Eulerian Motion Reconstruction for Water Scenery

• Water’s rapid particle flow makes persistent-object motion models unsuitable, motivating a periodic 4D representation from a single monocular view rather than multiple fixed viewpoints.Source: §1 · p. 2
• A static 3D Eulerian motion field advects canonical Gaussian splats for looping motion, while a time-varying residual captures non-periodic motion and appearance changes.Source: §1 · p. 2
• The approach initializes motion with pairwise 2D optical flow, then simulates Gaussian splats so each repeats at a different random time and trains the model end-to-end with rendering loss.Source: §1 · p. 2
• The framework outperforms existing 4D reconstruction baselines in reconstruction and animation quality, according to quantitative and qualitative evaluations.Source: §1 · p. 2

Looping Video Cinemagraph

• Prior 3D water-looping work uses multiview asynchronous videos, whereas this work reconstructs a loopable 3D representation from monocular capture.Source: §2 · p. 2

Novel View Synthesis for Dynamic Scenes

• Dynamic-scene reconstruction remains under-constrained from monocular input, and prior motion representations typically target low-frequency dynamics and persistent objects rather than water’s high-frequency flow.Source: §2 · pp. 2–3

Feed-forward 4D reconstruction

• Feed-forward methods offer fast inference and cross-scene generalization but struggle to generalize learned motion priors to stochastic fluids; generative methods generally produce requested-horizon videos, not persistent editable 4D scenes.Source: §2 · p. 3

3. Method

• The overall model combines canonical Gaussian splats, a static Eulerian motion field, and residual offsets to represent steady flow and stochastic water changes.The components are initialized separately and then optimized jointly end-to-end using rendering losses.Source: §3 · p. 3

3.1. Scene Representation

• The time-fixed field V assigns a water-particle velocity to each 3D location and is initialized by unprojecting adjacent-frame optical flow.Canonical Gaussians encode 3D position, scale, orientation, opacity, and color.Source: §3.1 · p. 3

3.2. Looping by Forward Euler Integration

• Canonical splats are advected with forward Euler integration and respawn every L frames, while independently sampled start times stagger their resets.Synchronized resets produce a visible discontinuity; staggered resets keep most Gaussians flowing at each frame.Source: §3.2 · pp. 3–4

3.3. Non-periodic Deformations

• The residual field D adds cycle-dependent offsets to Gaussian position, color, and opacity to fit non-periodic differences in the training video.At inference, the cycle number is set to 0 so the animation remains periodic.Source: §3.3 · pp. 4–5

3.4. Initialization

• Initialization fits a static Gaussian reconstruction, then lifts predicted 2D optical flow into 3D using rendered depth to pretrain V.The lifted flow yields position and scene-flow pairs for pretraining the Eulerian field.Source: §3.4 · p. 5

3.5. Joint Training

• Joint training propagates splats through V, adds residuals for the sampled frame, and optimizes rendered-image L1 and SSIM losses with an L2 penalty on D.The residual regularizer encourages D to remain small.Source: §3.5 · p. 5

4. Implementation

• The motion field and residual deformation field use triplanes, and Gaussian splatting uses a RayGS implementation.Canonical Gaussians are trained for 10k iterations, then fine-tuned with the fields for another 20k iterations; the chosen cycle length is L = 15.Source: §4 · p. 5
• The dataset contains seven water scenes filmed at 1168 × 880 and 30 fps for about 25 seconds, with camera translations and rotations for parallax.COLMAP is used to reconstruct a sparse point cloud and estimate camera parameters.Source: §4 · p. 5
• The method produces sharper water-region reconstructions than the baselines, while MovieS appears sharp in row 2 but has poor overall view-synthesis quality.The caption directs readers to supplementary videos for the overall comparison.Source: Figure 4 · p. 6
• The method achieves the best overall reconstruction accuracy and leads in spatial and temporal distribution alignment by a large margin.Table 1 reports full-image and water-region metrics.Source: Table 1 · p. 6

5.1. Quantitative Comparison

• The method outperforms baselines on most metrics, especially LPIPS and the distributional metrics FID, KID, and FVD.Metrics are reported for full images and water regions; the supplied passage gives no table cell values.Source: §5.1 · p. 7

5.2. User Study

• 97.6% of user-study judgments preferred the method overall; preference reached 98.9% for motion and was 86.3% against AmbGS for appearance.The study aggregated 2,100 judgments from 25 participants across seven scenes; Figure 5 encodes winrate across Appearance, Motion, and Reference settings.Source: §5.2 · p. 7, Figure 5 · p. 7

5.3. Qualitative Comparison

• The method captures more water detail and richer, more natural motion than the baselines, whose tracks are short, incoherent, or near-static.MoVieS can reconstruct good detail but often misses fast, subtle water motion; Figure 6 compares tracks across two scenes and methods.Source: §5.3 · p. 7, Figure 6 · p. 7, Figure 7 · p. 8

5.4. Ablations

• The full model has the best perceptual and distributional metrics among the component ablations, while removing the Eulerian field or flow initialization leaves little motion and removing the residual makes tracks less stochastic.The track visualization shows coherent water-following paths for the full model and reduced motion for variants without the Eulerian field or flow initialization.Source: §5.4 · p. 8, Table 3 · p. 8, Figure 8 · p. 8
• Removing random per-Gaussian start times worsens FID, KID, and FVD and produces an obvious discontinuity across loops.The variant rebirths all Gaussians at t = 0, unlike the full model’s staggered rebirths.Source: §5.4 · p. 8
• L = 15 balances motion range and reconstruction sharpness: larger cycle lengths represent more diverse motion but converge less well and lower reconstruction quality.The cycle-length ablation compares three settings qualitatively and reports corresponding metrics.Source: Table 2 · p. 7, §5.4 · p. 8

Limitations and future work

• The pipeline can fail on large reflective water surfaces because reflection-induced fake geometry produces incorrect depth for initializing the Eulerian motion field.The authors propose combining monocular depth-based SFM tools such as MegaSam [12] with physics-based reflection modeling [6].Source: §6 · p. 8

Looping artifacts

• Random start times spread water particles stochastically across loops and prevent full-width pulse bands at loop boundaries; per-frame PSNR changes by −0.19 dB while FVD rises significantly.The figure compares fixed-camera scanlines over 60 frames, or four 15-frame periods.Source: Figure 9 · p. 9

Cycle length

• Among cycle lengths 15, 25, and 35, L=25 yields more waterfall-state variation than L=15, while L=35 produces blurrier reconstruction due to worse convergence.Source: Figure 10 · p. 9

Limitations and future work

• The constant Eulerian motion field limits representation of stochastic and time-varying motion; preliminary fire results extend it to a time-varying model.Source: §6 · p. 9

Extensions

• Extensions show lake reconstruction with SFM tools such as MegaSam [12] and fire modeling using a time-varying Eulerian motion field.Source: Figure 11 · p. 9

Conclusion

• The conclusion reports qualitative and quantitative comparisons showing the approach reconstructs water motion effectively over baselines.Source: §6 · p. 9

Supplementary material

• Implementation details cover experimental setup, hyperparameters and architecture, and dataset capture details.Source: A. Overview · p. 10
• Additional ablations and results include per-component reconstruction visualizations.Source: A. Overview · p. 10
• The supplement provides the user-study interface and instructions and recommends viewing the accompanying video results for the reconstructed water dynamics.Source: A. Overview · p. 10

Additional implementation details

• The additional implementation details include experimental setup.Source: B. Additional Implementation Details · p. 10
• They also cover hyperparameters and architecture.Source: B. Additional Implementation Details · p. 10
• Dataset capture details are included in the supplementary material.Source: B. Additional Implementation Details · p. 10

Water Segmentation Mask

• Dynamic Gaussians receive per-Gaussian probabilities trained from water masks with binary cross-entropy, and probabilities above 0.2 identify dynamic Gaussians.Dynamic-mask values continue to be refined during dynamic training but are updated every 10 steps.Source: §B.1 · p. 10
• Dynamic Gaussians enter the Eulerian motion and residual deformation fields, while static Gaussians bypass them; both sets are concatenated for rendering.Source: §B.1 · p. 10

Animation run time

• Looping reuses per-cycle propagation across frames, making animation-time rendering with Algorithm 2 significantly faster than training-time Algorithm 1.Reported wall-clock timings use a single A6000 GPU and include deformation-only and deformation-plus-rasterization passes.Source: §B.1 · p. 10

B.2. Hyperparameters and Architecture

• The Eulerian motion field uses a 128-resolution triplane with 12 feature channels, decoded to velocity by a width-64, depth-2 MLP; the residual field uses 16 channels and decodes position, opacity, and color residuals.The residual decoder is a width-128, depth-2 MLP conditioned on position, cycle index, and cycle number.Source: §B.2 · p. 10
• Reusing per-cycle propagations across frames yields a ∼3–4× rendering speedup, with a larger gain over 30 cycles than over one cycle.The comparison covers training-time versus inference-time propagation, for both deformation-only and end-to-end rendering.Source: Table B1 · p. 10
• Optimization weights are 0.8 for L1, 0.2 for SSIM, 0.001 for offset L2 regularization, and 0.5 for dynamic-probability binary cross-entropy; each scene takes about 8 hours on 8 A6000 GPUs.Source: §B.2 · p. 10

B.3. Dataset Details

• The dataset comprises seven handheld fisheye water-scene captures rectified to pinhole cameras, with three held-out temporal test segments per scene and per-scene capture statistics.Reported statistics include maximum camera baseline, trajectory length, maximum rotation, and mean water-mask fraction.Source: §B.3 · p. 10

C.1. Component Ablation Visualization

• The evaluation depicts seven handheld fisheye captures rectified to pinhole cameras, with three held-out temporal test segments per scene; capture statistics include baseline, path length, rotation, and water-mask fraction.Source: Figure B1 · p. 11, Table B2 · p. 11
• The full model reconstructs the best water dynamism; training the motion field from scratch converges to very small motion and blurry strands, while removing opacity and color deformation yields blurry results.Source: Figure C2 · p. 11, §C.1 · p. 11
• Without the non-periodic residual, the looping representation cannot fit the non-looping input video and produces blurry results.Source: §C.1 · p. 11

D. User Study Details

• The user study asks forced-choice comparisons of appearance in fixed-time animation, motion in fixed-view animation, and held-out-frame reconstruction against ground truth.Source: D. User Study Details · p. 11