Fusion with Unknown Correlations and Covariance Intersection

Visual: two estimates with unknown cross-correlation flowing into covariance intersection, information weights, conservative covariance ellipse, and double-counting warning.

Multi-sensor fusion is only as good as its correlation accounting. If two estimates share a prior, process noise, map, feature track, or upstream sensor, then fusing them as independent double-counts information and makes covariance too small.

Covariance intersection (CI) is the conservative tool for the case where each input estimate has a mean and covariance, but the cross-correlation is unknown.

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Related Docs

• Bayesian Filtering and Error-State Kalman Filters
• Information Filters and Smoothers
• RTK-GPS, IMU, and Multi-Sensor Localization
• Gaussian Noise, Covariance, and Information
• Mahalanobis Distance, Chi-Square Gates, NIS, and NEES

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Where Unknown Correlation Comes From

Situation	Shared information
Track-to-track fusion	common process model, previous fused track, exchanged messages
GNSS/INS plus fused localization	same IMU or GNSS appears in both products
LiDAR map localization plus SLAM odometry	same map or scan-matching constraints
Multi-robot map fusion	prior map segments and loop-closure information
Perception output plus raw sensor fusion	detector output already used the same image, point cloud, or radar
Networked estimation	messages recirculate through graph cycles

If the lineage of each estimate is not tracked, independence is usually an unsafe assumption.

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Naive Independent Fusion

For two estimates (x_1, P_1) and (x_2, P_2), independent Gaussian fusion in information form is:

Y = P_1^-1 + P_2^-1
y = P_1^-1 x_1 + P_2^-1 x_2
P = Y^-1
x = P y

This is optimal only when the estimation errors are independent. If the errors are positively correlated, the result can be overconfident. In safety monitors, that can be worse than a visibly noisy estimate because the system believes it has more integrity margin than it really has.

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Covariance Intersection

CI replaces the independence sum with a convex information-weighted blend:

P_CI^-1 = omega P_1^-1 + (1 - omega) P_2^-1

x_CI = P_CI (
  omega P_1^-1 x_1 + (1 - omega) P_2^-1 x_2
)

0 <= omega <= 1

The weight is chosen to minimize a scalar size of the fused covariance, such as:

minimize log(det(P_CI))

or:

minimize trace(P_CI)

When the inputs are conservative and unbiased, CI gives a conservative fused estimate for any unknown cross-correlation. It is often less accurate than fusion with known correlations, but it avoids false precision.

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Weight Selection

Objective	Meaning	Common use
log det(P)	minimize uncertainty volume	multi-dimensional localization
trace(P)	minimize total marginal variance	tracking and simple implementation
largest eigenvalue	minimize worst-axis variance	safety envelopes and protection bounds
fixed omega	deterministic fallback	low-compute embedded systems

The chosen objective should match the downstream decision. A parking or docking planner may care about lateral covariance more than volume, while a map-fusion backend may care about determinant or trace.

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When CI Is the Right Tool

Use CI when:

• cross-covariance is unknown or too expensive to maintain,
• communication topology can create double-counting,
• an upstream module provides only mean and covariance,
• a safety monitor needs conservative bounds,
• two estimates cover the same state but have overlapping data lineage.

Avoid CI when:

• the exact joint covariance is available and trustworthy,
• the estimates are truly independent,
• one estimate is biased or faulted and should be rejected before fusion,
• the state vectors do not align and require careful common-state projection.

CI handles correlation uncertainty; it does not solve bias, outlier, frame, or timestamp errors.

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Partial Knowledge and Distributed Fusion

CI is the baseline for completely unknown correlation. If more structure is known, less conservative methods may be possible:

Method	Idea	Risk
Track lineage	avoid fusing duplicate raw evidence	requires metadata discipline
Split covariance	separate independent and common components	wrong split breaks consistency
Covariance union	cover disagreement or possible bias between estimates	can be very conservative
Inverse or ellipsoidal CI variants	exploit partial correlation structure	assumptions must be explicit
Factor-graph fusion	share raw factors instead of fused estimates	higher bandwidth and compute

The best fix is often architectural: move the fusion boundary upstream so that raw independent measurements are fused once in a common graph.

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Diagnostics

• Record source lineage for every fused estimate.
• Flag any fusion between a raw sensor product and a downstream estimate that already consumed that sensor.
• Compare covariance against empirical error with NEES when truth is available.
• Run replay with CI versus independent fusion and inspect covariance collapse.
• Check that CI output covariance is not smaller than both inputs in every arbitrary direction without a defensible reason.
• Monitor disagreement between input means. Large disagreement is an outlier or bias problem, not only a correlation problem.

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Failure Modes

Failure mode	Symptom	Mitigation
Double-counted GNSS	fused covariance shrinks during multipath	track lineage or use CI
Recirculated tracks	distributed nodes become overconfident after repeated exchange	CI, channel filters, or rumor control
Biased input	CI produces conservative covariance around the wrong mean	fault detection before fusion
Wrong state alignment	covariance looks reasonable in the wrong frame	transform mean and covariance with adjoints
Sequential CI order effects	output changes with fusion order	batch objective or deterministic ordering
Over-conservative CI	planner degrades unnecessarily	expose partial correlation or fuse raw factors

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Minimal Mental Model

If you do not know whether two estimates are independent, do not add their information matrices as if they are. CI fuses by choosing a conservative point inside the uncertainty geometry rather than pretending missing correlation data does not exist.

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Sources

• Julier and Uhlmann, "A Non-divergent Estimation Algorithm in the Presence of Unknown Correlations": https://doi.org/10.1109/ACC.1997.609105
• Julier and Uhlmann, "Using Covariance Intersection for SLAM": https://doi.org/10.1016/j.robot.2006.06.011
• Reinhardt, Noack, Arambel, and Hanebeck, "Minimum Covariance Bounds for the Fusion under Unknown Correlations": https://doi.org/10.1109/LSP.2015.2414972
• Noack et al., "A Quarter Century of Covariance Intersection: Correlations Still Unknown?": https://www.diva-portal.org/smash/get/diva2:1853630/FULLTEXT01.pdf
• Bakr and Lee, "Distributed Multisensor Data Fusion under Unknown Correlation and Data Inconsistency": https://doi.org/10.3390/s17102478
• Clark, Julier, Mahler, and Ristic, "Robust Multi-Object Sensor Fusion with Unknown Correlations": https://ieeexplore.ieee.org/document/5595888