Dynamic-HJSG: Cross-Domain Dynamic Structures

• Dynamic-HJSG is an umbrella term defining frameworks with dynamic, hierarchical, or joint structures across various domains.
• In dynamic view synthesis, it underpins Hybrid Gaussian Splatting through static–dynamic decomposition, enabling efficient temporal modeling and compact representations.
• It also encompasses multi-agent coordination via dynamic joint-state graph construction and Hamilton–Jacobi solvers using sparse-grid adaptivity for enhanced computational performance.

Searching arXiv for the cited work and closely related papers to ground the article. Dynamic-HJSG is not a canonical method name in the cited literature. The label is instead used interpretively for several dynamic, hierarchical, or joint formulations. In dynamic novel view synthesis, the closest explicit mapping is to Hybrid Gaussian Splatting (HGS), where “Dynamic-HJSG” denotes the dynamic aspects of HGS: Static–Dynamic Decomposition (SDD), temporal modeling with localized RBF-like parameterizations, and a two-stage training strategy for compact and efficient dynamic view synthesis (Zhang et al., 16 Dec 2025). In other sources, the same label is mapped to hierarchical anchor-driven Gaussian splatting, local-space temporal Gaussians, persistent dynamic Gaussians, dynamic human–scene joint structures, dynamic joint-state graph construction, or adaptive Hamilton–Jacobi solvers (Chen et al., 11 Jun 2025, Wu et al., 3 Jul 2025, Luiten et al., 2023, Wang et al., 27 Jan 2026, Zhou et al., 8 Sep 2025, Guo et al., 2020, Meng et al., 2023). This suggests that Dynamic-HJSG functions primarily as an umbrella descriptor rather than a single standardized algorithm.

1. Terminology and conceptual scope

Several cited papers explicitly state that they do not use the term “Dynamic-HJSG” and instead provide a nearest technical mapping. The HGS paper identifies the relevant mapping as the dynamic components of Hybrid Gaussian Splatting (Zhang et al., 16 Dec 2025). LocalDyGS, ADSR, Dyn-HSI, and DGH similarly treat the label as an interpretive shorthand for nearby frameworks rather than a formal title (Wu et al., 3 Jul 2025, Biswas et al., 29 Nov 2025, Wang et al., 27 Jan 2026, Wang et al., 18 Dec 2025). By contrast, “Dynamic-HJSG” appears as an explicit term in multi-agent coordination, where it denotes a dynamic graph-construction strategy for a homogeneous joint-state graph (Zhou et al., 8 Sep 2025).

Interpretation	Core structure	Representative source
Dynamic hybrid Gaussian splatting	Static–dynamic decomposition, localized temporal basis, alternating optimization	HGS (Zhang et al., 16 Dec 2025)
Dynamic hierarchical/joint Gaussian modeling	Local spaces, anchors, persistent Gaussians, 2D surfels, 4D HDR Gaussians	(Wu et al., 3 Jul 2025, Chen et al., 11 Jun 2025, Luiten et al., 2023, Zhang et al., 2024, Zhang et al., 26 Sep 2025)
Dynamic human–scene or semantic joint structure	Semantic assets, occupancy-conditioned motion, hierarchical memory, dynamic hair	(Biswas et al., 29 Nov 2025, Wang et al., 27 Jan 2026, Wang et al., 18 Dec 2025)
Dynamic joint-state graph construction	On-the-fly JSG expansion with homogeneity-based pruning	(Zhou et al., 8 Sep 2025)
Dynamic Hamilton–Jacobi sparse-grid/primal–dual solvers	Adaptive sparse-grid LDG and PDHG saddle-point schemes	(Guo et al., 2020, Meng et al., 2023)

A central consequence is terminological plurality. In Gaussian-splatting research, the phrase usually suggests dynamic, structured, or hierarchical parameterization. In graph planning, it denotes a precise search structure. In Hamilton–Jacobi numerics, it is naturally read as a dynamic Hamilton–Jacobi sparse-grid or saddle-point solver. Any encyclopedia treatment therefore has to distinguish these meanings rather than collapse them into a single lineage.

2. Hybrid Gaussian Splatting as the closest explicit mapping

In the dynamic view-synthesis context, the most direct definition of Dynamic-HJSG is the HGS formulation for compact dynamic 3D Gaussian Splatting (Zhang et al., 16 Dec 2025). HGS starts from the observation that prior dynamic 3DGS methods either use implicit deformation fields that increase model size and slow runtime and struggle with abrupt motion, or indiscriminately assign time-varying parameters to all Gaussians, including static content. Its solution is Static–Dynamic Decomposition: a unified Gaussian set is partitioned into static and dynamic primitives, static primitives share temporally invariant parameters, and only dynamic primitives carry time-varying $\mu_i(t)$, $r_i(t)$, and $\sigma_i(t)$.

The underlying 3DGS formulation remains explicit. Each anisotropic Gaussian primitive has mean $\mu_i \in \mathbb{R}^3$, covariance $\Sigma_i \in \mathbb{R}^{3 \times 3}$, scale $s_i \in \mathbb{R}^3$, rotation $R_i \in SO(3)$, color coefficients, and opacity. The density is

$$\alpha_i(x) = \sigma_i \exp\left(-\frac{1}{2}(x-\mu_i)^{\mathsf{T}}\Sigma_i^{-1}(x-\mu_i)\right),$$

and rendering uses projected 2D Gaussians followed by front-to-back alpha compositing,

$$I = \sum_{k} c_k\, \alpha_k \prod_{j<k}(1-\alpha_j).$$

What changes in HGS is the hybrid temporal parameterization. Dynamic primitives use an order-3 polynomial for position, an order-1 polynomial for rotation, and a Gaussian kernel in time for opacity:

$$\mu_i(t) = \sum_{k=0}^{3} b_{i,k}\,(t - \mu_i^{\tau})^k,$$

$r_i(t)$0

$r_i(t)$1

Static primitives set temporal coefficients to fixed values that make them temporally invariant, with shared fixed values not stored per primitive.

The decomposition is operationalized by Track Anything Model segmentation masks, which split multi-view videos into static versus dynamic regions. Gaussians initialized inside dynamic masks are labeled dynamic; others are labeled static; newly split or spawned Gaussians inherit the parent label. Compactness is further improved through interleaved-frame initialization, which initializes from every $r_i(t)$2-th frame to reduce the number of SfM points and initial primitives while preserving scene coverage.

Training is organized into two stages within alternating cycles. Stage 1 optimizes only the static primitives, refining geometry and appearance in static regions with stable gradients. Stage 2 freezes the updated static set and optimizes dynamic primitives, reducing inter-region gradient interference and suppressing flicker and temporal blur at interfaces. The loss follows 3DGS and combines a photometric $r_i(t)$3 term with D-SSIM; no extra temporal smoothness or boundary masks are introduced.

The reported efficiency gains are unusually large. On the Neural 3D Video dataset, HGS uses 6.87 MB, which is 63% smaller than STGS at 18.19 MB, up to 98% smaller than MixVoxels at 508 MB, and also smaller than 4DGS at 29.36 MB and ED3DGS at 31.41 MB. Reported rendering speeds include up to 125 FPS at 4K on RTX 3090, over 300 FPS at $r_i(t)$4 on RTX 3090, and 160 FPS at $r_i(t)$5 on RTX 3050. On Neural 3D Video it reports PSNR 32.36 dB and SSIM 0.952, comparable to STGS at 32.32/0.953. On Google Immersive it reports PSNR 29.60 dB and SSIM 0.925. An ablation further states that removing the two-stage strategy drops PSNR from 32.36 to 25.76 and SSIM from 0.952 to 0.910 (Zhang et al., 16 Dec 2025).

In this usage, Dynamic-HJSG denotes a compact dynamic Gaussian representation in which temporal degrees of freedom are explicitly restricted to dynamic content. The significance of that restriction is twofold: parameter efficiency and boundary coherence.

3. Hierarchical and joint Gaussian scene models

A broader Gaussian-splatting reading of Dynamic-HJSG encompasses several recent formulations that organize dynamics through locality, hierarchy, sparse controls, or persistent identity. These methods differ in parameterization, but they share a common attempt to avoid unconstrained per-Gaussian temporal updates.

LocalDyGS decomposes a global scene into seed-defined local spaces and generates time-dependent Temporal Gaussians only when motion occupies a local neighborhood (Wu et al., 3 Jul 2025). Each seed carries a static feature $r_i(t)$6 and a scale $r_i(t)$7; a global 4D hash encoding over $r_i(t)$8 produces a dynamic residual $r_i(t)$9; a learned weight field predicts $\sigma_i(t)$0; and fused features $\sigma_i(t)$1 are decoded into temporal Gaussian parameters. TGs with $\sigma_i(t)$2 are discarded, so activation and deactivation are explicit. The method trains for 30,000 iterations with Adam, uses $\sigma_i(t)$3 TGs per seed, and augments sparse SfM coverage with Adaptive Seed Growing. Reported performance includes PSNR 32.28, LPIPS 0.043, FPS 105, training time 0.58 h, and model size 100 MB on N3DV, as well as PSNR 30.58, SSIM 0.944, and LPIPS 0.173 on the VRU basketball court benchmark (Wu et al., 3 Jul 2025).

HAIF-GS replaces per-Gaussian deformations with sparse anchor-driven deformation (Chen et al., 11 Jun 2025). Its Anchor Filter predicts dynamic confidence to suppress updates in static areas, the Induced Flow-Guided Deformation module induces anchor motion through multi-frame feature aggregation and self-supervised bidirectional scene flow, and Hierarchical Anchor Propagation densifies anchors in regions of high temporal translation variance. Gaussian motion is obtained by anchor interpolation,

$\sigma_i(t)$4

with LBS-style updates for centers and rotations. On NeRF-DS it reports mean PSNR 24.63, MS-SSIM 0.9014, and LPIPS 0.1342; on D-NeRF it reports mean PSNR 42.00, SSIM 0.997, and LPIPS 0.010 (Chen et al., 11 Jun 2025).

Dynamic 3D Gaussians adopts a persistent oriented Gaussian representation in which color, opacity, and size are fixed over time while per-Gaussian pose follows time-varying $\sigma_i(t)$5 motion (Luiten et al., 2023). Local rigidity, rotational similarity, long-term isometry, and motion smoothness make dense 6-DOF tracking emerge from view synthesis without correspondence or optical flow. Reported performance on CMU Panoptic includes 28.7 PSNR at approximately 850 FPS splat rendering, dense 3D tracks with approximately 2.21 cm median error over 150 timesteps, and 2D tracking with approximately 1.57 px MTE and 100% survival across 371 tracks (Luiten et al., 2023).

Dynamic 2D Gaussians shifts the emphasis from appearance-first splatting to surface-aligned surfels for dynamic objects (Zhang et al., 2024). It combines sparse control-point deformation with planar 2D Gaussian primitives, Depth Distortion, and Normal Consistency, then filters rendered depth maps with object masks before TSDF fusion. On D-NeRF it reports PSNR 27.490, SSIM 0.957, and LPIPS 0.052; on DG-Mesh it reports Chamfer Distance 0.851 and EMD 0.132, with the full model improving the “bird” sample from CD 0.425 without depth filtering to CD 0.328 with all components (Zhang et al., 2024).

HDR-4DGS extends dynamic Gaussian splatting to HDR Dynamic Novel View Synthesis by combining a native 4D Gaussian model with a Dynamic Tone Mapper conditioned on exposure and temporal radiance context (Zhang et al., 26 Sep 2025). Its 4D compositing writes

$\sigma_i(t)$6

while the tone mapper uses

$\sigma_i(t)$7

On HDR-4D-Syn, the LDR-only version reports HDR PSNR 25.88, HDR SSIM 0.865, HDR LPIPS 0.076, LDR PSNR 33.16, and 40.80 fps; with LDR+HDR supervision it reports HDR PSNR 30.40 and 48.63 fps (Zhang et al., 26 Sep 2025).

Taken together, these works make Dynamic-HJSG intelligible as a family resemblance: dynamic Gaussian systems become tractable when motion is localized, hierarchically controlled, identity-preserving, or explicitly coupled to structure rather than distributed uniformly over all primitives.

4. Human–scene, semantic, and asset-centered interpretations

A second cluster of interpretations uses Dynamic-HJSG to denote hybrid or joint structures in which Gaussian rendering is only one part of a larger semantic or embodied pipeline.

ADSR, formally “Asset-Driven Sematic Reconstruction of Dynamic Scene with Multi-Human-Object Interactions,” treats Dynamic-HJSG as a dynamic, hybrid, joint semantic Gaussian Splatting framework (Biswas et al., 29 Nov 2025). It reconstructs dynamic scenes from a monocular video by generating high-fidelity meshes with TRELLIS, propagating rigid objects by $\sigma_i(t)$8 and humans by SMPL-based LBS, and then refining per-element alignment with Gaussian Splatting using corrective transforms $\sigma_i(t)$9. On HOI-M3 it reports object-wise Dist 0.0446, Comp 0.0539, F-score 0.6638, and CD 0.0493, compared with DreamScene4D at Dist 0.0734, Comp 0.0947, F-score 0.4801, and CD 0.0840. Whole-scene performance is reported as Dist 0.0843, Comp 0.1185, F-score 0.3754, and CD 0.1014 (Biswas et al., 29 Nov 2025).

Dyn-HSI interprets Dynamic-HJSG as a Dynamic Human–Scene Joint Graph abstraction even though the paper itself operates on voxel occupancy rather than an explicit graph (Wang et al., 27 Jan 2026). Its Vision–Memory–Control architecture combines Dynamic Scene-Aware Navigation, Hierarchical Experience Memory, and a Human–Scene Interaction Diffusion Model. The local scene state is represented by a $\mu_i \in \mathbb{R}^3$0 occupancy grid centered on the pelvis, text is encoded with CLIP, and the controller learns condition weights

$\mu_i \in \mathbb{R}^3$1

On LINGO it reports FID 0.092, MPJPE 0.040, foot skating 0.016, Traj. sim. 90.01%, Traj. err. 0.021, Goal err. 0.027, and Diversity 4.649. On Dyn-LINGO it reports Pene. Value 39.19, Rate 15.77%, Mean 1652, and Max 4525. The average inference time per frame is approximately 0.413 s (Wang et al., 27 Jan 2026).

Dynamic Gaussian Hair extends the idea to structured hair dynamics (Wang et al., 18 Dec 2025). DGH uses a coarse-to-fine temporal model: a pose-conditioned time-independent volumetric implicit deformation model predicts coarse displacements, a time-dependent refinement predicts per-point 3D flow vectors via cross-attention over volumetric features from previous frames, and a strand-guided optimization module attaches cylindrical Gaussians to hair segments and predicts dynamic color, scale, and opacity. Across 5 grooms it reports average PSNR approximately 27.0, SSIM approximately 0.871, LPIPS approximately 0.127, and average geometry errors approximately 0.0832 in $\mu_i \in \mathbb{R}^3$2 and 0.0266 in Chamfer distance (Wang et al., 18 Dec 2025).

These variants expand Dynamic-HJSG beyond pure scene rendering. The jointness may be semantic, as in ADSR; embodied and occupancy-conditioned, as in Dyn-HSI; or hierarchical along strands and tufts, as in DGH. A plausible implication is that the term becomes most useful when “dynamic,” “hybrid,” and “joint” all refer to different axes of structure rather than to a single representation choice.

5. Dynamic-HJSG as dynamic joint-state graph construction

The only source in which Dynamic-HJSG is an explicit algorithmic name is “Efficient Multi-Agent Coordination via Dynamic Joint-State Graph Construction” (Zhou et al., 8 Sep 2025). Here the setting is TCGRE, where homogeneous robots traverse an undirected graph with risky edges whose traversal costs can be reduced by pairwise support from teammates. The problem is reformulated through a simplified graph $\mu_i \in \mathbb{R}^3$3 containing risky-edge endpoints, support nodes, and all starts and goals, and then lifted into a Joint-State Graph whose states are robot location tuples $\mu_i \in \mathbb{R}^3$4.

In the full JSG, joint-edge costs are computed by matching risky-edge traversals and support-node occupancies between two joint-states. The paper proves that this matching is a Maximum Weighted Bipartite Matching problem, so Hungarian matching yields the optimal support assignment. Dynamic-HJSG then prunes the exponential construction by exploiting homogeneity and by generating the graph on the fly. Its central restriction is given by Lemma 1: it is sufficient to construct transitions where at most two robots change locations between joint-states. Under this restriction, the dynamically constructed H’JSG still contains the optimal joint-state path from start to goal.

The resulting complexity is summarized as

$\mu_i \in \mathbb{R}^3$5

where $\mu_i \in \mathbb{R}^3$6 is the number of visited joint-states, $\mu_i \in \mathbb{R}^3$7 the number of visited joint-edges, $\mu_i \in \mathbb{R}^3$8 the number of agents, and $\mu_i \in \mathbb{R}^3$9 the maximum degree of a super-node. This is contrasted with the static JSG, whose state and edge sets both scale as $\Sigma_i \in \mathbb{R}^{3 \times 3}$0.

Empirically, the reported summaries include sub-second median runtime of 100–102 ms across all team sizes, 98% completed runs, average runtime of approximately 120 ms on successful runs, timeout cases of approximately 2%, and effective runtime of approximately 132 ms. The reported completion probability at 60s is approximately 98%, compared with approximately 85% for HCES, approximately 65% for JSG, and approximately 40% for CES, with a log-rank test indicating statistically significant differences at $\Sigma_i \in \mathbb{R}^{3 \times 3}$1 (Zhou et al., 8 Sep 2025).

In this graph-planning meaning, Dynamic-HJSG is not a rendering method at all. It is a dynamic construction rule for a homogeneity-aware joint-state search space, and the “HJSG” acronym refers to a homogeneous joint-state graph rather than to Hamilton–Jacobi or Gaussian splatting.

6. Hamilton–Jacobi interpretations and terminological status

A separate technical lineage reads Dynamic-HJSG as a dynamic Hamilton–Jacobi sparse-grid or saddle-point solver. The adaptive sparse-grid LDG method for Hamilton–Jacobi equations in high dimensions combines sparse-grid multiresolution with local discontinuous Galerkin discretization and adaptive refinement based on hierarchical surpluses (Guo et al., 2020). For the bounded-domain HJ equation

$\Sigma_i \in \mathbb{R}^{3 \times 3}$2

the sparse-grid DG space replaces the full tensor-product complexity $\Sigma_i \in \mathbb{R}^{3 \times 3}$3 by $\Sigma_i \in \mathbb{R}^{3 \times 3}$4, while adaptivity refines regions where detail coefficients exceed a threshold. The semi-discrete LDG scheme reconstructs directional derivatives $\Sigma_i \in \mathbb{R}^{3 \times 3}$5 and uses a Lax–Friedrichs-type numerical Hamiltonian

$\Sigma_i \in \mathbb{R}^{3 \times 3}$6

Numerical tests are reported up to four dimensions, with adaptive sparse-grid LDG removing oscillations in post-kink regimes and achieving substantial DoF savings relative to full grids (Guo et al., 2020).

The PDHG-based HJ solver takes a different route by turning the time-implicit update into a saddle-point problem (Meng et al., 2023). For

$\Sigma_i \in \mathbb{R}^{3 \times 3}$7

the method introduces dual density $\Sigma_i \in \mathbb{R}^{3 \times 3}$8 and velocity $\Sigma_i \in \mathbb{R}^{3 \times 3}$9, then solves a primal–dual problem based on the convex conjugate $s_i \in \mathbb{R}^3$0. The paper gives the continuous saddle functional

$s_i \in \mathbb{R}^3$1

Its appeal is that non-smooth and spatiotemporally dependent Hamiltonians are handled through pointwise proximal updates of $s_i \in \mathbb{R}^3$2. The paper reports residuals below $s_i \in \mathbb{R}^3$3 and first-order accuracy for quadratic and $s_i \in \mathbb{R}^3$4 Hamiltonians in one and two dimensions (Meng et al., 2023).

These Hamilton–Jacobi meanings make clear that Dynamic-HJSG cannot be reduced to Gaussian splatting. In one domain, it names sparse-grid adaptivity around kinks and rarefactions; in another, it names a primal–dual treatment of implicit HJ updates. The commonality is structural rather than representational: both formulations exploit explicit decomposition to keep dynamic computation tractable.

A persistent misconception is therefore that Dynamic-HJSG denotes a single named architecture. The cited record does not support that view. HGS, LocalDyGS, ADSR, Dyn-HSI, and DGH all state, in effect, that the phrase is a mapping or interpretation rather than their formal method name (Zhang et al., 16 Dec 2025, Wu et al., 3 Jul 2025, Biswas et al., 29 Nov 2025, Wang et al., 27 Jan 2026, Wang et al., 18 Dec 2025). The most precise use is contextual. In dynamic view synthesis, it most closely denotes HGS-style static–dynamic factorization. In multi-agent planning, it denotes dynamic joint-state graph construction. In Hamilton–Jacobi numerics, it denotes dynamic sparse-grid or saddle-point solvers. The term is thus best treated as a cross-domain editorial label for dynamic hierarchical or joint structure, not as a singular standardized method.