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Void Shape Identification in a 2D Point Distribution

Netzer Moriya

Abstract

We introduce a new approach for identifying and characterizing voids within two-dimensional (2D) point distributions through the integration of Delaunay triangulation and Voronoi diagrams, combined with a Minimal Distance Scoring algorithm. Our methodology initiates with the computational determination of the Convex Hull vertices within the point cloud, followed by a systematic selection of optimal line segments, strategically chosen for their likelihood of intersecting internal void regions. We then utilize Delaunay triangulation in conjunction with Voronoi diagrams to ascertain the initial points for the construction of the maximal internal curve envelope by adopting a pseudo-recursive approach for higher-order void identification. In each iteration, the existing collection of maximal internal curve envelope points serves as a basis for identifying additional candidate points. This iterative process is inherently self-converging, ensuring progressive refinement of the void's shape with each successive computation cycle. The mathematical robustness of this method allows for an efficient convergence to a stable solution, reflecting both the geometric intricacies and the topological characteristics of the voids within the point cloud. Our findings introduce a method that aims to balance geometric accuracy with computational practicality. The approach is designed to improve the understanding of void shapes within point clouds and suggests a potential framework for exploring more complex, multi-dimensional data analysis.

Void Shape Identification in a 2D Point Distribution

Abstract

We introduce a new approach for identifying and characterizing voids within two-dimensional (2D) point distributions through the integration of Delaunay triangulation and Voronoi diagrams, combined with a Minimal Distance Scoring algorithm. Our methodology initiates with the computational determination of the Convex Hull vertices within the point cloud, followed by a systematic selection of optimal line segments, strategically chosen for their likelihood of intersecting internal void regions. We then utilize Delaunay triangulation in conjunction with Voronoi diagrams to ascertain the initial points for the construction of the maximal internal curve envelope by adopting a pseudo-recursive approach for higher-order void identification. In each iteration, the existing collection of maximal internal curve envelope points serves as a basis for identifying additional candidate points. This iterative process is inherently self-converging, ensuring progressive refinement of the void's shape with each successive computation cycle. The mathematical robustness of this method allows for an efficient convergence to a stable solution, reflecting both the geometric intricacies and the topological characteristics of the voids within the point cloud. Our findings introduce a method that aims to balance geometric accuracy with computational practicality. The approach is designed to improve the understanding of void shapes within point clouds and suggests a potential framework for exploring more complex, multi-dimensional data analysis.
Paper Structure (18 sections, 4 equations, 8 figures, 3 tables)

This paper contains 18 sections, 4 equations, 8 figures, 3 tables.

Table of Contents

  1. Abstract
  2. Introduction
  3. Problem Description: Maximal Internal Envelope (MIE)
  4. Proposed Methodology
  5. Void Identification
  6. Minimal Distance Scoring (MDS)
  7. Delaunay-Voronoi (DV) point selection
  8. Schematic Algorithm
  9. Void Shape Construction
  10. Polygon Construction
  11. Refining Criterion
  12. Complexity Analysis
  13. Detailed Considerations
  14. Overall Complexity
  15. Results - Void Identification and Shape Construction
  16. ...and 3 more sections

Figures (8)

  • Figure 1: Schematic Algorithm for Void Identification.
  • Figure 2: Schematic Algorithm for Void Shape Construction.
  • Figure 3: The 2D point distributions of a shell-like circle (left) and a moderately detailed shape (right).
  • Figure 4: Example of best segments selection based on MDS for the two clouds. Every segment is connecting CH pairs of the respective 2D points distribution.
  • Figure 5: Example of MIE points identification based on best segment and VD starting point on segment for the interion void polygon construction.
  • ...and 3 more figures