AI 中文总结
本研究提出保持邻近性的神经细分（PNS），在保留Loop细分结构约束的同时加入曲率门控修正，可提升局部脊线特征逼近能力且无高频伪影。
AI 中文摘要
经典细分方案因具有局部性、可重复性和分析可处理性而被广泛应用：单个模板定义了整个细化规则，且所得算子在迭代下的行为已被充分理解。然而，这种统一性意味着固定模板往往对局部几何特征拟合不足，例如曲率集中的尖锐脊线或柔和边缘。神经网格细化可适应此类特征，但无约束的顶点预测通常缺乏细分算子在将细化规则应用于自身输出后所需的结构行为。本研究中，我们提出保持邻近性的神经细分，简称PNS。PNS是一种可训练的细化规则，它在Loop细分的基础上，添加了以协变局部框架表示的小型有界曲率门控修正项。该构造设计为：对于任意有限的网络权重，该算子在刚性运动下完全等变，可精确重现平面输入，且始终处于Loop模板周围的二次邻近包络内。在平面价k的星形结构处，线性化算子与Loop一致，因此在该参考构型下继承了Loop的切空间特征子空间和Reif谱间隙。所有这些属性均为架构层面的，在任何训练发生前就已存在。实验表明，PNS在重复细分时，在保持在规定邻近包络内的同时，提升了对局部脊线特征的逼近能力。相比之下，无约束的神经基线方法虽在单步拟合中表现更强，但会产生高频伪影，且在迭代后偏离细分规则。本研究的核心结论是：可以在不放弃使细分成为几何处理基础原语的结构约束的前提下，将学习引入细分过程。
英文摘要
Classical subdivision schemes are widely used because they are local, repeatable, and analytically tractable. A single stencil defines the entire refinement rule, and the behaviour of the resulting operator under iteration is well understood. This uniformity, however, means that fixed stencils tend to underfit localised geometric features, such as sharp ridges or soft edges, where curvature is concentrated. Neural mesh refinement can adapt to such features, yet unconstrained vertex prediction usually lacks the structural behaviour required of a subdivision operator once the refinement rule is applied to its own output. In this work, we introduce Proximity-Preserving Neural Subdivision, or PNS for short. PNS is a trainable refinement rule that augments Loop subdivision with a small, bounded, curvature-gated correction expressed in a covariant local frame. The construction is designed so that, for any finite network weights, the operator is exactly equivariant under rigid motion, reproduces planar input exactly, and remains inside a quadratic proximity envelope around the Loop stencil. At planar valence-k stars, the linearised operator agrees with Loop, and it therefore inherits Loop's tangent eigenspaces and Reif spectral gap at that reference configuration. All of these properties are architectural and hold before any training takes place. Empirically, PNS improves the approximation of localised ridge features while remaining inside its prescribed proximity envelope under repeated subdivision. An unconstrained neural baseline, in contrast, achieves stronger one-step fitting but develops high-frequency artefacts and leaves the subdivision regime once iterated. The overall message of this work is that learning can be introduced into subdivision without abandoning the structural constraints that make subdivision useful as a geometry-processing primitive.
