We address the problem of comparing and aligning spatial point configurations in $\mathbb{R}^3$ arising from structured geometric patterns. Each pattern is decomposed into arms along which we define a normalized finite-difference operator measuring local variations of the height component with respect to the planar geometry of the pattern. This quantity provides a parametrization-independent local descriptor that complements global similarity measures. In particular, it integrates naturally with Wasserstein-type distances for comparing point distributions and with Procrustes analysis for rigid alignment of geometric structures.

翻译：我们研究了在$\mathbb{R}^3$中由结构化几何模式产生的空间点配置的比较与对齐问题。每个模式被分解为若干分支，沿每个分支我们定义一个归一化有限差分算子，用于度量高度分量相对于模式平面几何的局部变化。该量值提供了一个独立于参数化的局部描述符，可作为全局相似性度量的补充。特别地，它能自然地与用于比较点分布的Wasserstein型距离以及用于几何结构刚性对齐的Procrustes分析相结合。