Targetless Intrinsics and Extrinsic Calibration of Multiple LiDARs and Cameras with IMU using Continuous-Time Estimation

Chao Lu; Jiajun Zhu; Song Wu; Yuezhang Lv; Yunzhou Zhang

REVIEW 4 major objections 5 minor 36 references

Targetless Intrinsics and Extrinsic Calibration of Multiple LiDARs and Cameras with IMU using Continuous-Time Estimation

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A continuous-time bundle adjustment recovers intrinsics, extrinsics, and time offsets for multiple LiDARs, cameras, and IMUs in one targetless pass.

desk verdict Useful integration of existing targetless calibration tools, but the evidence is self-consistency rather than ground-truth accuracy, and the final staged optimization fixes the LiDAR map, undercutting the 'no accumulated errors' claim. read the letter →

arxiv 2501.02821 v1 pith:LWPAFMJO submitted 2025-01-06 cs.RO

Yuezhang Lv , Yunzhou Zhang , Chao Lu , Jiajun Zhu , Song Wu This is my paper · ORCID

classification cs.RO

keywords sensorcalibrationcontinuous-timeestimationbundleadjustmentLiDAR-cameraDAR-IMUtargetlessintrinsicsandextrinsicstimeoffset

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that all spatiotemporal parameters of a multi-sensor rig—camera and LiDAR intrinsics, 6-DOF extrinsics, and time offsets—can be recovered in a single joint optimization that needs no calibration boards and no overlapping fields of view. The method parameterizes the rig's motion with a continuous-time B-spline trajectory, builds common observations among cameras through structure-from-motion, among LiDARs through an adaptive voxel-plane map, and between LiDAR and camera by projecting LiDAR intensity into the image plane and matching features. A bundle adjustment over all residuals then refines every parameter simultaneously, so no sensor pair is calibrated in isolation and errors do not accumulate across pairs. On an autonomous vehicle with four LiDARs and six cameras, the reported calibration accuracy is 0.04 degrees and 0.5 cm for sensors with common view, and 0.08 degrees and 1 cm for sensors without common view.

What carries the argument

The load-bearing object is a continuous-time trajectory represented by two uniform B-splines—one for translation and one for rotation, with the rotation spline built through cumulative Lie-group increments $\mathrm{Exp}(\lambda \cdot \log(R))$—so that asynchronous measurements from every sensor can be evaluated at arbitrary timestamps without adding per-measurement pose variables. The second central mechanism is the adaptive voxel map: the LiDAR point cloud is recursively split into voxels, each voxel is fitted to a plane parameterized by the closest-point form $\varepsilon = n d$, and LiDAR points vote on these planes as residual factors, creating co-observations among different LiDARs. Finally, LiDAR-to-camera association is made by rendering the LiDAR intensity map onto the camera image plane and matching features, which turns visual pixels into metric 3D observations that constrain camera intrinsics and extrinsics jointly. These pieces combine into a single factor graph whose residuals $r_I$, $r_C$, $r_L$, and $r_{lc}$ are minimized together.

What would settle it

Take a rig with independently known ground-truth extrinsics, run the pipeline, then deliberately add a known scale error or constant drift to the base LiDAR odometry before map construction: if the camera intrinsics shift by more than the paper's claimed sub-0.5-pixel level, the frozen-map assumption is absorbing map error. A second decisive check is to compare the same rig's calibration against an independent target-based calibration in the same scene, where agreement within the claimed 0.04 degrees and 0.5 cm for common-view pairs would support the claim and larger disagreement would refute it.

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Extended reading notes

Core claim

The central claim is that a pipeline based on continuous-time estimation and bundle adjustment can simultaneously calibrate the intrinsic parameters, the 6-DOF extrinsic transforms, and the time offsets of any number of LiDARs, cameras, and IMUs, without overlapping fields of view, without calibration infrastructure, and without error accumulation from calibrating each sensor pair separately. The authors establish this by chaining three association mechanisms: structure-from-motion self-calibrates the cameras, an adaptive voxel map with plane bundle adjustment associates LiDAR frames and refines LiDAR-IMU extrinsics, and intensity-image projection of the LiDAR map creates direct LiDAR-camera correspondences that tie the camera intrinsics to metric map geometry. Joint nonlinear least squares over IMU, camera, LiDAR, and LiDAR-camera residuals then recovers all parameters at once. In the reported vehicle experiment, the paper states accuracies of 0.04 degrees and 0.5 cm for sensors with a common view and 0.08 degrees and 1 cm for sensors without a common view, with camera distortion errors within 0.5 pixels and LiDAR map stitching thickness within 2 cm.

Load-bearing premise

The final joint calibration treats the LiDAR map as optimal and never re-optimizes the map's planes, so any drift or distortion left in that map is absorbed into the camera intrinsics and extrinsics rather than corrected.

Editorial extensions

If this is right

The full parameter set—camera and LiDAR intrinsics, 6-DOF extrinsics, and time offsets—is estimated in one optimization, so errors do not compound across separately calibrated sensor pairs.
Cameras that never share a view and LiDARs with only a 30-degree mutual overlap are still constrained through the shared continuous-time IMU trajectory and the joint voxel map.
Because time offsets are estimated, the rig does not require precisely synchronized trigger signals to be calibrated.
On the reported vehicle data, common-view pairs calibrate to about 0.04 degrees and 0.5 cm, non-common-view pairs to about 0.08 degrees and 1 cm, and the method accepts any number of cameras and LiDARs in one pipeline run.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the paper is right, the frozen-LiDAR-map assumption in the final joint optimization is the main channel by which LiDAR odometry drift leaks into camera intrinsics and extrinsics; injecting known map errors into a test sequence would reveal how much of the claimed accuracy is inherited rather than directly measured.
The same continuous-time factors could in principle be run incrementally, turning this offline calibration into online recalibration whenever the rig moves through texture-rich structure, though the paper does not claim this extension.
Because camera-camera association relies on texture features and LiDAR-camera association on intensity edges, long corridors, repetitive walls, or low-light scenes are likely failure modes even though no calibration board is needed.
The claimed 'any number of sensors' generality depends on enough texture and structure to support structure-from-motion and feature matching; a rig with many cameras that see mostly sky or untextured ground may not satisfy that precondition.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper is a competent integration of existing targetless calibration components into a single pipeline: continuous-time B-splines, COLMAP for camera intrinsics, adaptive voxel-map BA for the LiDARs, and SuperPoint/SuperGlue matching between LiDAR intensity images and camera images. That combination is new and practically useful, and the test rig—4 solid-state LiDARs, 6 cameras, and an IMU on a real vehicle—is a serious environment. The calibration-harp check gives an independent, if narrow, confirmation that the camera distortion model is reasonable.

The main problem is that the evidence is mostly self-consistency, not accuracy. Table I reports repeatability across six sequences, not error against ground truth. There is no ground-truth extrinsic for the rig, so the numbers like 0.04 degrees and 0.5 cm are precision, not accuracy. The paper also excludes the z-axis translation for LiDAR-IMU and camera-IMU because the figure-eight motion is degenerate in that direction, yet claims 6-DOF calibration. That claim should be toned down. No baselines from other targetless methods are given except hand-eye calibration in two cases, and no code or data are released, which makes independent verification impossible.

The stress-test concern about Section IV.D is valid. The paper explicitly says it considers the LiDAR map to be optimal and there is no need to optimize the voxel plane in the final joint calibration. So the camera intrinsics and extrinsics are fitted against a fixed map built earlier. Any drift or local distortion in that map will be absorbed into the camera parameters. That directly weakens the 'without accumulating errors' claim. The pipeline is really staged, not fully simultaneous, and the final stage is a fit to a previously built map rather than a joint correction.

That said, the underlying math looks standard, the approach is sensible, and the real-world data are a plus. The paper needs to be much more careful about what is being measured and what is being claimed.

I would send this to peer review, because the integration is useful and the experimental setup is nontrivial, but I would push for heavy revision: release code and data, add baseline comparisons, and revise the abstract and Section IV.D to state the map-fix limitation explicitly.

Referee Report

4 major / 5 minor

Summary. The paper proposes a targetless, continuous-time calibration pipeline for multiple LiDARs, cameras, and an IMU. The method first initializes camera intrinsics via COLMAP/SfM and LiDAR-IMU extrinsics via hand-eye calibration, then refines the LiDAR map and LiDAR-IMU extrinsics using planar voxel-based bundle adjustment, and finally establishes direct LiDAR-camera associations by projecting LiDAR intensity images and matching them to camera images. The final joint optimization (Section IV.D) estimates camera intrinsics, camera-IMU and LiDAR-IMU extrinsics, and time offsets while keeping the LiDAR map fixed. Experiments on a self-driving vehicle with four LiDARs and six cameras report calibration consistency across six figure-eight sequences, camera distortion validation using a calibration harp, and various self-consistency metrics such as reprojection error, map entropy, point-cloud thickness, and epipolar error.

Significance. The paper addresses an important practical problem: joint spatiotemporal calibration of asynchronous, non-overlapping multi-sensor rigs without calibration targets. The continuous-time formulation and the use of adaptive voxel maps to create cross-LiDAR associations are reasonable building blocks. The pipeline is a natural combination of established components (Kalibr-style B-splines, BALM-like planar BA, COLMAP, SuperPoint/SuperGlue) applied to a multi-sensor setting. If the accuracy claims were supported by independent ground truth, the method would be a useful contribution to the calibration toolbox. However, as presented, the evaluation is mostly repeatability and self-consistency, and the fixed-map final stage undercuts the 'without accumulating errors' claim. The paper does not provide code or data, so reproducibility of the exact experiments is limited.

major comments (4)

[Section V.A, Table I] The reported 'accuracy' in Table I is not accuracy but repeatability: the six sequences all follow the same figure-eight trajectory, and no ground-truth extrinsic transforms are available for the real-world rig. The columns 'p' and 'RMSE' describe run-to-run means and dispersions, not errors against a known reference. Therefore the claims in Section V.A that 'calibration accuracy is 0.04 degrees and 0.5 cm' for common-view sensors and '0.08 degrees and 1 cm' for non-common-view sensors are unsupported. The authors should either obtain an independent ground truth (e.g., factory CAD, a separate target-based calibration, or a simulated rig with known transforms) for at least a subset of the extrinsic parameters, or explicitly re-label these numbers as repeatability and remove the accuracy wording.
[Section V.A and Section VI] The paper claims in the abstract and introduction that the method performs '6 DOF transformation' calibration, but Section V.A explicitly states: 'we do not estimate the z-axis translation' because the figure-eight motion makes it unobservable. This is a direct contradiction: the calibrated extrinsics appear to be 5-DOF (or the z-translation is held at the factory structural value). The authors must either (a) confine the claims to the actually estimated degrees of freedom, or (b) add a motion segment (e.g., nonzero pitch/roll excitation) that makes the z-axis translation observable, and report the corresponding results. As written, the 6-DOF claim is misleading and the limitation is not discussed in the conclusions.
[Section IV.D, Eq. (13)] The final joint calibration fixes the LiDAR map as optimal ('we consider the LiDAR map to be optimal and there is no need to optimize the voxel plane'). The LiDAR-camera residual in Eq. (13) projects these fixed map points through the camera model and optimizes camera intrinsics and camera-IMU extrinsics. However, the LiDAR map was built earlier from KISS-ICP odometry, single-LiDAR BA, and joint voxel-map BA (Sections IV.B and IV.C), before direct camera-LiDAR association. Any residual drift, distortion, or bias in that map will be absorbed into the camera intrinsics and extrinsics during the final stage. This directly weakens the paper's claim of 'without accumulating errors': the final stage is a fit to a previously constructed fixed map, so map errors propagate into the cross-modal parameters. The authors should either jointly optimize the voxel planes/map landmarks in the final bundle adjustment, or provide a quantitative sensitivity analysis (e.g., injecting known map perturbations and measuring the resulting calibration bias) to show that the map is sufficiently accurate for the claimed tolerances.
[Sections V.B–V.F] All extrinsic validation metrics are self-consistency measures computed from the same geometric quantities that the optimization minimizes: reprojection error (Table III), map entropy (Table V), point-cloud thickness (Table VI), and epipolar error (Table VII). Low values of these metrics are necessary but not sufficient evidence of calibration accuracy; they cannot detect common-mode errors such as a global scale error, a bias shared by all transforms, or a consistent shift between the LiDAR map and the camera poses. The only independent validation is the calibration-harp test of camera distortion in Section V.B, which checks distortion coefficients but not focal length, principal point, or any extrinsic parameter. The extrinsic accuracy claims therefore rest on circular evidence. Please add at least one independent cross-check, for example comparing a LiDAR-camera extrinsic against a target-based calibration result, or evaluating on a synthetic dataset with known ground-truth extrinsics.

minor comments (5)

[Section III.A, Eq. (5)] The notation 't∈ [ti − ti+1]' should read 't ∈ [ti, ti+1]' or 't ∈ [ti, ti+1)'; also the difference vector 'dij' is introduced but the index convention is unclear, since Eq. (4) uses 'di = xi − xi−1' and Eq. (5) uses 'dij = xi+j − xi+j−1'.
[Section IV.C] The sentence 'we do not project each frame of the camera, but select the frame with the largest parallax for projection' is vague: parallax is a property of a pair of poses, not of a single frame. Please clarify how the frame with 'largest parallax' is selected.
[References] Reference [14] is a duplicate of reference [1] (both list the same Furgale et al. paper), and the citation of [4] in Section II.A as the origin of continuous-time calibration is misleading because [4] is an observability-aware LiDAR-IMU calibration paper, not the original continuous-time calibration framework. Please correct the reference list and citations.
[Table I] The table uses inconsistent sensor names: the rows 'front lidar', 'back lidar', 'left lidar', 'right lidar' do not map clearly to the L_A–L_D labels used in the lidar-lidar and lidar-cam rows and in Table II. The units in the 'px (m±cm)' column are also confusing; clarify whether the '±' values are standard deviations, standard errors, or RMSE across the six runs.
[General] The manuscript contains many typographical and grammatical errors (e.g., 'covision', 'outer reference', 'the external translation of the sensor-IMU along the axis of rotation is insignificant'). A thorough language revision is needed before publication.

Circularity Check

1 steps flagged · score 2.0 of 10

Validation metrics are partly self-consistency checks, but the central calibration derivation is not circular.

fitted input called prediction [Section V.D (LiDAR-IMU Extrinsic Parameters Calibration Experiment), Table V]
"By measuring the accuracy of the deduced LiDAR point cloud map, the accuracy of the extrinsic parameters can be assessed. This paper adopts the Mean Map Entropy (MME) [37] as the map quality indicator."

The MME is computed on the LiDAR point-cloud map produced using the calibrated LiDAR-IMU extrinsics. That same point-to-plane consistency is exactly the cost minimized by the LiDAR plane BA residuals (Eq. 11) during calibration, including the joint voxel-map optimization in Sec. IV-B. Reporting low MME as evidence of extrinsic accuracy therefore partly restates the convergence of the optimizer on the training data rather than an independent measurement of the extrinsics; a biased extrinsic that the optimizer absorbed into the fixed map can still yield low map entropy. This is a self-consistency metric, not a prediction. The same pattern appears in the reprojection errors (Table III) and epipolar errors (Table VII), which are residuals of Eqs. (13) and (12).

full rationale

The calibration pipeline is a standard nonlinear least-squares estimation (Eq. 2) with residuals derived from independent sensor measurements; the central parameters (LiDAR-IMU, camera-IMU, LiDAR-camera, and intrinsic parameters) are estimated by minimizing physically motivated residuals, not by construction equal to their inputs. There is no load-bearing self-citation: references to prior LiDAR-IMU calibration work are not by the present authors and are used as building blocks, not to forbid alternatives. The explicit fixing of the LiDAR map in Sec. IV.D is a modeling assumption that can bias results if the map has drift, but it is not circular. The only mild circularity is in the validation: several reported metrics (map entropy, reprojection error, epipolar error) are computed from the same residuals or the same map used in optimization, so they partly measure self-consistency rather than external accuracy. However, the paper also provides an external target-based check of camera distortion (calibration harp, Sec. V.B) and compares tables against structural design values, so the central derivation is not a fit renamed as prediction.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced. The ledger mainly contains hand-chosen algorithmic hyperparameters and domain assumptions about the environment, rigidity, and the final LiDAR map being treated as optimal. The z-axis omission is an explicit ad-hoc simplification that constrains the generality of the claimed 6-DOF calibration.

free parameters (3)

B-spline order k = 3
Third-order B-spline is chosen in Section III-B without ablation or justification of the order.
Voxel map preset and minimum sizes = not specified
Section III-C describes adaptive voxelization but gives no numeric values for the initial or minimum voxel size.
Planarity threshold for voxel plane fitting = not specified
Section III-C determines a voxel is planar if its points 'can be fitted into a plane', but the fitting threshold is unstated and no sensitivity analysis is provided.

assumptions (5)

standard math Cumulative B-spline interpolation on Lie groups (Eq. 6) is valid and differentiable.
The paper relies on Sommer et al. [27] for the matrix-form cumulative B-spline on SO(3) without re-deriving or verifying it.
domain assumption All sensors are rigidly connected and observe the same 6-DOF motion.
Section III assumes LiDAR, camera and IMU are rigidly connected; this is standard for calibration but not validated in the experiments.
domain assumption The environment is static, texture-rich, and structured, and LiDAR intensity as gray values is sufficiently correlated with camera grayscale for SuperPoint/SuperGlue matching.
The abstract states the pipeline functions in texture-rich structured environments; Section IV.C relies on this for cross-modal matching, but failure modes are not analyzed.
domain assumption The LiDAR point cloud map after single-LiDAR BA is globally consistent enough to be treated as fixed ground truth in the final joint optimization.
Section IV.D states the LiDAR map is considered optimal and voxel planes are not re-optimized; this assumption is the weakest load-bearing premise.
ad hoc to paper The z-axis translation between sensor and IMU is unobservable in the test motion and is therefore not estimated.
Section V.A explains that the figure-eight motion makes translation along the rotation axis insignificant, so the authors omit it from estimation; this limits the 6-DOF claim.

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Cite this review

Pith. "Pith review of Targetless Intrinsics and Extrinsic Calibration of Multiple LiDARs and Cameras with IMU using Continuous-Time Estimation." pith.science (2026). https://pith.science/paper/LWPAFMJO

@misc{pith2026250102821,
  author       = {Pith},
  title        = {Pith review of: Targetless Intrinsics and Extrinsic Calibration of Multiple LiDARs and Cameras with IMU using Continuous-Time Estimation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LWPAFMJO}},
  note         = {Machine review of arXiv:2501.02821}
}

read the original abstract

Accurate spatiotemporal calibration is a prerequisite for multisensor fusion. However, sensors are typically asynchronous, and there is no overlap between the fields of view of cameras and LiDARs, posing challenges for intrinsic and extrinsic parameter calibration. To address this, we propose a calibration pipeline based on continuous-time and bundle adjustment (BA) capable of simultaneous intrinsic and extrinsic calibration (6 DOF transformation and time offset). We do not require overlapping fields of view or any calibration board. Firstly, we establish data associations between cameras using Structure from Motion (SFM) and perform self-calibration of camera intrinsics. Then, we establish data associations between LiDARs through adaptive voxel map construction, optimizing for extrinsic calibration within the map. Finally, by matching features between the intensity projection of LiDAR maps and camera images, we conduct joint optimization for intrinsic and extrinsic parameters. This pipeline functions in texture-rich structured environments, allowing simultaneous calibration of any number of cameras and LiDARs without the need for intricate sensor synchronization triggers. Experimental results demonstrate our method's ability to fulfill co-visibility and motion constraints between sensors without accumulating errors.

Figures

Figures reproduced from arXiv: 2501.02821 by the authors.

**Figure 2.** System Pipeline. Each block is explained in the corresponding subsection of IV with explicit reference back to this diagram. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** LiDAR BA calibration. a) Before calibration, initialization errors [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 5.** Figure 5: a) Original image. b) Projection and rendering of the LiDAR map [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: The self-driving vehicle is fitted with 4 solid-state LiDARs, 10 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 9.** Figure 9: Based on the calibrated extrinsic parameters between the LiDAR and [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]

**Figure 8.** Figure 8: Based on the camera imaging model, project the LiDAR point cloud [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 11.** Figure 11: Merging the images of FOV190 from a Bird’s Eye View (BEV) [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]

**Figure 10.** Figure 10: Merging LiDAR single-frame point clouds using the extrinsic [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Superpoint: Self- supervised interest point detection and description,

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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Lio-sam: Tightly-coupled lidar inertial odometry via smoothing and mapping,

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