用多分辨率四面体编码，精准提取神经SDF的零等值面。
TetraSDF: Analytic Isosurface Extraction with Multi-resolution Tetrahedral Grid
- 用四面体位置编码+ReLU MLP构建分段仿射SDF，保持解析性。
- 在多个基准上重建精度优于或媲美网格编码器，零等值面无误差。
- 适合需要精确几何表示的3D生成与重建任务，如工业建模。
从神经符号距离函数（SDF）中精确提取显式表面仍具挑战性。基于采样的等值面方法（如Marching Cubes）存在离散化误差，而连续分段仿射（CPWA）解析方法通常依赖普通ReLU MLP，难以有效学习高频SDF。本文提出TetraSDF，一种针对SDF的解析等值面提取框架，通过将ReLU MLP与多分辨率四面体位置编码结合，在保留网格编码器表达能力的同时实现零等值面的精确提取。其重心插值机制保持全局CPWA结构，可追踪编码器诱导的多面体复形内的ReLU线性区域。此外，引入基于编码器度量的固定解析输入预处理，降低方向偏差，提升训练稳定性。在多个基准测试中，TetraSDF的SDF重建精度达到或超过现有网格编码器水平，并能准确恢复网络零等值面为三角网格。
原文摘要 · Abstract (English)
Extracting an explicit surface that exactly matches the zero-level set of a neural signed distance function (SDF) remains challenging. Sampling-based isosurfacing methods such as Marching Cubes introduce discretization error. In contrast, continuous piecewise affine (CPWA) analytic approaches typically require plain ReLU MLPs, which limits the ability to learn high-frequency SDFs in practice. We present TetraSDF, an analytic isosurface extraction framework for SDFs that retains the expressiveness of grid-based encoders while enabling exact zero-level set extraction, by representing the SDF with a ReLU MLP composed with a multi-resolution tetrahedral positional encoder. Our positional encoder's barycentric interpolation preserves a global CPWA structure, allowing us to track ReLU linear regions within an encoder-induced polyhedral complex. We further introduce a fixed analytic input preconditioner derived from the encoder's metric to reduce directional bias, thereby stabilizing training. Across multiple benchmarks, TetraSDF matches or surpasses existing grid-based encoders in SDF reconstruction accuracy, while faithfully recovering the network's zero-level set as a triangle mesh.
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