Searching point patterns in point clouds describing local topography

TL;DR

This paper introduces a novel local descriptor for comparing and aligning 3D point patterns based on height variations, enhancing geometric analysis in topographical data.

Contribution

The work presents a new normalized finite-difference operator that serves as a local descriptor, integrating with Wasserstein distances and Procrustes analysis for improved pattern comparison and alignment.

Findings

Provides a parametrization-independent local descriptor

Enables natural integration with Wasserstein distances

Facilitates rigid alignment of geometric structures

Abstract

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.