[Paper Review] Strongly Near Voronoi Nucleus Clusters

James F. Peters, Ebubeki̇r İnan|arXiv (Cornell University)|Jan 29, 2016

Quantum many-body systems12 references6 citations

TL;DR

This paper introduces strongly near Vorono"{i} nucleus clustering in Vorono"{i} tessellations of plane surfaces, using strong proximity to identify clusters of adjacent Vorono"{i} regions around a central nucleus. The key contribution is proving that every collection of Vorono"{i} regions in such a tessellation forms a Zelins'kyi-Soltan-Kay-Womble convexity structure, with maximal nucleus clusters indicating high object concentration in applications like digital image analysis, fMRI, and tomography.

ABSTRACT

This paper introduces nucleus clustering in Voronoi tessellations of plane surfaces with applications in the geometry of digital images. A \emph{nucleus cluster} is a collection of Voronoi regions that are adjacent to a Voronoi region called the cluster nucleus. Nucleus clustering is a carried out in a strong proximity space. Of particular interest is the presence of maximal nucleus clusters in a tessellation. Among all of the possible nucleus clusters in a Voronoi tessellation, clusters with the highest number of adjacent polygons are called \emph{maximal nucleus clusters}. The main results in this paper are that strongly near nucleus clusters are strongly descriptively near and every collection of Voronoi regions in a tessellation of a plane surface is a Zelins'kyi-Soltan-Kay-Womble convexity structure.

Motivation & Objective

• To develop a novel clustering method based on strong proximity in Vorono"{i} tessellations for detecting high-concentration object regions in digital images.
• To define and analyze maximal nucleus clusters as indicators of dense object regions in tessellated surfaces.
• To establish a formal link between Vorono"{i} region collections and convexity structures using Zelins'kyi-Soltan-Kay-Womble axioms.
• To apply the framework to real-world imaging data, including satellite, fMRI, and tomography images, for object detection and cortical activity mapping.

Proposed method

• Defines a nucleus cluster as the set of Vorono"{i} regions strongly near a central region (the nucleus), where strong proximity means shared boundary points.
• Uses strong proximity ($A \mathop{\delta}\limits^{\doublewedge} B$) to identify adjacent Vorono"{i} regions based on common points or edges.
• Introduces descriptive strong proximity ($A \mathop{\delta_{\Phi}}\limits^{\doublewedge} B$) to compare regions based on feature vectors such as centroid, area, and gradient orientation.
• Applies the Zelins'kyi-Soltan-Kay-Womble convexity structure axioms to the family of all subsets of Vorono"{i} regions ($2^X$) to establish formal convexity.
• Employs the descriptive intersection ($A \mathop{\cap}\limits_{\Phi} B$) to identify regions with matching features, linking descriptive proximity to cluster formation.
• Validates the framework through geometric and topological analysis of tessellations, including examples from fMRI and tomography images.

Experimental results

Research questions

• RQ1 How can strong proximity be used to define meaningful clusters of adjacent Vorono"{i} regions in a tessellation?
• RQ2 What defines a maximal nucleus cluster, and how does it serve as an indicator of high object concentration in image data?
• RQ3 Can the collection of all Vorono"{i} regions in a tessellation be formally classified as a convexity structure under the Zelins'kyi-Soltan-Kay-Womble axioms?
• RQ4 How does descriptive strong proximity enhance the detection of similar regions in image analysis applications?
• RQ5 In what ways can nucleus clustering improve object detection and activity mapping in fMRI and tomography images?

Key findings

• Maximal nucleus clusters are defined as the largest collections of Vorono"{i} regions strongly near a common nucleus, serving as indicators of high object concentration in tessellated images.
• Strongly near nucleus clusters are shown to be descriptively near, meaning they share matching feature descriptions such as centroid, area, and gradient orientation.
• Every collection of Vorono"{i} regions in a tessellation forms a Zelins'kyi-Soltan-Kay-Womble convexity structure, satisfying axioms (C0) and (C1) for the power set $2^X$.
• The descriptive intersection $A \mathop{\cap}\limits_{\Phi} B$ is non-empty if and only if there exist regions $A$ and $B$ in the respective clusters that are descriptively near.
• The framework successfully identifies and classifies clusters in real-world applications, including detecting surface objects in satellite images and cortical activity in fMRI data.
• The method enables robust detection of fossil concentrations in 3D tomography images through the presence of maximal nucleus clusters.