Online Recursive Gaussian Fusion

• Online recursive Gaussian fusion is a family of sequential estimation processes that incorporates new Gaussian-derived observations into a compact state without reprocessing full data history.
• It leverages diverse Gaussian methodologies—such as Kalman filtering, Gaussian mixtures, and Gaussian processes—to achieve efficient, multi-sensor and multi-scale data fusion.
• This fusion technique underpins applications from atmospheric turbulence mitigation to decentralized sensor networks and 3D scene modeling through scalable recursive updates.

Online recursive Gaussian fusion denotes a family of sequential estimation and representation-update procedures in which new observations are assimilated into a Gaussian or Gaussian-derived state without reprocessing the full data history. In the literature considered here, the term spans several distinct instantiations: recursive image fusion supported by Gaussian mixture models and Kalman filtering for atmospheric turbulence mitigation (Anantrasirichai et al., 2018), information-form Gaussian-process updates for streaming and decentralized learning (Llorente et al., 22 Sep 2025), recursive Gaussian-process state-space estimation (Zheng et al., 2024), adaptive-resolution GP map fusion (Jin et al., 2021), sparse information-filter terrain mapping (Viset et al., 2022), Gaussian-mixture decentralized data fusion (Ahmed, 2019), and recurrent updates of 3D Gaussian scene primitives (Hu et al., 20 Dec 2025). This breadth suggests that the phrase is best understood as a technical pattern rather than a single standardized algorithmic recipe.

1. Scope and constituent meanings

Across the cited work, the adjective online consistently means that the update at time $t$ depends on the current observation and a compact state carried from time $t-1$, rather than on full-history recomputation. Recursive denotes explicit state recursions, typically in covariance form, information form, or exponential-average form. Fusion refers to the combination of multiple noisy sources of information—across time, across sensors or agents, across spatial scales, or across latent Gaussian primitives. Gaussian does not refer to a single object: it may denote Gaussian mixtures, Kalman-filter state models, Gaussian-process priors, Gaussian information states, or 3D Gaussian splats.

Domain	Gaussian object	Recursive mechanism
Atmospheric turbulence mitigation (Anantrasirichai et al., 2018)	GMM, Kalman filter, DT-CWT coefficients	Per-frame object, reference, and wavelet updates
Adaptive-resolution field mapping (Jin et al., 2021)	GP over cell averages	Kalman-style map update and Gaussian merging across scales
Recursive GP terrain maps (Viset et al., 2022)	Basis-weight GP weights	Information-filter recursions on local basis functions
Recursive GPSSM (Zheng et al., 2024)	Joint Gaussian over state and inducing variables	EKF-like prediction/correction and hyperparameter fusion
Decentralized and adaptive GP (Llorente et al., 22 Sep 2025)	RFF-based Gaussian linear model	Additive information updates, consensus, robust weighting
Decentralized GM fusion (Ahmed, 2019)	Finite Gaussian mixtures	Recursive Bayesian fusion via quotient approximations
Recurrent 3D scene modeling (Hu et al., 20 Dec 2025)	3D Gaussian scene primitives	State-to-state recurrent fusion with replay
Multi-sensor occupancy (Pavković et al., 24 Jul 2025)	Semantic 3D Gaussians	Block-wise recursive refinement within a frame
Multi-output distributed GP (Rao et al., 11 Apr 2026)	Gaussian over shared basis outputs	Recursive information updates and neighbour consensus

A common misconception is to reduce the topic to Kalman filtering alone. The literature shows a wider landscape: some methods are explicitly Bayesian and linear-Gaussian, some use Gaussian mixtures or Gaussian processes, and some use Gaussian primitives primarily as scene representations rather than as uncertainty distributions. Another misconception is that recursive fusion is necessarily temporal filtering in the strict state-space sense. GaussianFusionOcc, for example, is explicitly described as a single-frame multi-sensor method whose recursion occurs across fusion blocks within a frame rather than across time (Pavković et al., 24 Jul 2025).

2. Canonical update structures

The most compact formalization appears in information form. For RFF-based decentralized GP inference, the posterior over the finite-dimensional weight vector is encoded by an information matrix $D_t$ and information vector $\eta_t$, with online recursions

$$D_{c,t} = D_{c,t-1} + P_t, \qquad \eta_{c,t} = \eta_{c,t-1} + s_t,$$

where $P_t = \sigma_{\text{obs}}^{-2}\Phi_t\Phi_t^\top$ and $s_t = \sigma_{\text{obs}}^{-2}\Phi_t y_t$ (Llorente et al., 22 Sep 2025). The same additive structure reappears in multi-output recursive GP consensus, where each agent updates

$$\mathbf{\Omega}^{\mathbf g}_{i,t} = \mathbf{\Omega}^{\mathbf g}_{i,t-1} + \mathbf{J}_{i,t}^\top \mathbf{S}_{i,t}^{-1}\mathbf{J}_{i,t}, \qquad \boldsymbol{\xi}^{\mathbf g}_{i,t} = \boldsymbol{\xi}^{\mathbf g}_{i,t-1} + \mathbf{J}_{i,t}^\top \mathbf{S}_{i,t}^{-1}\mathbf{y}_{i,t},$$

and then fuses these natural parameters through consensus (Rao et al., 11 Apr 2026).

A second canonical pattern is covariance-form Gaussian conditioning. In adaptive-resolution GP map fusion, the posterior mean and covariance satisfy

$$\boldsymbol{\mu}^{+} = \boldsymbol{\mu}^{-} + \mathbf{\Gamma}\mathbf{v}, \qquad \mathbf{K}^{+} = \mathbf{K}^{-} - \mathbf{\Gamma}\mathbf{H}\mathbf{K}^{-},$$

with innovation $$\mathbf{v} = \mathbf{z} - \mathbf{H}\boldsymbol{\mu}^{-}$$, innovation covariance $t-1$0, and Kalman gain $t-1$1 (Jin et al., 2021). The same logic underlies recursive GPSSM prediction and correction, where a joint Gaussian over latent state and inducing variables is propagated by first-order linearization and updated by EKF-like measurement fusion (Zheng et al., 2024).

A third pattern is exponential averaging, used when the fused quantity is an image or transform coefficient rather than an explicitly parameterized posterior. In the turbulence-mitigation setting, the authors interpret the recursions as Gaussian-like because exponential moving averages arise as sequential Bayesian updates under Gaussian priors and Gaussian likelihoods with fixed variance ratio. This interpretation is stated explicitly for the object, reference, and lowpass image updates (Anantrasirichai et al., 2018).

Taken together, these forms imply a unifying principle: online recursive Gaussian fusion maintains a sufficient state—means and covariances, precisions and information vectors, or recursively blended latent coefficients—from which the effect of all past data can be reconstructed for prediction, while only the current observation enters the new update.

3. Transform-domain and image-sequence fusion

In atmospheric turbulence mitigation with moving objects, the recursive structure operates simultaneously on segmentation, tracking, geometric compensation, and wavelet-domain fusion. Moving objects are detected using an improved Gaussian mixture model, tracked with a Kalman filter, and recursively fused after affine warping: $t-1$2 The global reference frame is updated by

$t-1$3

with $t-1$4 chosen to approximate averaging over the last $t-1$5 frames (Anantrasirichai et al., 2018).

The same paper performs the actual restoration in the Dual-Tree Complex Wavelet Transform domain. Lowpass coefficients are updated by exponential averaging in background-consistent regions, while highpass coefficients are fused by separate rules for phase and magnitude. Phase is merged using $t-1$6 to give exponentially decaying weight to previous frames; magnitude is updated by a max-and-shrinkage rule,

$t-1$7

with $t-1$8 for weak coefficients. The resulting restored frame is reconstructed by inverse DT-CWT. The paper emphasizes that only $t-1$9, the previous fused states, and the current GMM and Kalman states are needed at each time step, not the full frame history (Anantrasirichai et al., 2018).

This case is significant because it makes clear that recursive Gaussian fusion need not be confined to posterior mean estimation. Here, Gaussian components enter as background-foreground segmentation and motion tracking modules, while the fused state itself is a recursively updated image representation in a transform domain.

4. Gaussian-process mapping, dynamics, and online control

A major branch of the literature realizes online recursive Gaussian fusion through Gaussian processes. In adaptive-resolution field mapping, the unknown field is modeled as a GP, but the maintained state is a vector of cell-averaged variables defined through an integral kernel. Online updates use Kalman-style formulas, while spatial-resolution changes are handled by linear Gaussian transformations: $D_t$0 This permits fusion both of measurements and of multiple fine cells into coarser cells, while maintaining theoretically sound correlations across resolutions (Jin et al., 2021).

A related scalability strategy appears in recursive GP terrain mapping with local finite-support basis functions. There, the sufficient statistics are an information vector $D_t$1 and information matrix $D_t$2, updated by

$D_t$3

Because only a bounded local subset of basis functions is active at each $D_t$4, each update affects a small sub-block, which yields temporal and spatial scalability and enables direct integration into an EKF SLAM system (Viset et al., 2022).

Recursive GPSSM work makes the Gaussian-fusion interpretation even more explicit. The method approximates the joint distribution by

$D_t$5

with $D_t$6 Gaussian. First-order linearization produces a linear-Gaussian transition and measurement model in the augmented state, so prediction and correction become EKF-style Gaussian updates. The same paper also recasts online hyperparameter adaptation as a Gaussian fusion step, in which changing the GP prior induces a Gaussian factor on the inducing variables and the posterior is updated by a Kalman-like rule (Zheng et al., 2024).

Online learning-based MPC with evolving GPs uses a different recursive mechanism: a GP posterior is updated by Cholesky add/delete recursions when new training points satisfy prediction-error or variance thresholds, and only if the candidate update does not increase the MPC value function at the current state. This combines recursive posterior prediction updates with finite-memory dataset management and yields recursive constraint satisfaction and input-to-state stability with respect to model-plant mismatch (Maiworm et al., 2019).

These GP-based methods show that online recursive Gaussian fusion can operate over fields, over latent inducing variables, or over learned dynamics models. In all cases, the common pattern is a fixed-size or bounded-size Gaussian state that accumulates past evidence through additive or Kalman-style recursions.

5. Decentralized and distributed formulations

In decentralized settings, recursive Gaussian fusion acquires a second meaning: fusion occurs not only across time but also across agents. For RFF-based decentralized GP learning, each agent computes local increments

$D_t$7

then uses additive consensus to approximate network-wide sums before updating its local information state. Robustness is introduced by residual-based diagonal weights $D_t$8, yielding weighted increments $D_t$9 and $\eta_t$0; dynamic adaptation is implemented either by a state-space model on the RFF weights or by a forgetting factor $\eta_t$1 applied to the information parameters (Llorente et al., 22 Sep 2025).

Consensus-based recursive multi-output GP adopts the same information-filter logic for vector-valued fields. Each node maintains the posterior over basis outputs $\eta_t$2, updates it with local measurements, and then performs neighbour-to-neighbour consensus on the precision matrix and precision-weighted mean. The paper emphasizes that this preserves inter-output correlations because consensus operates on the full $\eta_t$3-dimensional Gaussian state rather than on output-wise independent marginals (Rao et al., 11 Apr 2026).

A different but closely related tradition arises in decentralized Gaussian-mixture data fusion. Exact and approximate GM DDF both reduce to a quotient density in which a naive GM product is divided by a common-information term. Because the result is a “sum of quotients” mixture with non-Gaussian mixands, tractable recursive use requires new GM approximations, which the paper constructs through direct local sampling or indirect global sampling with weighted EM-like fitting (Ahmed, 2019). This extends online recursive fusion beyond single Gaussians to mixture-valued beliefs.

“Divide-and-Conquer Fusion” supplies a complementary perspective: when sub-posteriors are Gaussian,

$\eta_t$4

so product fusion reduces to classical information-form algebra. The same paper then generalizes this through GBF and divide-and-conquer SMC for non-Gaussian sub-posteriors, and identifies the progressive tree as the natural online structure in which a current fused posterior is recursively fused with the next arriving batch (Chan et al., 2021).

These decentralized formulations clarify that recursive Gaussian fusion is not merely sequential filtering at a single node. It is also a communication protocol: additive natural parameters make Gaussian states especially amenable to neighbour-to-neighbour consensus, product fusion, and hierarchical aggregation.

6. 3D Gaussian scene models and learned fusion operators

Recent work extends the notion from uncertainty-bearing Gaussian beliefs to Gaussian primitives as geometric scene elements. RecurGS maintains a single 3D Gaussian scene $\eta_t$5 and updates it across discrete scene states through

$\eta_t$6

where object-level changes are detected, matched, aligned with ICP plus Lie-algebra $\eta_t$7 refinement, and then fused through a voxelized, visibility-aware module. Replay supervision enforces consistency with previously rendered states, while new-region completion inserts Gaussians from per-state reconstructions into newly exposed regions. The resulting recurrent representation supports object-level manipulation and zero-shot novel-state synthesis without requiring additional scans (Hu et al., 20 Dec 2025).

GaussianFusionOcc occupies a nearby but importantly distinct position. It uses semantic 3D Gaussians

$\eta_t$8

as the scene representation for single-frame 3D semantic occupancy prediction. Camera, LiDAR, and radar features are fused by modality-agnostic deformable attention and an MLP-based fusion block, and Gaussian parameters are recursively refined across repeated blocks within the same frame. The paper explicitly states that it does not perform temporal Bayesian filtering or a Kalman-style recursive update over time; its recursiveness is intra-frame, with block outputs becoming block inputs (Pavković et al., 24 Jul 2025).

This distinction matters conceptually. RecurGS is a genuinely online recurrent fusion system over scene states, whereas GaussianFusionOcc is a learned recursive refinement operator over Gaussian primitives at a fixed time. The contrast shows that the term can cover both probabilistic recursion and learned recurrent geometry updates, provided that a Gaussian representation is incrementally fused rather than reconstructed from scratch.

7. Limitations, misconceptions, and research directions

Several limitations recur across the literature. In adaptive-resolution GP fusion, merging is irreversible, which is efficient in quasi-static environments but problematic if fine detail becomes interesting later; scalability to very large environments may still require additional sparse or low-rank structure (Jin et al., 2021). Recursive GPSSM accuracy depends on first-order linearization and moderate inducing-set size, and its online hyperparameter objective is approximate (Zheng et al., 2024). Decentralized RFF-GP fusion depends on RFF approximation quality, conditional independence assumptions, and sufficient consensus iterations, and exact Bayesian model averaging is not fully recovered in decentralized ensemble settings (Llorente et al., 22 Sep 2025).

In Gaussian-scene fusion, the limitations are different. RecurGS notes semantic association ambiguity, static-illumination assumptions, rigid-motion bias, and dependence on per-state reconstructions for new-region initialization (Hu et al., 20 Dec 2025). GaussianFusionOcc, by contrast, is limited chiefly by its single-frame design; the paper itself suggests that truly online operation would require persisting Gaussians over time, adding motion models, and possibly introducing explicit uncertainty (Pavković et al., 24 Jul 2025).

A further misconception is that recursive Gaussian fusion always yields exact centralized Bayes. The distributed literature is more nuanced. Exact equivalence may hold under shared feature maps, common hyperparameters, and converged consensus in information-form GP methods, or analytically in the Gaussian sub-posterior case, but GM quotient fusion and non-Gaussian divide-and-conquer fusion require approximations whose order and quality matter (Rao et al., 11 Apr 2026, Chan et al., 2021, Ahmed, 2019).

Taken together, these results suggest several durable research directions already visible in the cited work: reversible multiresolution fusion, continuous-time and non-rigid extensions, online hyperparameter adaptation, compressed or learned replay, dynamic communication graphs, and temporally persistent Gaussian-scene models. What unifies them is not a single algorithmic template, but a common commitment to representing accumulated evidence in a recursively updated Gaussian structure whose state remains compact enough for real-time assimilation.