5$	$167.9$
Ours	$28.7$	$81.6$	$215.8$
Verification (Ours)
Information	$1.1$	$2.6$	$6.0$
Diagnosis	$41.9$	$279.8$	$1663.5$

Table 9: Runtime [ms] on the head batch.

Figure 17: Cross-session analysis. Left : rotation difference between S1 and S2 on all observations and without gross errors (Trimmed: SODs above

100

mm removed; grey: inter-unit range). Right : residual range trend with level-

\alpha

thresholds.

Figure 18: Checkerboard target ( A – F : head pair; G , H : both pairs). A , B : camera and LiDAR factors. C , D : an accepted (green) and a rejected (red) board with residual ellipses and $p$ -values. E : eigenvalues of the residual covariance blocks. F : board consistency against depth. G : detection rate of an injected board bias. H : acquisition (dotted: precision target).

Split	$\|\mathcal{S}\|/\|\mathcal{V}\|$	[ ∘ ]	[mm]	[ ∘ ]	[mm]	[ ∘ ]	[mm]	$T$	$p$
		Precision		MDB		Deviation
S2a $\to$ S2b	$8/17$	$0.097$	$3.1$	$0.39$	$12.9$	$0.099$	$5.0$	$3.7$	$0.71$
S2b $\to$ S2a	$15/9$	$0.084$	$3.1$	$0.32$	$11.1$	$0.070$	$3.8$	$4.2$	$0.65$

Table 10: Cross-day hold-out validation of the chassis pair.

Figure 19: Stereo check on S5. Left :

\mathrm{NIS}_{k}/d_{k}

against the predicted board displacement (red: excluded). Right : rotation discrepancy to the stereo reference for each displacement threshold, with states,

10

th to

90

th bootstrap percentiles, and reference uncertainty (grey).

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 20: LiDAR points of held-out scenes projected with each extrinsic (rows) on three handheld windows and two Robot-A captures, colored by range. Insets: a near-vertical image edge (cyan) with the depth-discontinuity points (magenta) and their mean horizontal offset $\Delta$ [px]; tags: deviation from Ours (ref.). In each image, the inset edge is the candidate with the most depth-discontinuity points, chosen before any offset is computed.

Arc fraction	Free	Soft prior	Fixed radius
$1.00$	$1.95$	$1.95$	$1.93$
$0.45$	$4.17$	$3.98$	$2.01$
$0.33$	$6.94$	$5.72$	$2.15$
$0.22$	$16.73$	$7.31$	$2.76$

Appendix

Table 11: Circle-center RMSE [mm]

\downarrow

of three circle fits on synthetic partial arcs.

		Error [mm]		$\hat{s}$		$\hat{\rho}_{z}$
Dataset	$K$	B	RI	B	RI	B	RI
FS-B	$15$	$15.14$	$15.04$	$50.67$	$17.98$	$-0.12$	$-0.23$
Head S2	$21$	$6.85$	$6.44$	$14.69$	$3.43$	$-0.42$	$-0.31$
Chassis S2	$26$	$17.82$	$17.48$	$6.85$	$4.64$	$-0.17$	$+0.02$

Appendix

Table 12: Comparison of the beam and range-incidence LiDAR edge-noise models on real-world sets.

	Head ( $K{=}35$ )				Chassis ( $K{=}40$ )
$\alpha_{\mathrm{bias}}$	$\|\mathcal{S}\|$	$t$	$\sigma_{t}$ [mm]	$\mathrm{MDB}_{t}$ [mm]	$\|\mathcal{S}\|$	$t$	$\sigma_{t}$ [mm]	$\mathrm{MDB}_{t}$ [mm]
$0.002$	$30$	$-2.94$	$1.84$	$10.7$	$34$	$-2.62$	$2.06$	$11.9$
$0.005$	$29$	$-2.97$	$1.95$	$11.4$	$32$	$-2.57$	$2.02$	$11.7$
$0.01$	$28$	$-3.53$	$1.88$	$11.0$	$32$	$-2.57$	$2.02$	$11.7$
$0.02$	$26$	$-3.19$	$1.92$	$11.3$	$32$	$-2.57$	$2.02$	$11.7$
$0.05$	$26$	$-3.19$	$1.92$	$11.3$	$28$	$-2.14$	$1.96$	$11.4$

Appendix

Table 13: Sensitivity to the removal level on the combined Robot-A head and chassis batches.

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LV-Calib: LiDAR-Camera Extrinsic Calibration with Boundary-Response Modeling

Sheng Hong

Pen-Tung Sah Institute of Micro-Nano Science and Technology, Xiamen University, Xiamen 361102, China

Lidar-Camera Extrinsic Calibration Camera-LiDAR Fusion

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We present LV-Calib, a calibration framework for LiDAR-camera extrinsic estimation and LiDAR boundary-response calibration using a printable planar target. The target serves as a shared observation carrier: visual fiducials provide indexed image measurements, while circular reflectivity boundaries provide LiDAR-observable structural feature points. Instead of directly fitting boundary points as ideal geometric contours, LV-Calib automatically crops background points, estimates the target plane, and iteratively refines accurate LiDAR-side 3-D feature points from intensity and geometric constraints. The refinement explicitly handles the broadened and distorted transition band induced by finite beam footprint and mixed-intensity returns around black-white reflectivity discontinuities. Given these refined LiDAR features, we formulate a weighted reprojection-consistent extrinsic optimization with LiDAR feature alignment, where image observations are kept in the reprojection domain and LiDAR feature residuals are weighted by refinement confidence. Finally, using the estimated extrinsic and the extracted transition band, LV-Calib calibrates the LiDAR boundary response by estimating pitch-yaw-range residual statistics of boundary-overlap samples. Experiments on printed-board calibration data demonstrate sub-pixel reprojection accuracy, millimeter-level LiDAR feature consistency, and improved odometry performance. Code and calibration data will be released for reproducible evaluation.

Abstract

Lidar-Camera Extrinsic Calibration

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