On-Orbit Geometric Calibration

• On-orbit geometric calibration is the process of post-launch estimation and correction of sensor intrinsics and extrinsics to mitigate systematic distortions.
• It employs advanced imaging models, such as extended pinhole and rational models, to accurately map 3D scene points to 2D image coordinates.
• Optimization techniques like least-squares minimization, bundle adjustment, and EKF, combined with robust tie-point extraction, ensure sub-pixel registration and enhanced data integrity.

On-orbit geometric calibration is the rigorous, post-launch estimation and correction of the geometric parameters governing imaging sensor projection from scene coordinates to image coordinates, performed under operational flight conditions. This process directly addresses systematic and time-dependent distortions arising from instrument alignment, optical aberrations, platform attitude, timing, and environmental drift, thereby enabling accurate mapping between measured pixels and physical scene locations. In contemporary orbital remote sensing and astronomical missions, geometric calibration is indispensable for achieving mission-specified geolocation accuracy, multi-band registration, and scientific data integrity.

1. Geometric Sensor and Imaging Models

On-orbit geometric calibration is ultimately founded on parametric imaging models that relate scene points (ground, celestial, or planetary) to pixel locations via sensor intrinsics, extrinsics, and distortion fields. The classical extended pinhole model specifies the mapping from a 3D scene point $\mathbf X =(X,Y,Z)^T$ in camera coordinates to normalized image coordinates $(u,v)$ and then to pixel coordinates $(x,y)$ via

$$K = \begin{pmatrix} f & 0 & x_0 \ 0 & f & y_0 \ 0 & 0 & 1 \end{pmatrix}, \quad s\,(x,y,1)^T = K\,[X,Y,Z]^T, \quad s=Z$$

where $f$ is the focal length, and $(x_0,y_0)$ the principal point (Tulyakov et al., 2017). Off-axis telescopes and complex optical paths manifest non-radial distortions unsuitable for simple radial/tangential models; rational models, e.g., lifting pixel $\mathbf i=(i,j)$ to $\chi_6 = [i^2,ij,j^2,i,j,1]^T$ and applying rational functions $$(x,y,1)^T = (A_1^T\chi_6, A_2^T\chi_6, A_3^T\chi_6)^T$$, are adopted to capture general distortion fields (Tulyakov et al., 2017).

For earth-observation pushbroom imagers, geometric mapping is determined by the collinearity equations:

$$\mathbf X_g = \mathbf X_s + \lambda R \mathbf p_c, \quad \lambda = \frac{h - Z_s}{(R \mathbf p_c)_z}$$

with $\mathbf p_c = ((j-j_0)p_x,\, (i-i_0)p_y,\,-f)^T$, and attitude described by roll, pitch, yaw rotations (Garg et al., 22 Oct 2024).

Star sensors are subject to six-degree-of-freedom image plane displacement: principal point drift $(x_0,y_0,f_0)$, incline displacements $(\alpha,\beta)$, and in-plane rotation $(\psi)$:

$$R = R_z(\psi)R_y(\beta)R_x(\alpha),\qquad p' = R p - t$$

with the projected pixel location $$\begin{pmatrix} u\ v \end{pmatrix} = f \frac{(R_{11}x + R_{12}y + R_{13}f - x_0,\, R_{21}x + R_{22}y + R_{23}f - y_0)^T}{R_{31}x + R_{32}y + R_{33}f - f_0}$$ (Sun et al., 2018).

2. Calibration Data Sources and Tie-Point Generation

On-orbit data sources for geometric calibration reflect mission context. Astronomical imagers exploit celestial point-source catalogs (e.g., 2MASS, Tycho-2) as “calibration targets” with precisely known angular positions, detected via centroiding algorithms and catalog-matching (Astrometry.net, DoG filters) (Tulyakov et al., 2017, Sun et al., 2018). Terrestrial imagers acquire ground control points (GCPs) by matching features to geodetic reference products (Sentinel-3 OLCI, Landsat ETM), employing normalized cross-correlation or dense tie-point extraction using descriptors (ORB/BRIEF) (Baek et al., 5 Dec 2025, Garg et al., 22 Oct 2024, Garg et al., 25 Oct 2024).

Large-scale geostationary imagers (GHRC) utilize frame-to-frame overlap by raster scanning and inter-frame phase correlation to obtain tie points for mosaic error correction. Lunar imagers (PolCam) match high-phase-angle features to global reflectance maps, using tens or hundreds of thousands of matched points and digital elevation models (e.g., SLDEM) (Baek et al., 5 Dec 2025).

3. Optimization Procedures and Bundle Adjustment

Calibration proceeds by minimizing the total reprojection error between observed pixel locations $(x_{ij}^{\mathrm{obs}}, y_{ij}^{\mathrm{obs}})$ and modelled projections $(x_{ij}^{\mathrm{pred}}, y_{ij}^{\mathrm{pred}})$, over all images and tie points:

$$\min_{f,\{R_j\},D} \sum_{j=1}^{N_\mathrm{img} } \sum_{i=1}^{N_j} \| (x_{ij}^{\mathrm{obs}}, y_{ij}^{\mathrm{obs}}) - (x_{ij}^{\mathrm{pred}}, y_{ij}^{\mathrm{pred}}) \|^2$$

with unknowns including intrinsics, extrinsics (per-frame attitudes), and distortion parameters. Typical pipelines split optimization into three stages: rotation refinement (fixing $f$, refine $R_j$), global bundle adjustment (jointly refining $f$ and $R_j$), and distortion fitting (freeze $f$, $R_j$, optimize $A$) (Tulyakov et al., 2017, Baek et al., 5 Dec 2025).

For sensors lacking precise timing telemetry, line-wise observation times are reconstructed via a linear model:

$\mathrm{ET}_N = \mathrm{IST} + \mathrm{SR}\times N$

where imaging start time $\mathrm{IST}$ and scan rate $\mathrm{SR}$ are estimated within the bundle adjustment, using tie points to drive sub-pixel refinement (Baek et al., 5 Dec 2025).

Geostationary imagers (GHRC) perform on-orbit calibration by least-squares optimization of interior mirror orientations (roll/pitch biases), using the normal equations linearized by Taylor expansion (Garg et al., 25 Oct 2024). Star sensors employ an Extended Kalman Filter (EKF) over six degrees of freedom, processing angular residuals between measured and catalog star directions to drive continuous parameter estimation (Sun et al., 2018).

4. Error Sources, Correction Strategies, and Registration

Geometric error sources are attributed to platform-level uncertainties (attitude, ephemeris, thermal drift, control jitter), instrument-level misalignments (sensor-mount biases, scan-mirror distortions, encoder noise), and timing deficiencies (Garg et al., 25 Oct 2024, Baek et al., 5 Dec 2025). Correction strategies include:

• Band-to-band registration: misalignments across multispectral or polarimetric channels are quantified via tie-point offsets and stored as per-band lookup tables, with corrections applied during ground processing (Garg et al., 22 Oct 2024, Baek et al., 5 Dec 2025). Corrections achieve sub-pixel specification (typically ±0.25 pixel).
• Mosaic generation: frame overlap discrepancies are reduced via two-pass backtracking space-resection algorithms, where local frame orientation adjustments are estimated and propagated along image sequences, followed by a single resampling pass (Garg et al., 25 Oct 2024).
• Onboard vs. ground processing: resource-intensive geometric modelling, tie-point matching, and bundle adjustment are performed on ground segments, while only minimal preprocessing (binning, masking) executes in real time (Garg et al., 22 Oct 2024).

5. Validation Metrics and Achieved Performance

Calibration fidelity is quantified using root-mean-square error (RMSE) and percentile circular error (CE90). Cross-validation on independent targets and epochs assesses generalization. Reported results include:

Instrument	Pixel Error (px)	Geolocation (m/km)	Band Reg (px)	Reference
CaSSIS (TGO)	0.47 px (validation)	≈5 µrad	—	(Tulyakov et al., 2017)
GHRC (GSAT-29)	—	≈15 km (post-cal)	<0.25 px (98% CE)	(Garg et al., 25 Oct 2024)
OCM-3 (EOS-06)	<0.25 px (BBR)	0.06–0.15 km (median)	0.017–0.233 px	(Garg et al., 22 Oct 2024)
Star Sensor	—	≈0.23″ (post-EKF)	—	(Sun et al., 2018)
PolCam (Danuri)	0.87–1.48 px (RMS)	37–64 m RMS	<0.5 px centrals	(Baek et al., 5 Dec 2025)

The PolCam calibration with 160,256 tie points yielded orthorectified lunar maps with RMS residuals ≲1 pixel (≲43–64 m), achieving parity with Kaguya MI and LROC WAC geolocation (Baek et al., 5 Dec 2025). The CaSSIS bundle-adjusted model reproduced star locations to 0.47 px versus 3.56 px uncalibrated (Tulyakov et al., 2017). In OCM-3, inter-scene registration is held within ±0.35 px over six months, satisfying change-detection thresholds (Garg et al., 22 Oct 2024). EKF-calibrated star sensors show pointing improvements to ≈0.23″, exploiting CCD tracker capabilities (Sun et al., 2018).

6. Generalization, Best Practices, and Practical Considerations

Geometric calibration procedures generalize to any large-focal-length, off-axis telescope or earth observation imager provided target fields or ground truth reference data are available. Adopted principles include:

• Use of distortion models (rational, bicubic) capable of capturing complex, anisotropic effects (Tulyakov et al., 2017).
• Acquisition of calibration data under varied pointing attitudes and spatial coverage to ensure detector footprint is fully sampled (Tulyakov et al., 2017, Baek et al., 5 Dec 2025).
• Iterative bundle adjustment, robust outlier rejection, and independent validation across epochs and target types (Tulyakov et al., 2017).
• Open-sourcing of calibration software and tie-point datasets to accelerate future mission adoption (Tulyakov et al., 2017).
• For star sensors, regular calibration cadence matched to drift rates, robust innovation residual monitoring, and onboard algorithmic efficiency (EKF with 6×6 matrices) (Sun et al., 2018).

A plausible implication is that the dominant limiting factor for orbital geometric calibration lies as much in the density and quality of reference tie points as in sensor model sophistication. Routine application of these methods yields sub-pixel registration and systematic error mitigation necessary for scientific-grade remote sensing and survey instruments.

7. Coordinate Transformation and Product Generation

Orthorectified data products are generated via rigorous coordinate transformations: pixel-to-scene mapping uses refined sensor and timing models to compute line-of-sight vectors, which are intersected with planetary surfaces (e.g., SLDEM for lunar mapping via SPICE’s sincpt routine) (Baek et al., 5 Dec 2025). Geodetic outputs—latitude, longitude, elevation—are assembled into equidistant cylindrical map tiles and mosaicked, with bilinear interpolation filling between anchor points. Mosaic error correction algorithms ensure seamless multi-frame composition and high-fidelity geoscientific data products (Garg et al., 25 Oct 2024). Validation against legacy reference datasets (Kaguya MI, LROC WAC, Landsat) confirms geometric consistency and mission adequacy (Baek et al., 5 Dec 2025, Garg et al., 22 Oct 2024).