Camera Calibration Using Inaccurate and Asynchronous Discrete GPS Trajectory from Drones

R. Yang; Y. Bar-Shalom; H. A. J. Huang

eess.SYAug 27, 2026

Camera Calibration Using Inaccurate and Asynchronous Discrete GPS Trajectory from Drones

Authors: R. Yang, Y. Bar-Shalom, H. A. J. Huang

Organizations: DSO National Laboratories, 12 Science Park Drive, Singapore 118225 · Department of ECE, University of Connecticut, Storrs, CT 06269, USA

Abstract

This paper considers a stationary camera calibration problem, which estimates the camera orientation angles yaw, pitch and roll, using a drone trajectory recorded by a GPS. There are three challenges in using a GPS trajectory as ground truth for camera calibration. One, the altitude of GPS data is inaccurate with an unknown bias. Two, the GPS receiver and camera are not time synchronized, and there is an unknown time offset between the two systems. Three, the GPS trajectory is time-discrete and accurate interpolation is needed. This is actually an estimation problem since velocity is also needed. To address the first two challenges, we formulate the problem as a parameter estimation problem to estimate a vector consisting of the GPS altitude bias and time offset in addition to the camera yaw, pitch and roll biases. We then develop a special maximum likelihood estimator using the Iterated Least Squares algorithm which can work with a non-synchronized time-discrete GPS trajectory for the third challenge. Since the camera measurement errors are usually small, this requires a high calibration accuracy so that the residual bias error following the calibration should not be significant compared to the measurement error standard deviation. The calibration accuracy depends highly on the drone trajectory. This paper also recommends an appropriate drone trajectory which can yield a good calibration accuracy, namely, 14% of the measurement error standard deviation. Simulation tests are conducted to demonstrate the algorithm performance. The estimation results meet the Cramer-Rao Lower Bound (CRLB) since the Normalized Estimation Error Squared w.r.t.\ the CRLB is statistically acceptable.

Figures & tables

Fig. 1: Camera and image coordinate systems.

${\mathbf{x}}$	$[x\ y\ z]^{\prime}$ , a point in the common (ENU) coordinate system.
${\mathbf{x}}^{\mathrm{C}}$	$[x^{\mathrm{C}}\ y^{\mathrm{C}}\ z^{\mathrm{C}}]^{\prime}$ , a point in the camera coordinate system.
${\mathbf{x}}^{\mathrm{I}}$	$[x^{\mathrm{I}}\ y^{\mathrm{I}}]^{\prime}$ , a point in the image coordinate system.
${\mathbf{x}}^{\mathrm{s}}$	$[x^{\mathrm{s}}\ y^{\mathrm{s}}\ z^{\mathrm{s}}]^{\prime}$ sensor (camera) position.
$\alpha$	camera pointing azimuth or yaw (clockwise from N).
$\epsilon$	camera pointing elevation or pitch (up from horizontal).
$\rho$	camera roll (ideally zero), clockwise around the center of

TABLE I: Notations

Fig. 4: The LS fitting algorithms. (a) One-step LS fitting approach developed in [ 26 ] . (b) Two-step LS fitting approach used in the present paper.

Fig. 5: Test scenario 1. Target moves in constant velocity in a vertical rectangle (1,2,3,4) twice. The higher and lower horizontal edges are at altitudes 284m and 84m, respectively, and near and far vertical edges are at ranges 200m and 500m, respectively. The target speed is 12.5m/s. (a) Top view in the 3D common coordinates. (b) Trajectory in image coordinates.

Fig. 6: Test scenario 2. Target moves in constant velocity, makes a 180 o turn, and flies back in constant velocity. The target speed is 12.5m/s. (a) Top view in the 3D common coordinates. (b) Trajectory in image coordinates.

Scenario	Parameter	RMSE	$\sigma_{\rm CRLB}$
	$\alpha$ —yaw (mdeg)	0.23	0.21
	$\epsilon$ —pitch (mdeg)	0.87	0.82
1	$\rho$ —roll (mdeg)	2.90	2.81
	$\hbar$ — (mm)	4.69	4.35
	$\tau$ (ms)	0.27	0.22
	$\alpha$ —yaw (mdeg)	1.34	1.05

TABLE II: CRLB and RMSE from 100 runs

Fig. 7: Measured and GPS converted trajectories of scenario 1, where the actual and nominal yaw, roll and time difference are set to the same values as $\alpha=32^{\mathrm{o}}$ , $\rho=2.3^{\mathrm{o}}$ and $\tau=1.35$ s, respectively. The actual and nominal pitch are $\epsilon=4.1^{\mathrm{o}}$ and $2^{\mathrm{o}}$ , respectively. The actual and nominal GPS altitude bias are 10m and 0m, respectively.

Fig. 8: Measured and GPS converted trajectories of scenario 2, where the actual and nominal yaw, roll and time difference are set to the same values as $\alpha=32^{\mathrm{o}}$ , $\rho=2.3^{\mathrm{o}}$ and $\tau=1.35$ s, respectively. The actual and nominal pitch are $\epsilon=4.1^{\mathrm{o}}$ and $2^{\mathrm{o}}$ , respectively. The actual and nominal GPS altitude bias are 10m and 0m, respectively.

Fig. 9: NEES of 100 runs of the scenario 1.

Fig. 10: NEES of 100 runs of the scenario 2.

Fig. 11: Biased errors on all 5 $\times$ 5 grids. (a) The biased error caused by calibration error on yaw of 0.23mdeg. (b) The biased error caused by calibration error on pitch of 0.87mdeg. (c) The biased error caused by calibration error on roll of 2.9mdeg.

Fig. 12: Biased errors on all 5 $\times$ 5 grids. (a) Increases the yaw, pitch and roll by 0.23mdeg, 0.87mdeg and 2.9mdeg, respectively. (b) Reduces the yaw, pitch and roll by 0.23mdeg, 0.87mdeg and 2.9mdeg, respectively.

Calibration error			Bias
(mdeg)			(pixel)
$\alpha$	$\epsilon$	$\rho$	Min.	Max.	Mean	Stdv.	RMS
0.23	0	0	0.050	0.050	0.050	0.000	0.050
0	0.87	0	0.187	0.192	0.189	0.001	0.189
0	0	2.90	0.000	0.110	0.064	0.025	0.059
0.23	0.87	2.90	0.134	0.285	0.210	0.033	0.200

TABLE III: Biased error in pixel caused by the calibration error

Camera Calibration

Camera calibration estimates the geometric and optical parameters that connect image measurements to real-world positions and other sensors.

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OrthoTrack: Continuous 6-DoF UAV Trajectory Estimation Anchored in Public Orthophotos

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Continuous 6-DoF pose estimation is essential for autonomous UAV operations. Yet, existing visual odometry and SLAM methods accumulate drift and yield only relative, up-to-scale trajectories. Single-frame geo-localization, in turn, discards temporal continuity and remains too slow for real-time use. We present OrthoTrack, a training-free system that estimates continuous 6-DoF UAV trajectories using only publicly available orthophotos and surface models as a map prior. OrthoTrack matches keyframes against the orthophoto and lifts correspondences to metric 3D via the surface model. It then propagates these map-anchored correspondences to intermediate frames with optical flow, producing absolute, metrically scaled poses at every frame without GPS or post-hoc alignment. We also introduce the MovingDrone Dataset, a large-scale benchmark pairing photorealistic UAV sequences with dense 6-DoF ground truth and co-registered multi-modal geodata including multi-temporal orthophotos. On MovingDrone and real-world benchmarks, OrthoTrack runs in real time on a single GPU. It outperforms all baselines by a large margin, even those receiving oracle scale and alignment. By relying on publicly available geodata, OrthoTrack enables deployment to new regions without site-specific adaptation.

Abstract

Camera Calibration

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