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Convex digital solids.

C E Kim¹, A Rosenfeld

¹Department of Computer Science, University of Maryland, College Park, MD 20742; Department of Computer Science, Washington State University, Pullman, WA 99164.

IEEE Transactions on Pattern Analysis and Machine Intelligence

|April 14, 2012

Summary

This summary is machine-generated.

This study defines convexity for digital solids, proving it

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Area of Science:

Digital Geometry
Computational Topology
Computer Vision

Background:

Convexity is a fundamental geometric property with applications in various fields.
Defining and verifying convexity for discrete objects (digital solids) presents unique challenges.
Existing geometric properties for convex regions are not always sufficient for digital solids.

Purpose of the Study:

To introduce a precise definition of convexity for digital solids.
To establish a characterization of convex digital solids based on a specific property.
To develop an efficient algorithm for testing digital solid convexity.

Main Methods:

Formal definition of digital solid convexity.
Proof-based derivation of the chordal triangle property as a necessary and sufficient condition.
Comparative analysis of existing geometric properties versus the new criterion.
Algorithm design and implementation for convexity testing.

Main Results:

A novel definition of convexity for digital solids is established.
The chordal triangle property is identified as the definitive characteristic of convex digital solids.
It is demonstrated that certain previously considered geometric properties are only necessary, not sufficient, for digital solid convexity.
An efficient algorithm for determining the convexity of digital solids is presented.

Conclusions:

The chordal triangle property provides a complete and efficient characterization of convex digital solids.
The developed algorithm offers a practical tool for applications requiring convexity verification in digital geometry.
This work refines the understanding of convexity in the context of discrete spaces.