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Searching point patterns in point clouds describing local topography

Ewa Bednarczuk, Rafał Bieńkowski, Robert Kłopotek, Jan Kryński, Krzysztof Leśsniewski, Krzysztof Rutkowski, Małgorzata Szelachowska

TL;DR

A normalized finite-difference operator measuring local variations of the height component with respect to the planar geometry of the pattern provides a parametrization-independent local descriptor that complements global similarity measures.

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.

Searching point patterns in point clouds describing local topography

TL;DR

A normalized finite-difference operator measuring local variations of the height component with respect to the planar geometry of the pattern provides a parametrization-independent local descriptor that complements global similarity measures.

Abstract

We address the problem of comparing and aligning spatial point configurations in 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.
Paper Structure (18 sections, 17 equations, 2 figures, 2 tables)

This paper contains 18 sections, 17 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Data used
  3. Local topography data
  4. High resolution DEM
  5. Problem formulation
  6. Properties of the reference pattern $T$ of points
  7. Properties of the sought pattern $Q_\ell$
  8. Similarity measure
  9. Procrustes distance
  10. Objective function
  11. Algorithm
  12. Normalization
  13. Scaling of $z$
  14. Construction Steps / Algorithm
  15. Computational experiments
  16. ...and 3 more sections

Figures (2)

  • Figure 5.1: Patterns on map
  • Figure 5.2: Height profiles