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The Largest Empty Sphere Problem in 3D Hollowed Point Clouds

Netzer Moriya

TL;DR

A new approach for the adaptation of the Maximal Internal Envelope method extended to address the Largest Empty Sphere problem within unstructured 3D point clouds is introduced, often locating the Largest Empty Sphere in initial computational stages, suggesting a lower complexity than initially projected.

Abstract

We introduce a new approach for the adaptation of the Maximal Internal Envelope method, extended to address the Largest Empty Sphere problem within unstructured 3D point clouds. We explore the identification of the Largest Empty Sphere by computing Convex Hull vertices and employing a Voidness Score based on Minimal Distance Scoring for optimal segment selection. The integration of Delaunay triangulation and Voronoi diagrams facilitates the initial identification of potential Largest Empty Sphere candidates. Our analysis reveals the method's efficacy and efficiency, often locating the Largest Empty Sphere in initial computational stages, suggesting a lower complexity than initially projected.

The Largest Empty Sphere Problem in 3D Hollowed Point Clouds

TL;DR

A new approach for the adaptation of the Maximal Internal Envelope method extended to address the Largest Empty Sphere problem within unstructured 3D point clouds is introduced, often locating the Largest Empty Sphere in initial computational stages, suggesting a lower complexity than initially projected.

Abstract

We introduce a new approach for the adaptation of the Maximal Internal Envelope method, extended to address the Largest Empty Sphere problem within unstructured 3D point clouds. We explore the identification of the Largest Empty Sphere by computing Convex Hull vertices and employing a Voidness Score based on Minimal Distance Scoring for optimal segment selection. The integration of Delaunay triangulation and Voronoi diagrams facilitates the initial identification of potential Largest Empty Sphere candidates. Our analysis reveals the method's efficacy and efficiency, often locating the Largest Empty Sphere in initial computational stages, suggesting a lower complexity than initially projected.
Paper Structure (14 sections, 1 equation, 5 figures, 1 table)

This paper contains 14 sections, 1 equation, 5 figures, 1 table.

Table of Contents

  1. Abstract
  2. Introduction
  3. Problem Statement
  4. Mathematical Formulation
  5. Proposed Methodology
  6. A Short Description of the MIE method
  7. Key Steps and Equations
  8. Complexity Analysis
  9. Identification of the Largest Empty Sphere in the Shell-Like 3D Point Cloud
  10. Results - Largest Empty Sphere Selection
  11. Implementation and Analytical Insights
  12. Maximal Radius Analysis and LES Selection
  13. Computational Efficiency and Observations
  14. Conclusions

Figures (5)

  • Figure 1: Schematic Algorithm for the LES estimation.
  • Figure 2: The 3D point distributions of a shell-like sphere (left) and the two spheres construction with an internal cavity (right).
  • Figure 3: Example of best segments selection based on MDS for the two clouds. Every segment is connecting CH pairs of the respective 3D points distribution.
  • Figure 4: Example of MIE points identification based on best segment and VD starting point on segment for the LES selection.
  • Figure 5: The best sphere entitled LES for the shell-like sphere (left) and the two-spheres configuration (right)