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GaussianSpline: Adaptive B\'ezier Splines for Highly Dynamic 4D Gaussian Splatting

Tom Yachil, Andrei Sharf

arXiv:2610.35224v1cs.GRcs.CV

TL;DR

Dynamic 4D Gaussian Splatting must capture complex motion while avoiding unnecessary temporal capacity in near-static regions. GaussianSpline models each Gaussian with adaptive, non-uniform piecewise cubic Bézier splines and a reflection-based G1 continuity constraint. Experiments report improved reconstruction on challenging, non-uniform motion and competitive performance on general dynamic scenes.

  • Problem

    Dynamic Gaussian Splatting must model complex temporal evolution while balancing flexible motion trajectories in dynamic regions against unnecessary temporal degrees of freedom in near-static regions.

    Source: §1 · p. 2

  • Method

    GaussianSpline jointly learns per-Gaussian Bézier control points and non-uniform temporal partitions, using a reflection-based G1 constraint to produce smooth trajectories.

    Source: §1 · p. 2

  • Results

    GaussianSpline improves reconstruction quality on complex, highly non-uniform motion and maintains competitive performance on general dynamic scenes compared with state-of-the-art 4D Gaussian Splatting methods.

    Source: §1 · p. 2

  • Takeaways & Limitations

    With a fixed segment budget, trajectories allocate temporal resolution to fast motion and use broader intervals for slow motion while integrating into standard 3DGS optimization and densification.

    Source: §5 · p. 7

    A fixed, small number of spline segments can struggle with highly transient dynamics such as rapid collisions that produce large accelerations over very short intervals.

    Source: §5 · pp. 7–8

Abstract

from arXiv · show

We present GaussianSpline, a dynamic 4D Gaussian Splatting method that models per-Gaussian motion using adaptive, non-uniform piecewise cubic B\'ezier splines. Unlike approaches based on global deformations, fixed-order polynomials, or uniformly parameterized splines, GaussianSpline jointly learns the B\'ezier control points and the temporal partition of each Gaussian's trajectory. This adaptive representation allocates greater temporal capacity to highly dynamic regions while maintaining compact trajectories for near-static regions, enabling high-fidelity reconstruction of complex and rapidly changing motion. To jointly achieve globally smooth motion and local adaptivity, we enforce $G^1$ continuity between consecutive B\'ezier segments using a reflection-based constraint on their control points. This produces smooth, fully differentiable trajectories while preserving the flexibility of non-uniform temporal parameterization. Experiments on the Neural 3D Video benchmark, the Google Immersive dataset, as well as on synthetic scenes demonstrate that GaussianSpline substantially improves reconstruction quality in scenes with highly non-uniform and challenging dynamics, while maintaining competitive performance on general dynamic scenes compared with state-of-the-art 4D Gaussian Splatting methods.

1. Introduction

Dynamic Gaussian Splatting must represent both scene structure and complex motion across potentially millions of Gaussians, while balancing flexible trajectories in dynamic regions against unnecessary temporal degrees of freedom in static ones. GaussianSpline jointly learns per-Gaussian Bézier control points and non-uniform segment intervals, with a reflection-based G1 constraint for smooth motion; experiments report improved reconstruction on challenging dynamics and competitive performance on general scenes.

1. Introduction

  • Dynamic Gaussian Splatting must capture spatial structure and complex temporal evolution across potentially millions of Gaussians without over-allocating motion capacity to near-static regions.Highly dynamic regions need flexible trajectories, while near-static regions need fewer temporal degrees of freedom.Source: §1 · p. 2
  • GaussianSpline jointly learns each Gaussian’s Bézier control points and non-uniform segment intervals, allocating finer temporal resolution to rapid motion and broader intervals to slow motion.Each Gaussian receives an independent trajectory initialized with a minimal number of spline segments.Source: §1 · p. 2
  • A reflection-based G1 continuity constraint with direction-locked tangents and learnable magnitude scaling produces smooth, differentiable trajectories while retaining flexibility in motion speed and curvature.The constraint is applied across spline segments.Source: §1 · p. 2
  • GaussianSpline uses no auxiliary neural networks and integrates into the standard 3DGS optimization and densification pipeline.The stated aim is a compact, efficient representation for highly non-uniform dynamics.Source: §1 · p. 2
  • Experiments report improved reconstruction quality on scenes with complex, highly non-uniform motion, while maintaining competitive performance on general dynamic scenes against state-of-the-art 4D Gaussian Splatting methods.Evaluations include Neural 3D Video, Google Immersive, and synthetic scenes.Source: §1 · p. 2

2. Related Work

Prior dynamic-scene methods use neural deformation, structured spacetime representations, or explicit trajectories, each with trade-offs in computation, resolution, or temporal flexibility. GaussianSpline differs from spline approaches with fixed control-point distributions by learning locally refined distributions based on motion complexity.

2. Related Work

  • Dynamic NeRF methods can produce high-quality free-viewpoint video, but volumetric ray marching requires hundreds of network evaluations per pixel, making training take hours to days and playback far from real time.Grid- and plane-based factorizations reduce this cost but retain fixed spatial-temporal resolution.Source: §2 · p. 2
  • Dynamic 3DGS methods include per-timestep Gaussian tracking and MLP-based deformation, with MLP approaches requiring per-Gaussian render-time evaluations that add computational overhead.SC-GS combines deformation networks with sparse spatial constraints to reduce inference cost.Source: §2 · pp. 2–3
  • Structured 3DGS methods such as 4DGS and 4K4D use grid-based embeddings, but discrete resolution creates a trade-off between memory usage and reconstruction fidelity.These methods model dynamic Gaussian attributes using structured spacetime representations.Source: §2 · p. 3
  • Explicit trajectory methods include polynomial trajectories in Spacetime Gaussians and a polynomial–Fourier dual-domain representation in Gaussian-Flow, both avoiding neural networks.Fixed-degree polynomials assign uniform temporal capacity and may struggle with complex non-linear motion patterns.Source: §2 · p. 3
  • SplineGS uses piecewise cubic Hermite splines with motion-adaptive control-point pruning for monocular video, while Yoon et al.’s SplineGS uses NURBS motion bases for multi-view scenes; both use uniform spline parameterizations.GaussianSpline instead learns control-point distributions according to motion complexity.Source: §2 · p. 3
  • GaussianSpline jointly learns adaptive, locally refined control-point distributions, allocating temporal capacity where needed and improving reconstruction quality with fewer Gaussian splats.The contrast is with spline approaches that use uniform parameterizations and fixed control-point distributions.Source: §2 · p. 3

3. Method

GaussianSpline represents each Gaussian’s motion with a jointly optimized piecewise cubic Bézier trajectory whose segment intervals adapt to local motion. It combines this temporal adaptivity with G¹-continuous transitions and a dynamic 3D Gaussian Splatting representation.

4D Gaussian Splatting for dynamic scenes

  • Only Gaussian position and rotation vary over time; scale, opacity, and color are shared across timestamps, with temporal variation focused on translation and rotation.This restriction is described as a compact yet flexible representation for motions approximated by piecewise locally rigid deformations.Source: §3 · p. 3

Non-uniform Cubic Bézier Spline Trajectories

  • Each Gaussian follows a piecewise cubic Bézier spline defined by joints, interior control points, and non-uniform interval widths, with all spline parameters differentiable and jointly optimized.The learned intervals form a monotonic partition of the normalized timeline; the active segment is evaluated using a local parameter mapped to [0,1).Source: §3 · pp. 3–4
  • Learnable non-uniform intervals allocate finer temporal resolution to rapid motion and broader intervals to slow motion, improving fidelity over uniformly parameterized splines.The interval widths are optimized jointly with spline joints and control points; near-static trajectories can remain approximately flat and consume little temporal capacity.Source: §3 · pp. 4–5

Learnable motion speed

  • Shared joints provide C⁰ continuity, while a reflection-based G¹ constraint preserves tangent direction and lets a learnable scale vary motion speed across segments.The per-junction scales start at 1, recovering C¹ continuity, and are jointly optimized with the spline parameters.Source: §3 · p. 4

Per-Gaussian polynomial rotations

  • Per-Gaussian rotation uses a first-degree polynomial raw quaternion whose normalized form maps to a rotation matrix.The base quaternion and its linear velocity define the polynomial.Source: §3 · p. 5

Training

  • Training evaluates Gaussians at sampled timestamps, compares differentiably rasterized views with ground truth, and backpropagates photometric loss to spline and Gaussian parameters.Adaptive densification splits, clones, or prunes Gaussians early in training; afterward, the Gaussian set remains fixed while spline parameters and Gaussian attributes are optimized.Source: §3 · pp. 3, 5, Figure 2 · p. 4

4. Experiments and Results

Across synthetic and real dynamic scenes, GaussianSpline is compared with three trajectory-based baselines and ranks first or second across tests and metrics. Ablations favor non-uniform joint spacing over increasing spline order.

4. Experiments and Results

  • On checker sphere and soft contact, GaussianSpline preserves moving-object geometry and boundaries that competing methods blur, distort, or merge.The qualitative comparisons cover two synthetic scenes with known ground-truth 3D motion.Source: §4 · p. 6
  • GaussianSpline ranks first or second across all tests and metrics, with notably stronger performance on synthetic scenes.It leads PSNR on all three real scenes and checker sphere, and ranks second on soft contact; it leads SSIM on both synthetic scenes, coffee martini, and flames, and LPIPS on both synthetic scenes and flames.Source: Table 1 · p. 7, §4 · p. 7
  • In cook spinach, coffee martini, and flames, GaussianSpline retains sharper scene details than the compared methods.For cook spinach, adaptive partitioning concentrates temporal resolution around the dog's localized head motion; the coffee martini cap and tip and the flame structure, hand, and can are also better preserved.Source: §4 · pp. 6–7
  • Learnable non-uniform joint spacing consistently matches or beats uniform spacing across spline orders, while increasing spline order does not improve reconstruction quality.The lowest-order spline with non-uniform spacing achieves the best overall results, and the spacing gap narrows at higher orders.Source: §4 · p. 7, Table 2 · p. 7

5. Conclusions and limitations.

GaussianSpline combines adaptive per-Gaussian Bézier trajectories with smooth segment transitions and reports strong results, especially for complex, non-uniform motion. Its limitations include highly transient dynamics, uncoupled per-Gaussian trajectories, and a restricted evaluation set.

5. Conclusions and limitations.

  • Each Gaussian uses a piecewise cubic Bézier trajectory with learnable temporal partitions; a reflection-based G1 constraint smooths motion while retaining local speed flexibility, without auxiliary networks.The method integrates into the standard 3DGS optimization and densification pipeline.Source: §5 · p. 7
  • Non-uniform spacing matches or outperforms uniform spacing across spline orders, while higher order adds little; GaussianSpline leads PSNR on real N3DV and Google Immersive scenes and performs best overall on checker sphere.The reported transient-motion limitation and evaluation scope qualify these results.Source: §5 · p. 7
  • Highly transient motion, such as rapid collisions with large accelerations over very short intervals, can be difficult to capture with a fixed, small segment budget.Adaptive segment insertion guided by local curvature or reconstruction residuals is proposed as a future direction.Source: §5 · pp. 7–8
  • Trajectories are optimized independently without spatial motion priors or coupling, so rigid and articulated structures must be learned through individual Gaussian trajectories.Topological changes are handled implicitly through opacity and densification.Source: §5 · p. 8
  • Evaluation covers publicly available N3DV scenes, one Google Immersive scene, and two synthetic sequences, with each baseline using its default optimization schedule.The authors note that broader evaluation is desirable, while diverse densely sampled multi-view datasets with accurate ground truth remain scarce.Source: §5 · p. 8

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