GaussPDE: Graph-Based Partial Differential Equation-Driven Rendering for 3D Gaussian Splatting

1. INTRODUCTION

• GaussPDE evolves scalar fields over a propagation graph and modifies rendered appearance on static scenes, without requiring dynamic training data or material motion.The same evolving three-dimensional field is visualized from different viewpoints.Source: §1 · p. 1
• Unlike dynamic-scene methods that reconstruct temporal changes or physics-integrated Gaussian methods that synthesize motion and deformation, GaussPDE targets prescribed scalar-field evolution after reconstruction.This supports field visualization and propagation effects within captured scenes.Source: §1 · p. 1
• SuGaR regularizes primitives toward scene surfaces for mesh extraction, while Gaussian Grouping learns object identities for segmentation and editing; PDE-driven visualization additionally requires conductance specifying scalar-state exchange between primitives.Source: §1 · p. 1
• Reliable propagation is challenging because Gaussian scenes lack explicit adjacency, Euclidean neighborhoods can connect disconnected surfaces, and floaters or oversized splats can create unintended paths.Graph-based PDE behavior depends on the underlying connectivity.Source: §1 · p. 1
• GaussPDE combines covariance-aware connectivity with opacity, appearance, and lifted boundary conductance to define scene-conditioned scalar propagation.Camera-aware reconstruction and PDE-driven appearance updates are also part of the approach.Source: §1 · p. 1

NeRF

• Propagation is evaluated against an independent intrinsic-grid reference on two disconnected sheets with identical appearance, using matched in-sheet diffusion extent across methods.The comparisons include KNN, RBF, and the released Gaussian LBO implementation.Source: NeRF · p. 2
• GaussPDE performs PDE-driven dynamic rendering directly on static 3D Gaussian scenes without mesh extraction, voxelization, or further optimization during field evolution.The evolving field modifies Gaussian appearance while reconstructed geometry remains fixed.Source: NeRF · p. 2, Figure 2 · p. 2
• Camera-aware reconstruction regularization reduces camera-near floaters and oversized splats that can compromise PDE graph topology.CamReg is applied during reconstruction before the propagation graph is constructed.Source: NeRF · p. 2
• A boundary-aware Gaussian graph Laplacian uses covariance-aware neighborhoods and opacity, appearance, and boundary conductance for PDE evolution over irregular Gaussian primitives.The pipeline selects active Gaussians, constructs a sparse graph, and evolves scalar states while keeping graph connectivity fixed.Source: NeRF · p. 2

2. METHOD

• The pipeline applies CamReg during reconstruction, then selects active Gaussians, builds a sparse graph, and evolves scalar states while graph connectivity remains fixed.The graph and appearance interface can also accommodate wave or reaction–diffusion equations with appropriate state updates and numerical solvers.Source: §2 · p. 2, §2.3 · p. 3

2.1. Gaussian Graph Laplacian

• The graph uses covariance-aware neighborhoods and conductance factors for opacity, color similarity, and boundary scores; its fixed weights define a symmetric positive-semidefinite Laplacian with L1 = 0.Boundary scores from multi-view image or mask cues attenuate incident edges, providing soft suppression near selected object boundaries.Source: §2.1 · p. 2

2.2. Camera-Aware Regularization

• CamReg penalizes opaque primitives near training cameras and excessive projected footprints during reconstruction, targeting configurations that could become unintended graph nodes.Footprint penalties grow quadratically above the threshold, while pairs at or below it contribute zero.Source: §2.2 · p. 3

2.3. PDE Evolution and Rendering

• Mass-weighted heat diffusion evolves scalar states on active Gaussians, whose mapped colors update the zeroth-order spherical-harmonic coefficient while geometry, opacity, and higher-order coefficients remain unchanged.The masses are based on opacity and Gaussian volume; they are visualization weights, not calibrated material heat capacities or conductivities.Source: §2.3 · p. 3

3. EXPERIMENTS

• Experiments assess geometric separation against an independent synthetic PDE reference and examine the full visualization pipeline on reconstructed scenes.Scene examples span DTU, LLFF, and NeRF-Synthetic; reconstruction uses Splatfacto for 30K iterations, followed by CamReg, graph construction, and fixed-geometry PDE evolution.Source: §3 · p. 4, §3.1 · p. 4

3.1. Implementation and Datasets

• Scene graphs retain primitives above τα = 0.02, retrieve 50 Euclidean candidates, then select K = 10 Mahalanobis neighbors; operator comparisons share initialization and time-stepping settings.The controlled benchmark uses implicit Euler and selects states by diffusion extent.Source: §3.1 · p. 4

3.2. Controlled PDE Validation

• GaussPDE maintains low leakage across all four sheet-separation ratios, while KNN and RBF show substantial transfer and LBO leaks at g/h = 0.35.The comparison matches in-sheet diffusion extent against an independent grid reference.Source: §3.2 · p. 4

3.3. Scene Comparisons and CamReg Ablation

• On Ficus, GaussPDE colors the pot less than KNN and RBF when propagation is intended to remain within the plant region.This is a qualitative visualization-domain comparison, not a calibrated model of plant–pot heat transfer.Source: §3.3 · p. 4
• Scene-comparison images are interpreted qualitatively because response extent also depends on propagation rate and scalar-to-color mapping.The paper uses Sec. 3.2 for quantitative comparison at matched diffusion extent.Source: §3.3 · p. 4
• On the tested room and trex scenes, CamReg examples show fewer visible artifacts and a more localized PDE response than unregularized examples.The passage notes that effects on graph connectivity and propagation accuracy still need quantitative evaluation.Source: §3.3 · p. 4, Figure 4 · p. 4

• GaussPDE couples scalar-field evolution and novel-view rendering on static Gaussian scenes through camera-aware reconstruction, scene-conditioned conductance, and direct coupling of PDE states to appearance.Source: §4 · p. 4
• Controlled diffusion experiments demonstrate low cross-surface leakage and agreement with an independent grid reference.Source: §4 · p. 4
• Performance depends on reconstructed geometry and lifted visual cues; future work includes broader geometry and PDE validation, interactive source control, and extensions to geometry, opacity, and illumination dynamics.Source: §4 · p. 4