Weighted Covariance Intersection (WCI)

• WCI is a weighted form of covariance intersection that fuses multiple Gaussian estimates using a convex combination of precision matrices to maintain consistency under unknown cross-correlations.
• WCI formulates weight selection as an optimization problem—often via semidefinite programming—to minimize uncertainty measures like the trace or log-determinant of the covariance matrix.
• WCI is widely used in distributed sensor fusion and cooperative localization, though its conservatism has spurred alternative methods such as Bayesian and game-theoretic approaches.

Weighted Covariance Intersection (WCI) is the weighted form of covariance intersection, a fusion method for combining multiple Gaussian estimates when the cross-correlation between them is unknown or only partially known. In the standard formulation, for estimates $(x_i,P_i)$, WCI computes

$$P_{\text{WCI}}^{-1}=\sum_{i=1}^{N}\omega_i P_i^{-1},\qquad x_{\text{WCI}}=P_{\text{WCI}}\left(\sum_{i=1}^{N}\omega_i P_i^{-1}x_i\right),$$

with $\omega_i\ge 0$ and $\sum_i \omega_i=1$. The method is designed so that the fused covariance remains a valid upper bound for all cross-correlations consistent with the available information, which makes WCI a standard conservative rule in distributed estimation and sensor fusion (Pedroso et al., 20 Mar 2026).

1. Classical formulation and consistency

In the CI literature, WCI usually denotes the case in which the fused covariance is explicitly parameterized by weights and those weights are chosen, often heuristically and sometimes optimally, according to a scalar uncertainty criterion. For two full-state estimates, the classical formula is

$$P_{\text{CI}}^{-1}=\omega P_1^{-1}+(1-\omega)P_2^{-1},\qquad x_{\text{CI}}=P_{\text{CI}}\big(\omega P_1^{-1}x_1+(1-\omega)P_2^{-1}x_2\big),$$

with $\omega\in[0,1]$. For $N$ estimates, the weights lie on the simplex $$\Delta^N=\{\omega\in\mathbb{R}^N:1^\top\omega=1,\ \omega_i\ge 0\}$$ (Pedroso et al., 20 Mar 2026).

The central property of CI and WCI is consistency under unknown cross-correlation. In the positive definite, or Loewner, order, if each local estimate is consistent, then the CI-fused estimate remains consistent without requiring any cross-covariance terms. In information form, this is commonly written as

$$\Sigma_{\hat z'}^{-1}=\sum_{i=1}^N c_i \Sigma_i^{-1},\qquad \Sigma_{\hat z'}^{-1}\mathbb{E}[\hat z']=\sum_{i=1}^N c_i \Sigma_i^{-1}\mathbb{E}[\hat z_i],$$

with convex coefficients $c_i\ge 0$, $$P_{\text{WCI}}^{-1}=\sum_{i=1}^{N}\omega_i P_i^{-1},\qquad x_{\text{WCI}}=P_{\text{WCI}}\left(\sum_{i=1}^{N}\omega_i P_i^{-1}x_i\right),$$0. This information-space convex combination is the basic weighted mechanism underlying WCI (Chang et al., 2021).

The weight vector determines which conservative bound within the CI family is selected. Different $$P_{\text{WCI}}^{-1}=\sum_{i=1}^{N}\omega_i P_i^{-1},\qquad x_{\text{WCI}}=P_{\text{WCI}}\left(\sum_{i=1}^{N}\omega_i P_i^{-1}x_i\right),$$1 correspond to different worst-case fused covariances, and the usual design task is to choose $$P_{\text{WCI}}^{-1}=\sum_{i=1}^{N}\omega_i P_i^{-1},\qquad x_{\text{WCI}}=P_{\text{WCI}}\left(\sum_{i=1}^{N}\omega_i P_i^{-1}x_i\right),$$2 to minimize a scalar performance index $$P_{\text{WCI}}^{-1}=\sum_{i=1}^{N}\omega_i P_i^{-1},\qquad x_{\text{WCI}}=P_{\text{WCI}}\left(\sum_{i=1}^{N}\omega_i P_i^{-1}x_i\right),$$3. Typical choices stated in the literature are $$P_{\text{WCI}}^{-1}=\sum_{i=1}^{N}\omega_i P_i^{-1},\qquad x_{\text{WCI}}=P_{\text{WCI}}\left(\sum_{i=1}^{N}\omega_i P_i^{-1}x_i\right),$$4, $$P_{\text{WCI}}^{-1}=\sum_{i=1}^{N}\omega_i P_i^{-1},\qquad x_{\text{WCI}}=P_{\text{WCI}}\left(\sum_{i=1}^{N}\omega_i P_i^{-1}x_i\right),$$5, and $$P_{\text{WCI}}^{-1}=\sum_{i=1}^{N}\omega_i P_i^{-1},\qquad x_{\text{WCI}}=P_{\text{WCI}}\left(\sum_{i=1}^{N}\omega_i P_i^{-1}x_i\right),$$6, under the monotonicity assumption $$P_{\text{WCI}}^{-1}=\sum_{i=1}^{N}\omega_i P_i^{-1},\qquad x_{\text{WCI}}=P_{\text{WCI}}\left(\sum_{i=1}^{N}\omega_i P_i^{-1}x_i\right),$$7 (Pedroso et al., 20 Mar 2026).

2. Weight selection as an optimization problem

A standard interpretation of WCI is that it is optimal only within a specific parametric family of conservative bounds. In that sense, a “family-optimal” solution is optimal within the CI family rather than among all possible consistent fusion rules. Recent work formalizes this view through overlapping covariance intersection (OCI), which embeds standard CI and split covariance intersection (SCI) in a single semidefinite-programming framework (Pedroso et al., 20 Mar 2026).

For standard CI, the family-optimal weight selection problem can be written as the semidefinite program

$$P_{\text{WCI}}^{-1}=\sum_{i=1}^{N}\omega_i P_i^{-1},\qquad x_{\text{WCI}}=P_{\text{WCI}}\left(\sum_{i=1}^{N}\omega_i P_i^{-1}x_i\right),$$8

with fusion gain

$$P_{\text{WCI}}^{-1}=\sum_{i=1}^{N}\omega_i P_i^{-1},\qquad x_{\text{WCI}}=P_{\text{WCI}}\left(\sum_{i=1}^{N}\omega_i P_i^{-1}x_i\right),$$9

When specialized to multiple full-state estimates, this formulation recovers the state-of-the-art family-optimal CI solutions previously reported for standard CI (Pedroso et al., 20 Mar 2026).

This optimization view clarifies two points. First, WCI is not merely a heuristic convex combination; it is a robust design against all admissible cross-correlation structures. Second, once the criterion $\omega_i\ge 0$0 is fixed, weight selection can be posed as a convex SDP, rather than as an ad hoc line search. This has become particularly important in large-scale distributed estimation and cooperative localization, where off-the-shelf SDP solvers can be used for real-time or near real-time implementation (Pedroso et al., 20 Mar 2026).

3. Information-theoretic and structured generalizations

In Gaussian single-object estimation, WCI is also the single-object manifestation of a broader information-theoretic construction. In multiobject random finite set fusion, generalized covariance intersection (GCI) is the multiobject counterpart of CI/WCI and is exactly the weighted geometric mean of local densities. It solves a weighted information gain minimization problem,

$\omega_i\ge 0$1

and in the single-object Gaussian case reduces to the standard CI/WCI formulas above (Gao et al., 2019).

The same literature also identifies a dual construction based on weighted information loss,

$\omega_i\ge 0$2

whose unconstrained solution is the linear opinion pool, or arithmetic pooling. This dual rule is not standard WCI, but it is useful because it shows that WCI belongs to the weighted information-gain, or geometric-pooling, branch of conservative fusion, whereas arithmetic pooling belongs to a different design objective (Gao et al., 2019).

A separate generalization concerns structure in the state itself. Standard WCI is “monolithic” in the sense that it uses one global weight for the entire state. When probabilistic independence structure is available, tighter conservative bounds can be obtained by using multiple, non-monolithic weights. In one formulation, independent sub-states are fused with block-specific weights, and a general robust optimization over the unknown cross-covariance set yields tighter bounds than monolithic CI. On a simple problem, the non-monolithic CI solution and the general optimization scheme converge to the same solution; in large-scale target tracking, the non-monolithic algorithm yields a tighter bound and a more accurate estimate than original monolithic CI (Funk et al., 2023).

4. Sequential, split, and recursive forms

Several important variants of WCI address recursive fusion and implementation efficiency. In split covariance intersection filtering, each local covariance is decomposed into dependent and independent parts,

$\omega_i\ge 0$3

and the effective covariances are defined as

$\omega_i\ge 0$4

with fused covariance

$\omega_i\ge 0$5

The associated $\omega_i\ge 0$6-optimization problem

$\omega_i\ge 0$7

has been proved convex via convexity of $\omega_i\ge 0$8, which is a practically significant result because it guarantees that any local minimum is global and makes one-dimensional optimization straightforward (Li, 2021).

A second line of work concerns sequential CI. Enhanced sequential covariance intersection (ESCI) generalizes both batch CI and one-at-a-time SCI by allowing multiple estimate pairs to be fused at each sequential step. The recursion is

$\omega_i\ge 0$9

and it remains unbiased and consistent (Hu et al., 2021).

ESCI becomes especially notable when paired with an analytic weighting criterion. If the local importance of estimate pair $\sum_i \omega_i=1$0 is represented by a positive scalar $\sum_i \omega_i=1$1, then choosing

$\sum_i \omega_i=1$2

makes the final fusion result independent of the fusion order and of the number of estimates fused in each sequential step. Examples given for $\sum_i \omega_i=1$3 include $\sum_i \omega_i=1$4, $\sum_i \omega_i=1$5, $\sum_i \omega_i=1$6, $\sum_i \omega_i=1$7, and $\sum_i \omega_i=1$8 with diagonal $\sum_i \omega_i=1$9 for component-prioritized fusion (Hu et al., 2021).

5. Applications in distributed localization and multitarget fusion

WCI is especially prominent in cooperative localization. In distributed visual-inertial cooperative localization, CI is used so that each robot estimates only its own state and autocovariance while compensating for unknown correlations between robots. In one implementation, the CI prior covariance is taken as

$$P_{\text{CI}}^{-1}=\omega P_1^{-1}+(1-\omega)P_2^{-1},\qquad x_{\text{CI}}=P_{\text{CI}}\big(\omega P_1^{-1}x_1+(1-\omega)P_2^{-1}x_2\big),$$0

and fixed weights such as $$P_{\text{CI}}^{-1}=\omega P_1^{-1}+(1-\omega)P_2^{-1},\qquad x_{\text{CI}}=P_{\text{CI}}\big(\omega P_1^{-1}x_1+(1-\omega)P_2^{-1}x_2\big),$$1, $$P_{\text{CI}}^{-1}=\omega P_1^{-1}+(1-\omega)P_2^{-1},\qquad x_{\text{CI}}=P_{\text{CI}}\big(\omega P_1^{-1}x_1+(1-\omega)P_2^{-1}x_2\big),$$2, $$P_{\text{CI}}^{-1}=\omega P_1^{-1}+(1-\omega)P_2^{-1},\qquad x_{\text{CI}}=P_{\text{CI}}\big(\omega P_1^{-1}x_1+(1-\omega)P_2^{-1}x_2\big),$$3, and $$P_{\text{CI}}^{-1}=\omega P_1^{-1}+(1-\omega)P_2^{-1},\qquad x_{\text{CI}}=P_{\text{CI}}\big(\omega P_1^{-1}x_1+(1-\omega)P_2^{-1}x_2\big),$$4 were used in different update modes, with the authors noting that these weights could be found by minimizing the trace or determinant of the posterior covariance but were empirically tuned for consistent performance (Zhu et al., 2021).

A related multirobot localization architecture applies CI only in the communication update while keeping standard EKF prediction and observation updates locally. In information form, robot $$P_{\text{CI}}^{-1}=\omega P_1^{-1}+(1-\omega)P_2^{-1},\qquad x_{\text{CI}}=P_{\text{CI}}\big(\omega P_1^{-1}x_1+(1-\omega)P_2^{-1}x_2\big),$$5 fuses incoming estimates by

$$P_{\text{CI}}^{-1}=\omega P_1^{-1}+(1-\omega)P_2^{-1},\qquad x_{\text{CI}}=P_{\text{CI}}\big(\omega P_1^{-1}x_1+(1-\omega)P_2^{-1}x_2\big),$$6

$$P_{\text{CI}}^{-1}=\omega P_1^{-1}+(1-\omega)P_2^{-1},\qquad x_{\text{CI}}=P_{\text{CI}}\big(\omega P_1^{-1}x_1+(1-\omega)P_2^{-1}x_2\big),$$7

with convex coefficients $$P_{\text{CI}}^{-1}=\omega P_1^{-1}+(1-\omega)P_2^{-1},\qquad x_{\text{CI}}=P_{\text{CI}}\big(\omega P_1^{-1}x_1+(1-\omega)P_2^{-1}x_2\big),$$8. Under a weak connectivity condition on the union of observation links in the super-neighborhood of robot $$P_{\text{CI}}^{-1}=\omega P_1^{-1}+(1-\omega)P_2^{-1},\qquad x_{\text{CI}}=P_{\text{CI}}\big(\omega P_1^{-1}x_1+(1-\omega)P_2^{-1}x_2\big),$$9, the local covariance is bounded. Empirically, this architecture is less affected by blocked or partially unavailable communication than several local-state alternatives (Chang et al., 2021).

A more specialized 3D range-based distributed cooperative localization formulation introduces an explicit weighting matrix into CI. Instead of minimizing $\omega\in[0,1]$0, the method selects $\omega\in[0,1]$1 by

$\omega\in[0,1]$2

where $\omega\in[0,1]$3 is derived from INS error propagation over a future horizon $\omega\in[0,1]$4. This design addresses scale imbalance, correlation mismatch, and lack of task relevance in classical CI criteria. In the reported 3D simulations, agent 1 position RMSE was $\omega\in[0,1]$5 m for WCI versus $\omega\in[0,1]$6 m for CI-trace and $\omega\in[0,1]$7 m for CI-det, while distributed operation remained more robust and scalable than centralized fusion (Tu et al., 17 Aug 2025).

In multitarget tracking, WCI ideas reappear in random finite set form. One paper develops adaptive, information-driven GCI for labeled multi-Bernoulli filtering, in which each Bernoulli component receives sensor-specific adaptive weights computed from a Cauchy–Schwarz divergence between prior and posterior Bernoulli components. The resulting weighted GCI is explicitly described as a multi-object, RFS-based Weighted Covariance Intersection whose weights are label-dependent and time-varying (Wang et al., 2017).

6. Conservatism, criticisms, and alternatives

The principal criticism of WCI is conservatism. Because it is designed to remain valid for all admissible unknown cross-correlations, its covariance bound can be substantially larger than necessary. A game-theoretic alternative formulates fusion as a minimax problem over the fusion matrix $\omega\in[0,1]$8 and the unknown cross-covariance $\omega\in[0,1]$9,

$N$0

and produces a trace-consistent estimator that is less conservative than CI in the reported examples, although it relaxes full matrix consistency to trace-consistency (Leonardos et al., 2016).

A Bayesian alternative assumes a prior distribution on the full covariance matrix, derives the conditional distribution of the unknown off-diagonal blocks given the known diagonal blocks, and uses Monte Carlo integration to approximate the MMSE fusion rule. In the two-node case the conditional distribution of the cross-covariance is an inverted matrix variate $N$1-distribution. The reported simulations show that this Bayesian method works better than the popular covariance intersection method, but it replaces CI’s worst-case guarantee with a prior-dependent MMSE construction (Weng et al., 2013).

From a set-membership viewpoint, the Convex Combination Ellipsoid (CCE) method has also been proposed as a less conservative covariance estimate than CI. CCE uses the same scalar weight $N$2 as CI and the same fused center, but scales the CI base covariance by a factor $N$3. The paper emphasizes two properties that CI lacks in general: CCE does not introduce additional uncertainty that was not already present in the prior estimates, and its ellipsoid is contained in the union of the prior ellipsoids while still containing their intersection (Zamani et al., 2023).

These alternatives do not eliminate the role of WCI. Rather, they clarify its place: WCI remains the canonical conservative fusion rule when unknown cross-correlations must be handled without modeling assumptions, whereas structured, Bayesian, game-theoretic, or set-membership alternatives are used when tighter bounds are sought and additional assumptions or different optimality criteria are acceptable (Leonardos et al., 2016).