eKalibr-Inertial: Event Visual–Inertial Calibration

• eKalibr-Inertial is a continuous-time spatiotemporal calibrator for event-based visual–inertial systems that uses uniform cubic B-splines to represent motion trajectories.
• It leverages an event-only front end to extract asymmetric circle grid patterns for robust estimation of camera–IMU extrinsics, temporal offset, and inertial parameters.
• Its joint batch optimization refines calibration parameters—with metrics showing minimal standard deviations—ensuring high repeatability and precision under dynamic conditions.

Searching arXiv for the specified topic and closely related calibration literature. eKalibr-Inertial is a continuous-time spatiotemporal calibrator for event-based visual–inertial systems that estimates camera–IMU extrinsics, temporal offset, IMU intrinsics, gravity, and continuous-time trajectory states from event streams and inertial measurements. In the formulation introduced for event-based visual–inertial calibration, the sensor pair consists of one event camera and one IMU rigidly mounted together, calibration is performed with an asymmetric circle grid board, and all time-varying states are represented by uniform cubic B-splines so that orientation and position can be queried at arbitrary timestamps and temporal calibration is intrinsically supported by the trajectory model (Chen et al., 7 Sep 2025).

1. Problem setting and calibration target

The method addresses the spatiotemporal calibration problem for event-based visual–inertial fusion. The stated goal is to “accurately estimate spatiotemporal parameters—camera–IMU extrinsic transform and temporal offset—so that event streams and inertial measurements can be fused optimally for high-dynamic, high-DHR motion estimation.” In this setting, fusion quality depends on precise extrinsic alignment $T_{CI} \in SE(3)$ and temporal alignment $\Delta t$ between the camera clock and IMU clock. The world frame is defined by the circle grid board, the camera is frame $c$, and the IMU is frame $b$ (Chen et al., 7 Sep 2025).

A central design choice is that the system is event-only at the visual front end. It extracts circle grid patterns directly from raw events using normal-flow clustering and spatiotemporal ellipse fitting, and therefore does not require LED boards or event-to-image reconstruction. The circle grid is asymmetric in order to remove $180^\circ$ rotational ambiguity. This places eKalibr-Inertial within the line of work initiated by eKalibr and eKalibr-Stereo, but specialized to event-based visual–inertial calibration rather than event-only intrinsic calibration or event-based stereo spatiotemporal calibration.

The scope of estimation is broader than camera–IMU extrinsics alone. The method jointly refines B-spline control points for rotation and position, the camera–IMU transform, the temporal offset, gravity, and IMU intrinsics. At the same time, camera intrinsics are not estimated in this stage: they are fixed and assumed to have been pre-calibrated via eKalibr. A common misconception is therefore to treat eKalibr-Inertial as a monolithic event-camera calibration pipeline; in the formulation reported here, it is specifically a spatiotemporal calibrator whose visual model uses already calibrated camera intrinsics.

2. Continuous-time state representation and sensor models

The continuous-time representation uses uniform cubic B-splines of order $k=4$ for both position and rotation. For position, with control points $X_{pos}=\{p_i \in \mathbb{R}^3,\tau_i\}$ and uniform spacing $\Delta\tau_{pos}$,

$$p(\tau)=p_i+\sum_{j=1}^{k-1}\lambda_j(u)\bigl(p_{i+j}-p_{i+j-1}\bigr), \qquad u=\frac{\tau-\tau_i}{\Delta\tau_{pos}}.$$

For rotation, with control points $X_{rot}=\{R_i\in SO(3),\tau_i\}$ and uniform spacing $\Delta t$0,

$\Delta t$1

Analytic derivatives provide $\Delta t$2, $\Delta t$3, and

$\Delta t$4

The event camera follows a pinhole projection model with radial–tangential distortion. For a 3D point $\Delta t$5 in camera coordinates, normalized coordinates are $\Delta t$6 and $\Delta t$7, with

$\Delta t$8

$\Delta t$9

$c$0

Using

$c$1

the projection is

$c$2

The circle grid model assumes known 3D circle centers $c$3 and board geometry; their 2D projections are ellipses.

The IMU model includes scale and nonorthogonality through upper-triangular mapping matrices and constant biases over the short calibration window: $c$4 The continuous-time kinematics entering residual construction are

$c$5

Temporal alignment is modeled as

$c$6

The continuous-time formulation is not merely a smoothing convenience. It supports asynchronous event and IMU timestamps natively, and it makes temporal calibration part of the state-estimation problem rather than an external synchronization step.

3. Event-based front end and staged initialization

The event-based front end is inherited from eKalibr and eKalibr-Stereo. It begins with normal flow estimation on the surface of active events, followed by polarity-consistent clustering of events. Spatiotemporal ellipse fitting on matched cluster pairs yields ellipse parameters and centers, and the centers are synchronized and organized into ordered grid patterns

$c$7

To maintain correspondence continuity when the board is only partially observed, eKalibr-Inertial uses the incomplete grid tracking module from eKalibr-Stereo: for each circle tracked in three consecutive SAEs, a three-point Lagrange polynomial predicts its next location, and nearest-neighbor association to newly extracted ellipse centers forms the next incomplete patterns (Chen et al., 7 Sep 2025).

Initialization is a defining component of the method. The paper describes a “rigorous and efficient three-stage initialization tailored to event data,” but the sequence as written comprises four concrete steps. First, the rotation B-spline is initialized from gyro data by solving

$c$8

starting from ideal intrinsics $c$9 and $b$0.

Second, visual–inertial alignment recovers the camera–IMU rotation and temporal offset by rotation-only hand–eye alignment. PnP on each detected grid pattern gives camera rotations $b$1 and translations $b$2, and the method estimates $b$3 and $b$4 by minimizing residuals built from relative rotations. If $b$5 is large, specifically greater than $b$6 ms, a coarse estimate is obtained by cross-correlation of angular-velocity norms and then refined in hand–eye alignment.

Third, translation and gravity are initialized through accelerometer integration. The method uses

$b$7

together with first- and second-order integration quantities

$b$8

and kinematic relations for $b$9 and $180^\circ$0 to recover $180^\circ$1 and $180^\circ$2.

Fourth, the position spline is initialized from PnP camera positions by fitting IMU world positions to the camera trajectory: $180^\circ$3

This initialization sequence is important for interpretation. The method does not begin with a single global nonlinear solve from arbitrary starting values; rather, it constructs a sequence of states in which rotation, temporal offset, translation, gravity, and position are brought into a jointly consistent regime before batch refinement.

4. Joint batch optimization and estimated quantities

After initialization, eKalibr-Inertial performs a continuous-time batch optimization with visual, gyroscope, and accelerometer residuals. The visual reprojection residual for a correspondence $180^\circ$4 is

$180^\circ$5

The inertial residuals are

$180^\circ$6

The optimization objective is

$180^\circ$7

where the information matrices are derived from sensor noise and $180^\circ$8 is the Huber loss. The nonlinear least-squares solver is Ceres, using Levenberg–Marquardt or Gauss–Newton, and the local support of B-splines yields sparse Jacobians (Chen et al., 7 Sep 2025).

The parameter partition is explicit:

Category	Contents
Optimized	$180^\circ$9, $k=4$0, $k=4$1, $k=4$2, $k=4$3, $k=4$4, $k=4$5
Fixed	$k=4$6, board geometry, pattern indexing

The gravity vector is modeled as a two-DoF direction with constant norm. IMU intrinsics include accelerometer and gyroscope mapping matrices and biases. This parameterization shows that the method is both extrinsic and intrinsic on the inertial side, but not intrinsic on the camera side.

A practical misunderstanding is to equate the batch stage with a purely visual refinement. In fact, the cost couples reprojection, gyroscope, and accelerometer terms against the same continuous-time trajectory, and the optimized output includes trajectory splines in addition to calibration parameters. The resulting products are $k=4$7, $k=4$8, $k=4$9, $X_{pos}=\{p_i \in \mathbb{R}^3,\tau_i\}$0, and the trajectory splines $X_{pos}=\{p_i \in \mathbb{R}^3,\tau_i\}$1 and $X_{pos}=\{p_i \in \mathbb{R}^3,\tau_i\}$2.

5. Experimental results and operating characteristics

The reported experimental platform is a stereo rig with two hardware-synchronized DAVIS346 event cameras of resolution $X_{pos}=\{p_i \in \mathbb{R}^3,\tau_i\}$3 and their built-in IMUs. Three asymmetric circle grids were used: $X_{pos}=\{p_i \in \mathbb{R}^3,\tau_i\}$4, $X_{pos}=\{p_i \in \mathbb{R}^3,\tau_i\}$5, and $X_{pos}=\{p_i \in \mathbb{R}^3,\tau_i\}$6, with radius rate $X_{pos}=\{p_i \in \mathbb{R}^3,\tau_i\}$7 and spacing $X_{pos}=\{p_i \in \mathbb{R}^3,\tau_i\}$8 mm. For each board, the protocol collected $X_{pos}=\{p_i \in \mathbb{R}^3,\tau_i\}$9 sequences of approximately $\Delta\tau_{pos}$0 s with motion excitation (Chen et al., 7 Sep 2025).

Quantitative calibration results are reported with respect to the left IMU frame. For the left camera, the estimated extrinsics were roll $\Delta\tau_{pos}$1, pitch $\Delta\tau_{pos}$2, yaw $\Delta\tau_{pos}$3, translation $\Delta\tau_{pos}$4 cm, and temporal offset $\Delta\tau_{pos}$5 ms, with reference approximately $\Delta\tau_{pos}$6 ms. For the right camera, the reported values were roll $\Delta\tau_{pos}$7, pitch $\Delta\tau_{pos}$8, yaw $\Delta\tau_{pos}$9, translation $$p(\tau)=p_i+\sum_{j=1}^{k-1}\lambda_j(u)\bigl(p_{i+j}-p_{i+j-1}\bigr), \qquad u=\frac{\tau-\tau_i}{\Delta\tau_{pos}}.$$0 cm, and $$p(\tau)=p_i+\sum_{j=1}^{k-1}\lambda_j(u)\bigl(p_{i+j}-p_{i+j-1}\bigr), \qquad u=\frac{\tau-\tau_i}{\Delta\tau_{pos}}.$$1 ms. For the right IMU with respect to the left IMU, the estimated extrinsics were roll $$p(\tau)=p_i+\sum_{j=1}^{k-1}\lambda_j(u)\bigl(p_{i+j}-p_{i+j-1}\bigr), \qquad u=\frac{\tau-\tau_i}{\Delta\tau_{pos}}.$$2, pitch $$p(\tau)=p_i+\sum_{j=1}^{k-1}\lambda_j(u)\bigl(p_{i+j}-p_{i+j-1}\bigr), \qquad u=\frac{\tau-\tau_i}{\Delta\tau_{pos}}.$$3, yaw $$p(\tau)=p_i+\sum_{j=1}^{k-1}\lambda_j(u)\bigl(p_{i+j}-p_{i+j-1}\bigr), \qquad u=\frac{\tau-\tau_i}{\Delta\tau_{pos}}.$$4, translation $$p(\tau)=p_i+\sum_{j=1}^{k-1}\lambda_j(u)\bigl(p_{i+j}-p_{i+j-1}\bigr), \qquad u=\frac{\tau-\tau_i}{\Delta\tau_{pos}}.$$5 cm, and $$p(\tau)=p_i+\sum_{j=1}^{k-1}\lambda_j(u)\bigl(p_{i+j}-p_{i+j-1}\bigr), \qquad u=\frac{\tau-\tau_i}{\Delta\tau_{pos}}.$$6 ms.

The interpretation given in the source is that small standard deviations indicate high repeatability, temporal offsets in the $$p(\tau)=p_i+\sum_{j=1}^{k-1}\lambda_j(u)\bigl(p_{i+j}-p_{i+j-1}\bigr), \qquad u=\frac{\tau-\tau_i}{\Delta\tau_{pos}}.$$7–$$p(\tau)=p_i+\sum_{j=1}^{k-1}\lambda_j(u)\bigl(p_{i+j}-p_{i+j-1}\bigr), \qquad u=\frac{\tau-\tau_i}{\Delta\tau_{pos}}.$$8 ms range are consistently recovered, and overall accuracy is comparable to frame-based calibrators. This suggests that the event-only front end is sufficiently stable for precision spatiotemporal calibration under real-world motion.

The practical operating procedure is also explicit. Camera intrinsics should be pre-calibrated via eKalibr. Board specifications, camera intrinsics, IMU noise levels, and spline knot spacing $$p(\tau)=p_i+\sum_{j=1}^{k-1}\lambda_j(u)\bigl(p_{i+j}-p_{i+j-1}\bigr), \qquad u=\frac{\tau-\tau_i}{\Delta\tau_{pos}}.$$9 must be configured. Data collection should use $X_{rot}=\{R_i\in SO(3),\tau_i\}$0–$X_{rot}=\{R_i\in SO(3),\tau_i\}$1 s sequences with sustained motion excitation, keeping the board in view and covering diverse viewpoints; purely planar motion should be avoided and all axes should be excited. Best practices further recommend asymmetric grids, sufficient event rates, moderate continuous motion, and avoiding stationary phases typical of LED-based methods.

The method assumes a rigid mount between camera and IMU, a constant temporal offset over the short calibration window, constant IMU biases over that window, fixed gravity magnitude with optimized direction, accurate board geometry, and adequate pattern visibility. Reported limitations include sensitivity to very low event rates or poor contrast, reduced observability under insufficient motion excitation, degradation if camera intrinsics are poor, and the need for coarse temporal initialization when $X_{rot}=\{R_i\in SO(3),\tau_i\}$2 is large.

6. Relation to adjacent calibration and inertial-alignment literature

Within the eKalibr lineage, eKalibr-Inertial extends eKalibr and eKalibr-Stereo by moving from event-only intrinsic calibration and event-based stereo spatiotemporal calibration to event-based visual–inertial spatiotemporal calibration. In relation to frame-based Kalibr, the distinction stated in the source is direct: frame-based Kalibr relies on intensity images and frame timestamps, whereas eKalibr-Inertial extracts circle grid patterns directly from raw events and performs continuous-time optimization without LED boards or event-to-image reconstruction (Chen et al., 7 Sep 2025).

Relative to iKalibr, the difference is one of sensing scope and target usage. iKalibr is a unified targetless spatiotemporal calibration framework for resilient integrated inertial systems supporting IMU, radar, LiDAR, and camera, with one-shot multi-sensor calibration and continuous-time inertial trajectories (Chen et al., 2024). By contrast, eKalibr-Inertial uses a circle grid calibration target and is tailored to event-based visual–inertial systems. A plausible implication is that the two systems occupy complementary positions: eKalibr-Inertial emphasizes event-only target-based visual front ends, whereas iKalibr emphasizes targetless multi-sensor generality.

On the inertial side, eKalibr-Inertial estimates IMU intrinsics and gravity in batch, but related work has emphasized online or extended-window refinement under weak excitation. “Joint On-Manifold Gravity and Accelerometer Intrinsics Estimation for Inertially Aligned Mapping” proposes a fixed-lag factor-graph estimator for accelerometer intrinsics and gravity direction, with velocity-agnostic residuals and on-manifold optimization on $X_{rot}=\{R_i\in SO(3),\tau_i\}$3 (Nemiroff et al., 2023). The source explicitly notes that such a method complements Kalibr and eKalibr-Inertial by providing online refinement of accelerometer intrinsics and gravity alignment during normal operation. This suggests an architectural division between offline spatiotemporal calibration sessions and runtime inertial alignment maintenance.

The multi-IMU literature broadens this comparison further. “Online Multi-IMU Calibration Using Visual-Inertial Odometry” presents a centralized EKF-based framework with online intrinsic and extrinsic calibration for unsynchronized IMUs using MSCKF camera constraints (Hartzer et al., 2023). That line of work differs from eKalibr-Inertial in estimation architecture—filtering rather than continuous-time batch optimization—but it shares the goals of extrinsic estimation, intrinsic bias estimation, and robustness to asynchronous inertial sensing. A plausible implication is that eKalibr-Inertial and online multi-IMU filters can be seen as offline and online endpoints of the same broader calibration problem.

7. Nomenclature, misconceptions, and term ambiguity

The designation “eKalibr-Inertial” is not unique across arXiv. In a distinct 2026 paper on inverse problems, the same label is used for “a second-order inertial interacting particle formulation of Ensemble Kalman Inversion,” described as a heavy-ball reformulation of continuous-time EKI with damping, attraction, and short-range repulsion (Herty et al., 4 Jun 2026). That usage belongs to optimization and inverse problems, not event-based visual–inertial calibration. In the context of sensor calibration, however, eKalibr-Inertial denotes the continuous-time spatiotemporal calibrator for event-based visual–inertial systems described above.

Several misconceptions can be resolved directly from the formulation. First, the method is not based on event-to-image reconstruction; the front end extracts circle grids directly from raw events. Second, it is not an LED-board method and does not require stationary sensors; the reported robustness explicitly supports high-speed motions necessary for visual–inertial calibration. Third, it does not optimize camera intrinsics in the presented formulation; those are fixed and pre-calibrated via eKalibr. Fourth, its temporal calibration model is a constant offset $X_{rot}=\{R_i\in SO(3),\tau_i\}$4 over the short calibration window, not a general nonstationary clock-drift model. Fifth, the method supports one-shot multi-camera multi-IMU calibration, but that support remains within the target-based, circle-grid-driven event-calibration paradigm (Chen et al., 7 Sep 2025).

Taken together, these properties define eKalibr-Inertial as a continuous-time, target-based, event-only front-end calibrator for spatiotemporal event-camera–IMU alignment. Its technical identity lies in the combination of raw-event circle-grid extraction, staged visual–inertial initialization, spline-based trajectory modeling, and joint batch refinement of extrinsics, timing, gravity, and IMU intrinsics.