title: >- [Paper Note] KineticGS: Momentum-driven Coherent 4D Gaussian Splatting for Monocular Dynamic Scene Reconstruction description: >- [ECCV 2026][3D Vision][Dynamic Scene Reconstruction] Models Gaussian particles as momentum-carrying entities with energy-aware activation and space-filling curve synchronization to achieve physically coherent 4D reconstruction from monocular videos (+0.91 dB dynamic PSNR). tags: - ECCV 2026 - 3D Vision - Dynamic Scene Reconstruction - 4D Gaussian Splatting - Monocular Video date: 2026-09-19 content_hash: b7c812d14b2c1b3a

KineticGS: Momentum-driven Coherent 4D Gaussian Splatting for Monocular Dynamic Scene Reconstruction¶

Conference: ECCV 2026
Paper: ECCV Official
Code: https://github.com/wyunqing/KineticGS.git
Area: 3D Vision
Keywords: Dynamic Scene Reconstruction / 4D Gaussian Splatting / Monocular Video / Momentum-Driven / Motion Coherence

TL;DR¶

Addressing trajectory fragmentation and non-physical deformations caused by the lack of physical priors in monocular dynamic reconstruction, KineticGS models Gaussians as momentum-carrying particles within a hierarchical framework spanning momentum-guided sampling, learnable energy-flow activation, and holistic kinetic synchronization, yielding +0.91 dB PSNR gains across dynamic object regions.

Background & Motivation¶

Reconstructing 4D dynamic geometry and temporal evolution of real-world environments from casual monocular videos is foundational for immersive AR/VR, autonomous robotic perception, and realistic digital avatars. However, single-view dynamic reconstruction represents an inherently ill-posed inverse problem, as the system must disentangle non-rigid motions, complex trajectories, and frequent self-occlusions from severely constrained 2D observations. While neural radiance fields (NeRF) pioneered continuous scene synthesis, their volumetric formulations entail heavy sampling costs and sluggish rendering. The advent of 3D Gaussian Splatting (3DGS) and its 4D extensions enabled real-time differentiable rendering through explicit primitives, yet existing 4DGS variants predominantly rely on appearance-based photometric supervision without enforcing physical motion principles.

The core tension lies in the absence of kinetic priors to govern Gaussian deformation trajectories over time. Although several recent methods introduce pre-computed 2D optical flow or monocular depth priors, these signals are inherently noisy and degrade sharply under rapid motion, severe occlusions, or textureless regions. Under such unconstrained conditions, appearance-only Gaussians exhibit disjointed motion and abrupt transparency shifts upon reappearing after occlusion, manifesting as severe motion drift and visual flickering. Conversely, physics-based simulation methods like PhysGaussian embed Newtonian mechanics via continuum mechanics solvers, but they strictly require precise material parameters and external boundary forces that remain fundamentally unidentifiable in monocular videos.

The key insight of this paper is to bypass explicit material parameter identification and instead treat classical momentum continuity—the physical principle that momentum carries inertial persistence—as the primary inductive bias. Core idea: KineticGS formulates a hierarchical momentum-driven 4D Gaussian Splatting framework, modeling Gaussians as momentum-carrying particles sampled via a geometry-velocity momentum field, regulated by smooth double-sigmoid energy activation windows under occlusion, and synchronized via space-filling curve neighborhoods to achieve physically coherent dynamic reconstruction.

Method¶

Overall Architecture¶

KineticGS deconstructs monocular dynamic reconstruction into a three-level hierarchy: Gaussian primitives, momentum-carrying particles, and holistic motion entities. Given an input monocular video, the framework first estimates depth-based geometric confidence (acting as physical mass) and non-rigid residual optical flow (acting as physical velocity) to formulate a spatial momentum field, adaptively sampling dynamic particles in the canonical reference space. Next, each particle maintains a learnable energy-flow activation window that modulates temporal visibility and continuous momentum release across time, even when partially occluded. Finally, particles are serialized along space-filling curves to construct semantically coherent neighborhoods where kinetic synchronization constraints enforce aligned velocity and activation timing before driving differentiable neural Gaussian rendering.

%%{init: {'flowchart': {'rankSpacing': 24, 'nodeSpacing': 28, 'padding': 6, 'wrappingWidth': 400}}}%%
flowchart TD
    A["Monocular Video Input<br/>RGB Frame Sequence"] --> B["Momentum-Guided Particle Initialization<br/>Depth Confidence × Residual Flow Momentum Field"]
    B --> C["Learnable Energy-Flow Activation<br/>Double-Sigmoid Temporal Momentum Window"]
    C --> D["Holistic Kinetic Synchronization<br/>Space-Filling Curve Neighborhood Coherence"]
    D --> E["Neural Gaussian Deformation & Rendering<br/>Temporally Coherent 4D Output"]

Key Designs¶

1. Momentum-Guided Particle Initialization: Allocating representation capacity to high-energy zones

Conventional dynamic Gaussian methods uniformly distribute anchors across static voxels, starving complex non-rigid motions of representational capacity. KineticGS grounds particle allocation in classical mechanics ($p = mv$). In monocular video, mass and velocity cannot be directly measured; the framework interprets depth-based geometric confidence as physical mass $m(u, v)$—reflecting geometric stability against optimization jitter—and residual optical flow (total flow $\mathbf{f}(u, v)$ minus rigid camera-induced flow $\mathbf{f}_{\text{geom}}(u, v)$) as non-rigid velocity. The spatial momentum distribution is formulated as: $$ \mathbf{p}(u, v) = m(u, v) \otimes \big(\mathbf{f}(u, v) - \mathbf{f}_{\text{geom}}(u, v)\big) $$ Normalizing the squared momentum magnitude $|\mathbf{p}(u, v)|^2$ yields an energy density distribution for importance sampling. High-momentum pixels are back-projected via reference depth into 3D space to form dynamic particles equipped with learnable neural features and scales. During early training, a flow-depth consistency loss $\mathcal{L}_{\text{flow}}$ penalizes divergence between predicted particle positions and flow-propagated coordinates, establishing a solid kinetic anchor scaffold.

2. Learnable Energy-Flow Activation: Preserving momentum persistence through smooth temporal windows

When dynamic objects undergo occlusion or temporary disappearance, frame-wise discrete optimization causes Gaussians to vanish abruptly and re-instantiate with fragmented trajectories. Based on momentum persistence, KineticGS equips each particle with a continuous activation window $\mathbf{w}_i^t$ governed by a double-sigmoid formulation: $$ \mathbf{w}_i^t = \sigma\left(\frac{t - (t_i^{\text{ct}} - t_i^{\text{cr}})}{\tau}\right) \cdot \sigma\left(\frac{(t_i^{\text{ct}} + t_i^{\text{cr}}) - t}{\tau}\right) $$ where activation center $t_i^{\text{ct}}$ and temporal span $t_i^{\text{cr}}$ are predicted by a spatial MLP conditioned on particle coordinates, and $\tau > 0$ controls boundary smoothness. During deformation updates, positional displacements are modulated by a binary step gate $\mathcal{H}(\mathbf{w}_i^t) \in \{0, 1\}$, restricting spatial motion to active energy periods, while attributes such as features, scales, and grid offsets are softly modulated by $\mathbf{w}_i^t$. This prevents non-physical spatial drift while ensuring smooth visual continuity as objects re-emerge.

3. Holistic Kinetic Synchronization: Enforcing object-scale momentum alignment via space-filling curves

Particles within the same physical entity should evolve with coordinated momentum. However, under monocular depth ambiguities, standard Euclidean KNN grouping frequently connects spatially adjacent points across object boundaries or depth discontinuities. KineticGS resolves this by projecting particle coordinates onto space-filling curves (randomly alternating between Hilbert and Z-order curves during training). Space-filling curves preserve spatial locality within a 1D continuous ordering, naturally isolating distinct motion bodies without explicit semantic segmentation. Within curve-derived neighborhoods $\mathcal{N}$, kinetic synchronization penalizes relative displacement disparities and activation misalignment: $$ \mathcal{L}{\text{coh}} = \sum|^2 $$ By tightly coupling spatial displacement and temporal activation centers, this regularizer acts as a discrete momentum conservation field, suppressing erratic individual divergence and maintaining macro-object integrity.}} |\Delta \mathbf{l}_i - \Delta \mathbf{l}_j|_2^2 + \alpha |t_i^{\text{ct}} - t_j^{\text{ct}

Loss & Training¶

The total optimization objective integrates photometric rendering supervision with momentum-based regularization terms: $$ \mathcal{L}{\text{total}} = \mathcal{L}}} + \lambda_{\text{flow}}\mathcal{L{\text{flow}} + \lambda}}\mathcal{L{\text{reg}} + \lambda $$ where }}\mathcal{L}_{\text{coh}$\mathcal{L}_{\text{flow}}$ provides early geometric flow alignment; $\mathcal{L}_{\text{reg}}$ enforces a uniform reference prior over particle activation center and span histograms using KL divergence, preventing temporal collapse; and $\mathcal{L}_{\text{coh}}$ guarantees kinetic neighborhood synchronization. A gradual warm-up scheduler activates physical loss terms progressively, with training executed on a single NVIDIA RTX 3090 GPU.

Key Experimental Results¶

Main Results¶

Quantitative evaluations are conducted on the real-world handheld DyCheck-iPhone dataset and the controlled non-rigid HyperNeRF-interp benchmark. Table 1 reports dynamic object regions (extracted via U2-Net), and Table 2 evaluates full-scene averages across all sequences.

Dataset / Setting	Metric	KineticGS (Ours)	MoDec-GS [17] (Prev. SOTA)	4DGaussians [41]	Deformable 3DGS [48]	Gain vs. Prev. SOTA
DyCheck-iPhone (Dynamic Object Avg)	mPSNR (dB) ↑ mSSIM ↑ mLPIPS ↓	20.12 0.840 0.136	19.21 0.832 0.139	18.43 0.826 0.137	17.22 0.807 0.151	+0.91 dB +0.008 -0.003
- Apple sequence (Dynamic)	mPSNR (dB) ↑	24.98	23.96	21.98	19.46	+1.02 dB
- Block sequence (Dynamic)	mPSNR (dB) ↑	20.55	19.62	18.29	15.65	+0.93 dB
- Paper-windmill (Dynamic)	mPSNR (dB) ↑	22.06	20.58	20.03	18.73	+1.48 dB
- Teddy sequence (Dynamic)	mPSNR (dB) ↑	20.53	19.72	18.67	17.09	+0.81 dB
DyCheck-iPhone (Full Scene Avg)	mPSNR (dB) ↑ mSSIM ↑ mLPIPS ↓	14.65 0.342 0.511	14.42 0.336 0.524	14.24 0.325 0.532	12.60 0.303 0.590	+0.23 dB +0.006 -0.013
HyperNeRF-interp (Full Scene Avg)	mPSNR (dB) ↑ mSSIM ↑ mLPIPS ↓	27.92 0.828 0.221	27.78 0.827 0.219	27.59 0.802 0.287	25.90 0.764 0.291	+0.14 dB +0.001 +0.002

Ablation Study¶

Ablation studies analyze the incremental contribution of each kinetic module on DyCheck-iPhone dynamic object regions (Table 6a), alongside neighborhood construction strategies on the full scene (Fig. 7a).

Configuration	Momentum Init.	Energy Activation	Kinetic Sync.	mPSNR (dB) ↑	mSSIM ↑	mLPIPS ↓	Description / Gain
Voxel Baseline	✗	✗	✗	19.17	0.829	0.141	Standard uniform voxel anchor layout
+ Momentum-Guided Init.	✓	✗	✗	19.68	0.836	0.138	Concentrates primitives on high-motion zones (+0.51 dB)
+ Learnable Energy Activation	✓	✓	✗	19.76	0.837	0.134	Smooth temporal window prevents flicker (+0.59 dB)
KineticGS Full Model	✓	✓	✓	20.12	0.840	0.136	Complete physical momentum hierarchy (+0.95 dB)

Neighborhood Grouping Mechanism (Full Scene DyCheck)	PSNR (dB) ↑	SSIM ↑	Note
Voxel Baseline	14.41	0.335	Standard anchor configuration
KNN Neighborhoods ($k=8$)	14.53	0.337	Euclidean metric crosses depth discontinuities
Space-filling Curves (Ours)	14.65	0.342	Alternating Hilbert/Z-order serialization preserves boundaries

Key Findings¶

Disproportionate gains in dynamic zones: Global scene metrics dilute dynamic improvements due to dominant static backgrounds (+0.23 dB globally vs. +0.91 dB in dynamic regions). In high-speed scenes like Paper-windmill, KineticGS surpasses the second-best model by +1.48 dB.
Occlusion resilience in per-frame metrics: Per-frame PSNR curves during knife slicing sequences show that while baseline 4DGS exhibits sharp drops during occlusion, KineticGS maintains stable rendering trajectories via its smooth momentum-carrying activation windows.
Exceptional storage compactness: The anchor-particle hierarchical representation requires only ~25% of the memory footprint demanded by unconstrained 4DGaussians.

Highlights & Insights¶

Bridging visual observables to physical dynamics: Reinterpreting monocular depth confidence as inertial mass and residual non-rigid optical flow as velocity enables physical motion priors without requiring unidentifiable material constants.
Space-filling curves as semantic-free boundary protectors: Employing Hilbert and Z-order curves provides a lightweight, unsupervised spatial serialization that circumvents Euclidean cross-object bleeding in monocular depth fields.
Double-sigmoid energy shields: Continuous temporal activation windows decouple particle existence from instantaneous visibility, preventing cold-start artifacts and visual disintegration upon reappearance.

Limitations & Future Work¶

Reliance on a single canonical reference frame: Relying on the first frame as the global canonical space degrades performance when handling multiple entities entering/exiting the field of view over extended durations.
Sensitivity to severe optical flow corruption: In scenes with severe specular reflections or heavy motion blur, corrupted optical flow residuals can propagate inaccurate momentum priors to the initialization phase.
Future directions: Investigating streaming adaptive canonical spaces that update incrementally over long video streams, along with unsupervised cycle-consistent motion refinement.

vs 4DGaussians [41]: 4DGaussians learns unconstrained spatio-temporal MLPs without physical regularization, causing severe fragmentation under occlusion. KineticGS introduces momentum anchors and space-filling curve synchronization, outperforming it by 1.69 dB PSNR on dynamic regions with 75% lower storage.
vs MoDec-GS [17]: MoDec-GS explores hierarchical motion decomposition but lacks momentum-guided sampling and explicit physical synchronization. KineticGS demonstrates that physics-grounded initialization substantially sharpens deformation boundaries.
vs PhysGaussian [43]: PhysGaussian integrates Newtonian dynamics via Material Point Methods (MPM), necessitating known material stiffness and external forces. KineticGS abstracts momentum from visual residual kinematics, making it directly viable for unconstrained monocular videos.

Rating¶

Novelty: ⭐⭐⭐⭐☆ Elegantly adapts momentum conservation principles into Gaussian Splatting and resolves monocular neighborhood mixing via space-filling curves.
Experimental Thoroughness: ⭐⭐⭐⭐☆ Rigorous evaluation across real and synthetic benchmarks, separating dynamic region analysis from global scene metrics.
Writing Quality: ⭐⭐⭐⭐⭐ Well-structured, mathematically rigorous, with clear conceptual analogies between physics and vision.
Value: ⭐⭐⭐⭐☆ Provides a practical, highly efficient, and physically grounded paradigm for monocular 4D dynamic scene synthesis.