热场签名：从点云到光滑几何
Heat Field Signatures: From Point Clouds to Smooth Geometry
- University of Texas at Dallas（德克萨斯大学达拉斯分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
热场签名通过将点云提升为多尺度热场，直接计算闭式几何签名，在多种基准上超越现有方法，并实现旋转不变性。
AI中文摘要：
将多尺度几何分析直接应用于不规则点云仍然困难：诸如局部维度、各向异性、密度变化和几何转变等量通常通过显式的邻域、流形或图构造来估计，或者留给神经网络从坐标中推断。我们引入了热场签名（HFS），它将点云提升为多尺度光滑环境热场族，为从离散样本到几何分析提供了直接接口。从该场出发，HFS直接根据成对距离计算闭式全局和局部签名，捕获热集中、内在维度、各向异性和尺度转变。我们进一步引入了热维度谱（HDS），这是多尺度几何组成的紧凑摘要。HFS可用作闭式描述符、轻量级学习表示或神经点云模型的几何特征通道。在涵盖亚细胞、神经元、树木和蛋白质数据的合成及真实世界基准上，HFS优于强点云和多参数持久性基线，同时大幅降低端到端成本。在SCOP蛋白质折叠分类中，HFS仅使用坐标就将最强深度基线提高了近24个百分点，而独立HFS表示在构造上完全旋转不变。更广泛地，HFS将经典热场转变为现代点云学习中多尺度几何分析的实用接口。
英文摘要：
Bringing multiscale geometric analysis directly to irregular point clouds remains difficult: quantities such as local dimension, anisotropy, density variation, and geometric transitions are typically estimated through explicit neighborhood, manifold, or graph constructions, or left for neural networks to infer from coordinates. We introduce Heat Field Signatures (HFS), which lift a point cloud to a multiscale family of smooth ambient heat fields, providing a direct interface from discrete samples to geometric analysis. From this field, HFS computes closed-form global and local signatures directly from pairwise distances, capturing heat concentration, intrinsic dimension, anisotropy, and scale transitions. We further introduce the Heat Dimension Spectrum (HDS), a compact summary of multiscale geometric composition. HFS can be used as a closed-form descriptor, a lightweight learned representation, or a geometric feature channel for neural point-cloud models. Across synthetic and real-world benchmarks spanning subcellular, neuronal, tree, and protein data, HFS outperforms strong point-cloud and multiparameter-persistence baselines while substantially reducing end-to-end cost. On SCOP protein-fold classification, HFS improves over the strongest deep baseline by nearly $24$ percentage points using coordinates alone, while standalone HFS representations are exactly rotation-invariant by construction. More broadly, HFS turns a classical heat field into a practical interface for multiscale geometric analysis in modern point-cloud learning.
