Eulerian Motion Reconstruction for Water Scenery

Chuhan Chen; Yen-Chi Cheng; Ayush Saraf; Rajvi Shah; Tuotuo Li; Johannes Kopf; Chen Gao; Hung-Yu Tseng

cs.CVSep 29, 2026

Eulerian Motion Reconstruction for Water Scenery

Authors: Chuhan Chen, Yen-Chi Cheng, Ayush Saraf, Rajvi Shah, Tuotuo Li, Johannes Kopf, Chen Gao, Hung-Yu Tseng, +3 more

Organizations: Carnegie Mellon University · University of Illinois Urbana-Champaign · Meta

Abstract

Reconstructing and animating water scenery from nature produces compelling and immersive visual experiences. Previous work examined this task from the perspective of 2D video textures, with the goal of creating a looping video. In our work, we tackle the problem from a 3D perspective, creating a looping 4D dynamic reconstruction which can be interactively rendered from novel viewpoints from a single non-looping 2D source video. We represent motion as a 3D static \textit{Eulerian} motion field that advects canonical Gaussian splats that are cyclically reborn at fixed time periods, supervised using rendering losses. To model non-periodic and stochastic dynamics present in real-world scenes, we add a non-periodic, time-varying residual term to capture deviations from the static Eulerian motion field. We show quantitatively and qualitatively that our framework enables photorealistic animation of water scenes better than prior art.

Figures & tables

Figure 2 : Method Overview. (Stage 1) Given a monocular input video, we first create a static reconstruction in the form of Canonical Gaussian Splats (Sec. 3.4 ). (Stage 2) We use depth computed from the static reconstruction and camera poses to lift predicted 2D optical flow to 3D, and use (3D position, 3D scene flow) pairs at every frame to pre-train a static Eulerian motion field (Sec. 3.4 ). (Stage 3) To synthesize input images at each time step, we advect Canonical Gaussian Splats using our motion field through forward Euler integration with our looping algorithm (Sec. 3.2 , Alg. 1 ) and additionally apply a non-periodic residual field $D$ to account for the non-periodic nature of water (Sec. 3.3 ). Advected splats are rendered into synthesized images; they are compared to the ground truth via rendering losses that are back-propagated to optimize Canonical Gaussian Splat parameters and the motion field (Sec. 3.5 ). Given the trained loopable model, we can generate infinite looping animations by repeating every $L$ frames using Alg. 2 and setting $p_{n}=0$ .

Panel of Figure 3 — Figure 3 : Why per-Gaussian random start times? Five Gaussians at different canonical positions (dashed circles) are advected by a downward Eulerian motion field $\mathbf{V}$ (gray arrows) with cycle length $L{=}15$ . Each is $t_{n}=(i-t^{0}_{n})\bmod L$ steps below its canonical (rebirth) position and snaps back on rebirth ( $t_{n}{=}0$ ). (a) With a shared start time, every Gaussian has the same $t_{n}=i\bmod L$ and resets simultaneously at $i=0,15,\dots$ : the whole scene moves as one rigid clump (red band) and teleports back to the top in a single frame (jump artifact). (b) With random start times ( $t^{0}_{n}=0,3,6,9,12$ for the red , blue , green , violet , orange Gaussians) the resets are staggered across the cycle, so at any frame only a few Gaussians reset while the rest flow smoothly.

Panel of Figure 4 — Figure 4 : Qualitative comparison of reconstruction compared to baselines. Our method produces much sharper result than the baselines in the water regions. While MovieS appears sharp in row2, its overall view synthesis quality is poor ( refer to videos in the supp. material ).

	Full image						Water region
Method	PSNR ↑	SSIM ↑	LPIPS ↓	FID ↓	KID ↓	FVD ↓	PSNR ↑	SSIM ↑	LPIPS ↓	FID ↓	KID ↓	FVD ↓
MoVieS [ 15 ]	15.19	0.331	0.524	100.87	0.027	1564.90	15.35	0.291	0.526	87.21	0.014	1157.00
MoSca [ 11 ]	19.87	0.509	0.565	117.54	0.037	828.87	23.31	0.723	0.577	104.51	0.024	534.47
4DGS [ 39 ]	23.01	0.693	0.397	84.00	0.026	634.19	25.71	0.803	0.436	87.58	0.022	417.40
AmbGS [ 34 ]	22.52	0.706	0.344	61.32	0.014	536.55	25.49	0.806	0.392	69.15	0.014	314.24
Ours	23.05	0.716	0.315	39.63	0.007	210.01	25.37	0.798	0.375	46.50	0.008	156.69

Table 1: Quantitative comparison of methods.

\downarrow

(

\uparrow

) indicates lower (higher) is better. We report metrics over the full image as well as restricted to the water region, which isolates the dynamic content our method targets. FID, KID and FVD follow the distribution-based evaluation protocol of Lift4D [ 18 ] . Our method achieves the best overall reconstruction accuracy in comparison to the baselines and leads the baselines in spatial and temporal distribution alignment by a large margin.

Panel of Figure 5 — Figure 5 : User study. Bar height is the percentage of times users rated our method’s visual quality higher than the competing method (“Winrate”), for three settings: Appearance (camera moves, scene frozen), Motion (camera fixed, scene advances), and Reference (both advance, shown alongside the real footage). Aggregated over $25$ participants and $7$ scenes ( $2100$ judgments). Our method was preferred over all baseline methods in all settings.

Panel of Figure 6 — Figure 6 : Scene flow comparison. We visualize tracks formed by canonical Gaussian Splats by iteratively querying our Eulerian motion field with residual applied, on two scenes (rows), against 4DGS [ 39 ] , AmbGS [ 34 ] , MoSca [ 11 ] and MoVieS [ 15 ] . For the baselines, which do not expose an explicit Eulerian motion field, we extract the corresponding tracks by querying each method’s own motion representation at the same set of points and time steps. Our Eulerian motion field yields long, coherent trajectories that follow natural water flow, whereas the baselines produce short, incoherent, or near-static tracks.

Figure 7 : Visualizing temporal progression of water. We compare our method to the baselines at different time steps. Our method reproduces natural flow of water, while 4DGS [ 39 ] AmbGS [ 34 ] and MoSca [ 11 ] only demonstrate slight shift in color over time.

Cycle length	PSNR ↑	SSIM ↑	LPIPS ↓	FID ↓	KID ↓	FVD ↓
$L=35$	22.78	0.702	0.329	42.18	0.008	247.76
$L=25$	22.87	0.708	0.321	39.35	0.007	210.96
$\mathbf{L=15}$ (ours)	23.05	0.716	0.315	39.63	0.007	210.01

Table 2 : Cycle-length ablation. Effect of the cycle length

L

on reconstruction quality. Metrics follow the same protocol as Tab. 3 .

Method	PSNR ↑	SSIM ↑	LPIPS ↓	FID ↓	KID ↓	FVD ↓
w/o Eulerian motion field	23.42	0.733	0.319	50.86	0.011	422.66
w/o flow initialization	22.50	0.704	0.325	55.44	0.012	473.29
w/o non-periodic residual	22.97	0.713	0.321	47.22	0.010	309.37
Static reconstruction	23.11	0.723	0.319	60.88	0.015	522.97
w/o Alg. 1	22.86	0.702	0.329	47.01	0.010	334.30
Full model	23.05	0.716	0.315	39.63	0.007	210.01

Table 3 : Ablations of our method. Each row removes a single component of our full model or replaces it with a static reconstruction. Best per column is highlighted in red (bold) and second best in orange.

Panel of Figure 8 — Figure 8 : Ablation track visualization. Tracks formed by iteratively querying each ablation variant’s motion representation on two scenes, shown at the final frame of a 15-frame propagation. The full model produces coherent water-following trajectories; model with the Eulerian motion field or flow initialization removed barely reconstructs and water flow motion, and removing the non-periodic residual produces less natural and stochastic tracks.

Panel of Figure 9 — Figure 9 : Random start time prevents ”looping artifacts” For each scene we fix the camera, advance time for 60 frames (4 periods with 15 frames in each period), and stack a single horizontal scanline (red, in the sampled-row frames (b)) into an $x$ – $t$ image (c); time runs downward. GT (a) is the reference. Without random start times (bottom row per scene, “w/o Alg. 1”) the Gaussians pulse as one body, resulting in the slice being crossed by horizontal bands that span the full width at a single instant at loop boundaries. With random start times (top row per scene, “Ours”) water particles spread stochastically across loops. Per-frame metrics barely register this ( $-0.19$ dB PSNR), while FVD rises significantly.

Panel of Figure 10 — Figure 10 : Effect of changing cycle length $L=(15,25,35)$ . Columns vary the cycle length $L$ ; rows show two time steps in the loop. It can be observed that L=25 produces better dynamics (more variation in the waterfall states) compared to L=15, while L=35 results in blurrier overall reconstruction due to worse convergence.

Figure 11 : Extensions. With the help of SFM tools such as MegaSam [ 12 ] , our method can be extended to reconstruct motion of water surfaces, e.g . , lakes. We also demonstrate that our Eulerian motion field can represent other media such as fire by extending it to a time-varying model.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

	Alg. 1	Alg. 2	Speedup
15 frames ( $L=15$ , 1 cycle)
Deformation only	0.673 s (44.88 ms/frame)	0.224 s (14.93 ms/frame)	$3.01\times$
End-to-end (incl. raster)	0.719 s (47.95 ms/frame)	0.270 s (17.98 ms/frame)	$2.67\times$
450 frames ( $L=15$ , 30 cycles)
Deformation only	23.483 s (52.19 ms/frame)	5.605 s (12.46 ms/frame)	$4.19\times$
End-to-end (incl. raster)	28.232 s (62.74 ms/frame)	7.072 s (15.72 ms/frame)	$3.99\times$

Appendix

Table B1 : Animation run time. Wall-clock time to render 15 frames (

L=15

, 1 cycle) and 450 frames (

L=15

, 30 cycles), for both the pure deformation pass and end-to-end (deformation + rasterization), using the training-time propagation (Alg. 1) versus our inference-time propagation (Alg. 2). Reusing per-cycle propagations across all frames of a cycle yields a

\sim\!3

–

4\times

speedup, with a larger gain as the number of cycles grows.

Panel of Figure 15 — Figure B1 : The seven captured water scenes , each shown at the middle frame of its sequence. Captions give the number of frames (train/test), the maximum camera baseline, and the maximum rotation between viewing directions. Every scene is a hand-held fisheye capture rectified to a pinhole camera; test frames are three held-out temporal segments per scene.

Scene	Frames	Train	Test	Baseline	Path	Rotation	Water
Greenacre Park	750	660	90	7.0 m	15 m	111 ∘	34%
Garden	750	660	90	6.8 m	11 m	101 ∘	39%
Parley Falls	750	660	90	8.4 m	17 m	83 ∘	39%
Creek	750	660	90	3.4 m	11 m	93 ∘	38%
Sculpture	750	660	90	10.4 m	14 m	105 ∘	44%
Waterstep	800	710	90	12.1 m	16 m	101 ∘	63%

Appendix

Table B2 : Per-scene capture statistics. Baseline is the maximum distance between any two camera centres, Path the total trajectory length, Rotation the maximum angle between any two optical axes, and Water the mean fraction of pixels inside the water mask.

Figure C2 : Component ablations. Our full model reconstructs the best dynamism of water with all the proposed components. Without initializing the motion field or learning a position-based deformation field from scratch converges to very small motion (water particles become blurry strands), while no opacity and color deformation prevents our looping representation from fitting to the non-looping input video, producing blurry results.

Panel of Figure 18 — Figure D3 : User-study interface. (a) instructions shown at the start of the study and (b) forced-choice questions asked on every comparison page.

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Eulerian Motion Guidance: Robust Image Animation via Bidirectional Geometric Consistency

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Optical Flow Diffusion Models Geometric Guidance Motion

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Recent advancements in image animation have utilized diffusion models to breathe life into static images. However, existing controllable frameworks typically rely on Lagrangian motion guidance, where optical flow is estimated relative to the initial frame. This paper revisits the same optical-flow primitive through a more local supervision design: we use adjacent-frame Eulerian motion fields to guide generation, where the motion signal always describes a short temporal hop. This shift enables parallelized training and provides bounded-error supervision throughout the generation process. To mitigate the drift artifacts common in adjacent frame generation, we introduce a Bidirectional Geometric Consistency mechanism, which computes a forward-backward cycle check to mathematically identify and mask occluded regions, preventing the model from learning incorrect warping objectives. Extensive experiments demonstrate that our approach accelerates training, preserves temporal coherence, and reduces dynamic artifacts compared to reference-based baselines. The code, model, and data have been made available at https://nguyentthong.github.io/eulerian/

Abstract

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