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Reconstruction and Simulation of Elastic Objects with Spring-Mass 3D Gaussians

Licheng Zhong, Hong-Xing Yu, Jiajun Wu, Yunzhu Li

arXiv:2403.09434v3cs.CV

TL;DR

Spring-Gaus addresses the challenge of reconstructing and simulating elastic objects when visual methods do not estimate their physical properties. It integrates a 3D Spring-Mass model with 3D Gaussian kernels, and evaluations on synthetic and real-world datasets demonstrate accurate reconstruction, simulation, and future prediction. The framework is currently limited to elastic objects because its spring lengths remain fixed during simulation.

  • Problem

    Visual methods can model object appearance and geometry but do not adequately identify physical dynamics for reconstructing and simulating elastic objects.

    Source: §1 · pp. 1–2

  • Method

    Spring-Gaus integrates a learnable 3D Spring-Mass model with 3D Gaussian kernels and separates appearance and physics learning.

    Source: §1 · pp. 2–3

  • Results

    Evaluations on synthetic and real-world datasets demonstrate reconstruction and simulation of geometry, appearance, and physical dynamic properties, including short-term future prediction.

    Source: §1 · p. 3, §4.3 · p. 13, §5 · p. 14

  • Takeaways & Limitations

    Spring-Gaus provides simulatable digital assets of elastic objects and supports prediction under different environmental conditions and initial states.

    Source: §1 · p. 3, §5 · p. 14

    Spring-Gaus is constrained to elastic objects because its spring lengths are fixed at the start of dynamic simulation.

    Source: D Limitations and Future Work · p. 20

Abstract

from arXiv · show

Reconstructing and simulating elastic objects from visual observations is crucial for applications in computer vision and robotics. Existing methods, such as 3D Gaussians, model 3D appearance and geometry, but lack the ability to estimate physical properties for objects and simulate them. The core challenge lies in integrating an expressive yet efficient physical dynamics model. We propose Spring-Gaus, a 3D physical object representation for reconstructing and simulating elastic objects from videos of the object from multiple viewpoints. In particular, we develop and integrate a 3D Spring-Mass model into 3D Gaussian kernels, enabling the reconstruction of the visual appearance, shape, and physical dynamics of the object. Our approach enables future prediction and simulation under various initial states and environmental properties. We evaluate Spring-Gaus on both synthetic and real-world datasets, demonstrating accurate reconstruction and simulation of elastic objects. Project page: https://zlicheng.com/spring_gaus/.

1 Introduction

Spring-Gaus addresses the difficulty of identifying elastic-object dynamics from vision by integrating an expressive, efficient 3D Spring-Mass model with 3D Gaussians. It reconstructs appearance, geometry, and physical properties from multi-view videos and supports future prediction and simulation.

1 Introduction

  • Existing dynamic 3D representations capture temporal appearance and geometry but generally do not estimate physical properties or predict future object dynamics.Physics-based alternatives can be limited by known material models, global parameters, computational cost, or representation resolution.Source: §1 · pp. 1–2
  • The approach targets expressive modeling of collisions, deformations, and bouncing while remaining efficient for gradient-based inverse physical-parameter optimization.Its differentiable design supports inverse optimization of physical parameters.Source: §1 · p. 2
  • Spring-Gaus is a 3D object representation integrating a 3D Spring-Mass model to represent elastic-object dynamics through learnable mass points and springs, including learnable topology and physical parameters.It can model complex deformation and motion for heterogeneous elastic objects and supports efficient inverse optimization of physical parameters because it is differentiable.Source: §1 · p. 2
  • The reconstruction pipeline decouples appearance and geometry optimization from physical-dynamics reconstruction for more effective optimization.It requires only a few multi-view videos for physical parameter identification and is designed to remain robust to geometry-reconstruction quality.Source: §1 · pp. 2–3
  • Spring-Gaus is evaluated on synthetic and real-world datasets, demonstrating accurate reconstruction and simulation of elastic objects.The evaluation also covers future prediction and simulation under varying initial states and environmental conditions.Source: Figure 1 · p. 2, §1 · p. 3

2 Related Work

Prior work develops neural, Gaussian, and physics-based representations for reconstructing or simulating dynamic objects. Spring-Gaus differs by targeting elastic-object system identification from raw videos with an expressive and efficient Spring-Mass model.

3D Object Representations

  • Existing 3D representations include implicit neural functions, voxel grids, and 3D Gaussian Splatting, but Spring-Gaus focuses on reconstructing simulatable 3D objects.Related methods include NeRF successors, DirectVoxGO, and DreamGaussian.Source: §2 · p. 3
  • PhysGaussian supports forward simulation of reconstructed Gaussian objects, while Spring-Gaus additionally focuses on system identification from raw videos and inverse physical reconstruction.The comparison distinguishes forward simulation from inverse reconstruction of physical objects.Source: §2 · p. 3

Dynamic Scene Reconstruction

  • Dynamic NeRF and Gaussian methods model temporal scene changes, while PAC-NeRF learns physical parameters with an MPM-based representation for dynamic reconstruction and prediction.PAC-NeRF is identified as the most relevant comparison in this group.Source: §2 · p. 4
  • PAC-NeRF assumes known material models, whereas Spring-Gaus uses an expressive and efficient 3D Spring-Mass model for real elastic-object reconstruction and simulation.The stated contrast concerns adaptability to complex real objects.Source: §2 · p. 4

Physics-Informed Learning

  • DANO provides differentiable reconstruction and simulation for rigid NeRF objects from a few videos, whereas Spring-Gaus addresses elastic objects with different physical-modeling challenges.The paper frames elastic-object modeling as requiring a redesign of the physical models used for rigid objects.Source: §2 · p. 4

3 Approach

Spring-Gaus combines 3D Gaussian appearance and geometry with a differentiable 3D Spring-Mass model, reconstructing elastic objects from multiview videos while enabling simulation and physical-parameter optimization. Its pipeline separates static reconstruction, Gaussian refinement, dynamic simulation, and staged optimization.

3.1 Appearance and Geometry Representation

  • 3D Gaussian kernels represent appearance and shape through centers, isotropic covariance controlled by a scalar scale, color, and opacity, then project and render into images.The splatting process projects kernels into camera space before rendering them in image space.Source: §3.1 · p. 5

3.2 Physics-Based Dynamics with 3D Spring-Mass Model

  • Spring-Gaus integrates learnable masses, springs, topology, and physical parameters with 3D Gaussian kernels to model heterogeneous elastic objects.The model is designed to remain expressive while supporting efficient gradient-based physical-parameter estimation.Source: §3.2 · p. 5
  • Volume sampling creates anchor points, which connect to nearest neighbors through springs characterized by stiffness and damping to simulate object dynamics.Kernel centers inherit simulated motion from anchors through inverse-distance-weighted interpolation.Source: §3.2 · pp. 5–7
  • Anchor dynamics combine spring, damping, and gravity forces, use semi-implicit Euler integration with boundary conditions, and support nonlinear spring behavior through p_k.When p_k is zero, the spring force becomes Hooke’s law; positive values produce a nonlinear spring-length dependence.Source: §3.2 · pp. 6–7
  • The soft vector modulates connected springs because larger n_k makes the simulated object more rigid while smaller n_k produces softer behavior.A learnable κ controls this modulation across anchors.Source: §3.2 · pp. 7–8

3.3 Optimization

  • Optimization uses a weighted L1 and D-SSIM rendering loss, stages static reconstruction, Gaussian refinement, and dynamic reconstruction, and updates physical parameters at selected keyframes.Gaussian appearance parameters are refined after interpolation-induced drift, while differentiable simulation and rendering optimize the physical parameters.Source: §3.3 · pp. 8–9

4 Experiment

Spring-Gaus is evaluated on synthetic and real-world data for reconstruction, future prediction, and simulation under changed physical and environmental conditions. The experiments emphasize geometric accuracy, faithful dynamics, and adaptation to new conditions.

4.2 Implementation Details

  • The real-world pipeline registers separately reconstructed static and dynamic scenes because sparse views are insufficient for direct dynamic Gaussian reconstruction.Registration optimizes scale, translation, and rotation, while mask-center and perceptual losses reduce sensitivity to color discrepancies.Source: §4.2 · p. 10, Figure 3 · p. 10, §4.3 · p. 13

4.1 Datasets

  • Spring-Gaus evaluates synthetic sequences using 20 training frames, 10 future-prediction frames, and comparisons with PAC-NeRF, Dynamic 3D Gaussians, and 4D Gaussian Splatting.The synthetic dataset contains fourteen models, ten viewpoints, and thirty frames per sequence.Source: §4.1 · p. 9, §4.3 · p. 11

4.3 Qualitative and Quantitative Results

  • On synthetic data, Spring-Gaus maintains geometry and appearance while reconstructing reasonable dynamics relative to PAC-NeRF [18], Dynamic 3D Gaussians [26], and 4D Gaussian Splatting [44].Dynamic reconstruction is quantitatively evaluated with Chamfer Distance and Earth Mover’s Distance.Source: §4.3 · p. 11, Table 2 · p. 11, Figure 4 · p. 12
  • Spring-Gaus outperforms PAC-NeRF across CD, EMD, PSNR, and SSIM for future prediction while maintaining faithful reconstructions close to ground truth.The reported results also support efficient gradient-based identification of the physical parameters.Source: Table 1 · p. 11, §4.3 · p. 13, Figure 5 · p. 13

4.4 Generalization to New Conditions

  • Real-world predictions closely follow observations, and the method supports simulation with altered ground position, ground stickiness, object properties, initial velocities, and gravity.These demonstrations treat the reconstructed object as a digital asset for unseen environmental conditions.Source: §4.4 · p. 13, §4.3 · p. 13, Figure 6 · p. 14

5 Conclusion

Spring-Gaus acquires simulatable digital assets of elastic objects from video by combining Gaussian appearance representation with physical dynamics learning. Experiments show reconstruction and short-term prediction under varied environmental conditions.

5 Conclusion

  • Spring-Gaus integrates a 3D Spring-Mass model with 3D Gaussians to reconstruct geometry, appearance, and physical dynamic properties.Appearance and physics are learned separately to avoid optimization interference.Source: §5 · pp. 14–15
  • The method predicts short-term future dynamics under different environmental conditions, demonstrating identification of physical properties from observational data.The reported evaluation covers both synthetic and real-world datasets.Source: §5 · pp. 14–15

A Data flow

The pipeline separates appearance learning from physical learning and transfers simulated anchor motion back to Gaussian centers. This design clarifies how visual reconstruction and dynamics simulation interact.

B Data flow

  • Appearance learning optimizes Gaussian kernels for static image rendering, while physics learning optimizes physical parameters over sampled anchors.Anchors A are sampled from Gaussian centers X.Source: Figure 8 · p. 18, A Data flow · p. 18
  • After simulation, Gaussian centers at time t are interpolated from the updated anchor positions A_t.The interpolation reconnects simulated particle motion with the rendered Gaussian representation.Source: Figure 8 · p. 18, A Data flow · p. 18

B More Implementation Details

Implementation uses a fixed particle-and-spring configuration with staged velocity and Gaussian optimization. An ablation indicates that per-particle physical parameters are needed for heterogeneous objects.

B More Implementation Details

  • Spring-Gaus uses 2048 anchors, 256 spring neighbors per anchor, mass m_0 = 1, damping ζ_0 = 0.1, and 300 dynamic-reconstruction iterations.The experiments use one RTX 3090 GPU and a nonlinear spring force with p_k = 0.5.Source: B More Implementation Details · p. 18
  • The initial velocity vector v_0 is independently optimized from a few frames captured before environmental interaction.This follows the practice used by PAC-NeRF.Source: B More Implementation Details · p. 18
  • Uniform Gaussian scaling during static reconstruction produces a more evenly distributed point cloud and anchor set, improving dynamic simulation.During refinement, each Gaussian receives its own scaling scalar s_i rather than retaining the shared s_0.Source: B More Implementation Details · pp. 18–19
  • Optimizing one global physical parameter cannot accurately model heterogeneous objects with complex physical properties.The ablation motivates optimizing physical parameters for individual particles.Source: Figure 9 · p. 19

C Ablation Study

The ablation study highlights per-anchor physical parameter optimization and the soft vector η as key design choices for modeling heterogeneous elastic objects.

C Ablation Study

  • Optimizing parameters for each anchor point increases the model’s freedom to represent object physics.Source: Table 3 · p. 19
  • The soft vector η provides a more flexible formulation by regulating the quantity and intensity of springs linked to anchor points.Source: Table 3 · p. 19, C Ablation Study · p. 20
  • Spring-Gaus models heterogeneous objects with distinct physical properties across segmented sections, unlike PAC-NeRF’s difficult per-particle optimization.Its sparse-anchor formulation enables individual optimization of physical parameters for each anchor point.Source: C Ablation Study · p. 20

D Limitations and Future Work

The authors identify scope limitations involving material behavior, interaction settings, and evaluation coverage, and propose corresponding directions for future work.

D Limitations and Future Work

  • Fixed spring lengths constrain Spring-Gaus to elastic objects, motivating dynamic spring-length and connectivity changes for plastic deformation.Source: D Limitations and Future Work · p. 20
  • The current method models a single object colliding with the ground, leaving multi-object interaction and more complex boundary conditions for future work.Source: D Limitations and Future Work · p. 20
  • A more comprehensive real-world dataset should include higher spatial and temporal resolution, diverse objects and materials, varied interactions, and challenging visual conditions.Source: D Limitations and Future Work · p. 20

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