title: >- [Paper Note] Parametric SDF for Dynamic Surface Reconstruction description: >- [ECCV 2026][3d_vision][Dynamic Surface Reconstruction] Generalizes static SDF fields into parametric polynomial-Fourier temporal trajectories with differentiable iso-surfacing and PBR rendering for high-fidelity dynamic mesh recovery. tags: - ECCV 2026 - 3d_vision - Dynamic Surface Reconstruction - Parametric SDF - Differentiable Rendering date: 2026-09-19 content_hash: 0414cf4a2d907b68

Parametric SDF for Dynamic Surface Reconstruction¶

Conference: ECCV 2026
Paper: ECCV Official
Code: https://timchonggao.github.io/psdf
Area: 3D Vision
Keywords: Dynamic Surface Reconstruction, Parametric SDF, Differentiable Rendering, FlexiCubes, PBR Material Decomposition

TL;DR¶

By reformulating static Signed Distance Functions into time-dependent parametric trajectories parameterized by hybrid polynomial-Fourier bases, p-SDF couples differentiable iso-surfacing with physically based differentiable rendering to achieve robust, high-fidelity dynamic surface reconstruction and material disentanglement under large non-rigid deformations and topological changes.

Background & Motivation¶

Reconstructing high-fidelity dynamic surfaces from multi-view video captures is a cornerstone problem in computer vision and graphics, powering simulations, visual effects, augmented reality, and robotics. Recent advances in neural implicit representations (such as NeRF) and 3D Gaussian Splatting have revolutionized novel view synthesis, achieving photorealistic visual quality across continuous viewpoints. However, because these radiance-centric representations focus almost entirely on optimizing view-dependent appearance, their extracted underlying geometries often suffer from severe surface noise, geometric flickering, and disconnected floaters, making them ill-suited for downstream physics simulation or interactive mesh editing.

Existing dynamic surface reconstruction methodologies generally encounter fundamental bottlenecks tied to their underlying paradigms. Lagrangian formulations track dynamic deformations by mapping a canonical static template to target frames using deformation fields; however, their reliance on a fixed-topology template and the oversmoothing tendency of coordinate MLPs fundamentally prevents them from handling topological changes (such as tearing or merging) and extreme non-rigid motions. Conversely, incremental optimization approaches optimize dynamic scenes sequentially, but suffer from catastrophic error accumulation when view coverage is sparse. On the other hand, Eulerian spatio-temporal volume factorizations (such as 4D tensor planes) bypass explicit tracking and can accommodate topological variations, but their lack of temporal inductive biases frequently results in severe temporal flickering and degenerate sliver triangles.

This paper tackles the challenge by observing the physical evolution of dynamic geometry from a continuous Eulerian perspective: while 3D objects undergo non-rigid deformation and topological shifts, the Signed Distance Function (SDF) value at any fixed spatial location evolves along a continuous and well-behaved 1D temporal trajectory. The core idea is to parameterize static grid-based SDFs into compact, time-dependent parametric curves using hybrid polynomial and Fourier basis functions (p-SDF), turning dynamic surface tracking into a smooth 1D temporal signal fitting problem optimized end-to-end via differentiable iso-surfacing and physically based rendering.

Method¶

Overall Architecture¶

The p-SDF framework takes calibrated multi-view RGB video streams, foreground masks, and camera parameters as input, producing time-evolving explicit triangular meshes alongside disentangled PBR material properties and environmental illumination. At each timestamp \(t\), the system queries analytical parametric basis functions at each node of a bounded 3D spatial grid to instantiate an instantaneous static SDF field; differentiable iso-surfacing via FlexiCubes is then executed to extract an explicit triangular mesh \(M_t = (V_t, F_t)\); simultaneously, a temporal tri-plane appearance field is sampled at visible surface points to predict diffuse albedo, metallic, roughness, and self-occlusion attributes; finally, these geometric and material attributes are rendered through a physically based differentiable rasterizer, allowing photometric and regularizing gradients to propagate end-to-end to update all trajectory coefficients, appearance hash grids, and lighting parameters.

%%{init: {'flowchart': {'rankSpacing': 24, 'nodeSpacing': 28, 'padding': 6, 'wrappingWidth': 400}}}%%
flowchart TD
    A["Multi-view Video Stream & Timestamp t"] --> B["Parametric SDF Trajectory Evaluation<br/>Polynomial & Fourier Basis Interpolation"]
    B --> C["Differentiable Iso-surfacing<br/>FlexiCubes Mesh Extraction Mt"]
    A --> D["Temporal Tri-plane Appearance Field<br/>4D Decomposed Spatio-Temporal Hash Grids"]
    C --> E["Visible Surface Position & Normal Rasterization"]
    D --> F["PBR Attribute Decoding<br/>Albedo / Metallic / Roughness / Occlusion"]
    E --> G["Differentiable PBR Shading & Rendering<br/>Disentangled Illumination & Materials"]
    F --> G
    G --> H["End-to-End Gradient Backpropagation<br/>Photometric Loss + Trajectory Smoothness"]

Key Designs¶

1. Parametric SDF Trajectory Representation: Decoupling Smooth Drift from High-Frequency Dynamics To circumvent the limitations of canonical deformation MLPs and computationally brittle numerical ODE flow integration, p-SDF represents dynamic scene geometry across a bounded 3D spatial grid (default resolution \(96^3\)). Rather than storing a static scalar distance value at each node \(p_i\), the representation defines the signed distance as a continuous analytical function of time \(s(p_i, t)\) using a hybrid basis formulation:

\[s(p_i, t) = \sum_{n=1}^{N_p} \mu_{i,n} t^n + a_{i,0} + \sum_{n=1}^{N_f} \left( a_{i,n} \cos(\omega_n t) + b_{i,n} \sin(\omega_n t) \right)\]

where the polynomial basis (\(N_p = 6\)) models low-frequency global translations and smooth rigid-like drifting trends, while the Fourier basis (\(N_f \in [18, 100]\)) captures rapid periodic deformations and high-frequency geometric transitions. The entire continuous dynamic geometry is compactly parameterized by a coefficient matrix \(W \in \mathbb{R}^{N^3 \times (N_p + 2N_f + 1)}\). This analytical trajectory structure intrinsically embeds temporal continuity without relying on numerical integration, providing robustness against topological transitions while eliminating drift accumulation.

2. Differentiable Iso-surfacing: Enabling Direct Explicit Mesh Supervision Upon evaluating per-node SDF values at timestamp \(t\), the framework leverages FlexiCubes to extract the explicit surface mesh \(M_t = (V_t, F_t)\). Unlike standard Marching Cubes or post-hoc Poisson surface reconstruction, FlexiCubes incorporates differentiable cell configurations and dual-marching-cubes-based gradient propagation, enabling gradients computed from 2D rendered image discrepancies to flow directly into surface vertex positions \(V_t\) and backward into grid trajectory parameters \(W\). This end-to-end coupling directly optimizes the geometric boundary, eliminating the internal cavities and needle-like sliver triangles commonly produced by point-cloud or Gaussian surfel meshing pipelines.

3. Temporal Tri-plane Appearance Field: Low-Rank Dynamic PBR Material Modeling Because RGB appearance signals undergo sharp, non-smooth variations at stationary coordinates due to specular highlights, moving shadows, and dynamic occlusions, appearance cannot be cleanly approximated by smooth low-order temporal curves. To resolve appearance-geometry ambiguities, the framework introduces three multi-resolution spatio-temporal hash grids covering orthogonal domains: \(G_{xyt}\), \(G_{xzt}\), and \(G_{yzt}\). For any extracted surface coordinate \((x, y, z)\) at time \(t\), features are retrieved via parallel trilinear interpolation, concatenated, and processed by a lightweight MLP decoder \(\Phi_{\text{app}}\) to predict physically based rendering attributes \((c, m, r, o)\)—representing diffuse albedo, metallic, roughness, and ambient occlusion. This decomposed representation prevents 4D memory explosion while enabling clean disentanglement between surface geometry and material reflectance under dynamic lighting.

Loss & Training¶

The entire network is trained end-to-end under a unified objective balancing photometric accuracy and physical priors:

\[\mathcal{L}_{\text{total}} = \mathcal{L}_{\text{photo}} + \lambda_1 \mathcal{L}_{\text{curve}} + \lambda_2 \mathcal{L}_{\text{app}}\]

where \(\mathcal{L}_{\text{photo}}\) measures multi-view color reconstruction error (\(L_1\) and perceptual loss, combined with silhouette mask supervision), \(\mathcal{L}_{\text{curve}}\) enforces spatial Laplacian smoothness across basis coefficients of neighboring grid nodes to ensure spatial coherence in motion trajectories, and \(\mathcal{L}_{\text{app}}\) regularizes material roughness and environment illumination. The model is trained on a single NVIDIA RTX 4090 GPU for 5,000 iterations using the Adam optimizer with exponential learning rate decay, converging in approximately 1 to 3 hours per scene.

Key Experimental Results¶

Main Results¶

The method is comprehensively evaluated across the synthetic non-rigid benchmark SynMotion-360, the real-world human dynamic capture CMU Panoptic Studio, and the multi-view tabletop dynamic dataset DiVa-360. Geometric fidelity is measured via Chamfer Distance (CD), F1-Score, Edge Chamfer Distance (ECD), Edge F1-Score (EF1), and mesh structural integrity metrics—Aspect Ratio sliver rate (AR) and Minimum Angle sliver rate (MA). Novel view synthesis quality is reported using PSNR, SSIM, and LPIPS.

Dataset	Metric	Ours (p-SDF)	Prev. SOTA (AT-GS / NeuS2)	Gain / Analysis
SynMotion-360	CD ↓	1.157	1.203 (NeuS2) / 1.602 (AT-GS)	Outperforms implicit SDF baselines and substantially surpasses Gaussian surfels
SynMotion-360	ECD ↓	3.170	3.220 (NeuS2) / 3.227 (AT-GS)	Superior sharp edge reconstruction along thin moving boundaries
SynMotion-360	EF1 ↑	0.596	0.593 (NeuS2) / 0.566 (AT-GS)	Highest boundary fidelity and edge precision across non-rigid sequences
CMU Panoptic	CD ↓	0.026	1.041 (DG-Mesh) / 6.525 (AT-GS)	Avoids sequential error drift under sparse views; achieves an order-of-magnitude error reduction
CMU Panoptic	F1 ↑	0.896	0.788 (DG-Mesh) / 0.687 (AT-GS)	Reconstructs fine human limbs and thin chair legs missed by baselines (+10.8% over DG-Mesh)
CMU Panoptic	AR ↓	4.983	19.359 (AT-GS) / 21.631 (NeuS2)	Sliver triangles reduced to less than one-fourth of baseline rates
CMU Panoptic	MA ↓	0.678	11.353 (AT-GS) / 12.563 (NeuS2)	Severe small-angle triangle degeneracy drops from over 11% to below 0.7%
DiVa-360	AR ↓	3.126	19.234 (AT-GS) / 22.530 (NeuS2)	End-to-end explicit mesh extraction delivers clean, regular surfaces on real objects
DiVa-360	MA ↓	0.396	11.206 (AT-GS) / 12.976 (NeuS2)	Degenerate faces virtually eliminated on real-world captures
DiVa-360	PSNR ↑	27.584	27.418 (AT-GS) / 26.087 (NeuS2)	Establishes state-of-the-art photometric reconstruction on real tabletop scenes
DiVa-360	LPIPS ↓	0.049	0.067 (AT-GS) / 0.056 (NeuS2)	Lowest perceptual error with sharpest texture detail recovery

Ablation Study¶

The paper systematically analyzes basis function configurations, spatial grid resolutions, and temporal Fourier bandwidths (\(N_f\)) on dynamic sequences:

Configuration Variant	PSNR (dB) ↑	Key Empirical Findings
Polynomial only (\(N_p=6, N_f=0\))	25.91	Severely underfits dynamic high-frequency motions, resulting in heavy motion blur and geometric collapse
Fourier only (\(N_p=0, N_f=36\))	35.90	Captures cyclic variations well, but lacks baseline drift stability, producing low-frequency spatial oscillations
Full Mixed Basis (\(N_p=6, N_f=36\))	36.34	Combining low-order polynomials with harmonic bases delivers optimal spatio-temporal accuracy
Grid resolution \(64^3\)	Lower	Coarse discretization fails to resolve thin structures (e.g., antlers, chair legs) due to spatial undersampling
Grid resolution \(96^3\) (Default)	Optimal	Striking an ideal balance between reconstruction fidelity and GPU memory footprint (19.7 GB VRAM)
Grid resolution \(128^3\)	Marginal Gain	Substantially escalates memory and extraction overhead with diminishing returns in geometric precision

Key Findings¶

Crucial Role of Hybrid Basis Formulation: Eliminating Fourier bases causes PSNR to plummet from 36.34 dB to 25.91 dB (-10.43 dB), demonstrating that low-degree polynomials alone cannot resolve complex non-rigid deformation. The combination ensures smooth baseline tracking without sacrificing high-frequency temporal responsiveness.
Order-of-Magnitude Improvement in Mesh Regularity: Metrics quantifying degenerate triangles (AR and MA) demonstrate dramatic improvements (e.g., AR drops from 19.36 to 4.98 on CMU Panoptic, and MA drops from 11.21 to 0.40 on DiVa-360). Because FlexiCubes directly couples 2D image gradients to mesh vertex positions, p-SDF eliminates the jagged, needle-like sliver faces inherent to heuristic post-processing methods like TSDF fusion or Poisson reconstruction.
Geometry-Appearance Decoupling vs. Gaussian Overfitting: While AT-GS achieves marginally higher synthetic PSNR (32.67 dB vs 32.15 dB) via adaptive Gaussian densification, its geometric Chamfer Distance is significantly worse (1.602 vs 1.157). Gaussian primitives overfit image pixels by scattering arbitrary floaters in free space; in contrast, p-SDF enforces physically plausible surface manifolds while securing superior perceptual fidelity (LPIPS 0.028 vs 0.045).

Highlights & Insights¶

Formulating 4D Dynamics as 1D Grid Signals: Instead of learning complex 3D non-rigid coordinate deformation fields or integrating neural velocity vectors over time, the method exploits the continuous temporal variation of SDF values at stationary spatial locations, reducing 4D dynamic surface reconstruction to parallel 1D curve fitting.
End-to-End Explicit Surface Differentiable Rendering: Integrating FlexiCubes with an nvdiffrast-based differentiable PBR pipeline creates a direct gradient pathway from multi-view image loss into explicit mesh vertices, providing explicit surface regularities and avoiding the noisy artifacts of purely implicit fields.
Seamless Handling of Severe Topological Changes: By representing surfaces as the zero-level set of a continuous time-dependent field rather than deforming a fixed-topology canonical mesh, the representation effortlessly models complex topological tearing, merging, and hole closures without numerical instability.

Limitations & Future Work¶

GPU Memory Footprint: Storing full temporal basis coefficients across a dense \(96^3\) grid requires 19.7 GB of peak VRAM during training. Extending the method to arbitrarily long sequences will necessitate temporal chunking, dynamic octrees, or sparse hash representations.
Absence of Point-to-Point Vertex Correspondences: As an Eulerian representation where surfaces are extracted anew per frame, p-SDF does not inherently provide dense Lagrangian vertex tracking trajectories across time, requiring downstream non-rigid registration for correspondence tracking.
Sensitivity to Multi-View Completeness: While temporal basis curves provide a smoothness prior that prevents catastrophic collapse, geometry degradation can still emerge under extreme monocular or heavily occluded camera configurations.

vs NeuS2 / AT-GS (Incremental Optimization): Incremental methods optimize sequences frame-by-frame via historical state propagation, making them highly vulnerable to cumulative error drift under complex dynamic motions; p-SDF optimizes all temporal frames jointly using global parametric trajectories, guaranteeing temporal stability without drifting.
vs DG-Mesh / Dynamic 2DGS (Canonical Global Deformation): Canonical template approaches rely on a fixed topology deformed by continuous coordinate MLPs, failing completely when surfaces tear, interact, or change genus; p-SDF extracts geometry dynamically from zero-level sets, making topological transitions seamless.
vs SDFFlow (Neural Vector Integration): SDFFlow tracks SDF dynamics by integrating learned neural velocity fields, which is computationally prohibitive and prone to error accumulation; p-SDF evaluates analytical basis curves in \(O(1)\) time, delivering superior scalability and clean gradient backpropagation.

Rating¶

Novelty: ⭐⭐⭐⭐⭐ [Generalizing Eulerian grid SDFs into analytical polynomial-Fourier parametric temporal curves provides an elegant, effective solution for dynamic surface modeling]
Experimental Thoroughness: ⭐⭐⭐⭐⭐ [Evaluated across synthetic non-rigid datasets, real-world human motion, and tabletop captures, reporting extensive geometric, topological, and perceptual metrics]
Writing Quality: ⭐⭐⭐⭐⭐ [Thoroughly motivated, clearly structured, mathematically precise, with insightful pipeline diagrams and qualitative analyses]
Value: ⭐⭐⭐⭐⭐ [Offers a robust, principled foundation for dynamic 3D surface reconstruction and physically grounded downstream relighting applications]