TetraSDF: Precise Mesh Extraction with Multi-resolution Tetrahedral Grid

TL;DR

TetraSDF addresses the challenge of extracting meshes that exactly match the zero-level set of neural SDFs by preserving CPWA structure through a multi-resolution tetrahedral positional encoder with barycentric interpolation. It introduces a fixed input preconditioner to whiten encoder-induced bias and develops an analytic mesh-extraction pipeline that jointly tracks encoder-induced polyhedral cells and ReLU MLP linear regions. The method yields highly self-consistent meshes with improved SDF fidelity, outperforming grid-based encoders and analytic baselines across multiple datasets while maintaining practical runtime and memory efficiency. This work enables precise, scalable mesh extraction for neural implicit surfaces, facilitating accurate geometry reconstruction in 3D scenes and geometry-centric AI applications.

Abstract

Extracting meshes that exactly match the zero-level set of neural signed distance functions (SDFs) remains challenging. Sampling-based methods introduce discretization error, while continuous piecewise affine (CPWA) analytic approaches apply only to plain ReLU MLPs. We present TetraSDF, a precise analytic meshing framework for SDFs represented by a ReLU MLP composed with a multi-resolution tetrahedral positional encoder. The encoder's barycentric interpolation preserves global CPWA structure, enabling us to track ReLU linear regions within an encoder-induced polyhedral complex. A fixed analytic input preconditioner derived from the encoder's metric further reduces directional bias and stabilizes training. Across multiple benchmarks, TetraSDF matches or surpasses existing grid-based encoders in SDF reconstruction accuracy, and its analytic extractor produces highly self-consistent meshes that remain faithful to the learned isosurfaces, all with practical runtime and memory efficiency.

Figures (6)

• Figure 1: Overview of TetraSDF. A preconditioned input $\mathbf{x}$ is mapped by the multi-resolution tetrahedral positional encoder to barycentrically interpolated features within its containing polyhedral cell $\mathcal{C}_{\mathbf{x}}$ (\ref{['mtd:tet-enc', 'sec:precondition']}). These cells form the encoder-induced polyhedral complex (the initial skeleton), from which we start edge subdivision (\ref{['sec:grid-skeleton']}). We then perform grid-aware edge subdivision that jointly tracks polyhedral cells and ReLU MLP linear regions to obtain the candidate vertex and edge sets $(\mathcal{V}, \mathcal{E})$ (\ref{['sec:region-indicators', 'sec:perturbation']}), and finally extract mesh faces from $(\mathcal{V}, \mathcal{E})$ to obtain the final mesh (\ref{['sec:face-extraction']}). In (a)–(c), we zoom into a single edge subdivision iteration for a neuron.
• Figure 2: Subdivision of a cube cell into six congruent tetrahedra.
• Figure 3: Qualitative comparison on Thingi10K corresponding to \ref{['tab:baselines_cd']}.
• Figure 4: Qualitative effect of the input preconditioner $\mathbf{A}^*$ on the Stanford Bunny. The preconditioner improves the network’s SDF accuracy, especially in high-curvature regions (e.g., the ears).
• Figure 5: Qualitative comparison under the Small setting. Sampling-based methods often over-fragment triangles with surface artifacts, while our method avoids both by construction.

Theorems & Definitions (13)

• Definition : Tetrahedral networks
• Remark : Piecewise affine property of tetrahedral networks
• Definition : Sign vectors
• Definition : multi-resolution tetrahedral grid
• Definition : Tetrahedral positional encoder
• Definition : Polyhedral cells
• Theorem 1: Affine property of barycentric interpolation within a tetrahedron
• proof
• Lemma 1: Affine property of levelwise feature concatenation
• proof