Searching point patterns in point clouds describing local topography

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.