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On the cellular convexity of complexes.

¹Department of Computer Science, University of Maryland, College Park, MD 20742.

IEEE Transactions on Pattern Analysis and Machine Intelligence

|August 27, 2011

Summary

This study introduces a new geometric definition for cellular convexity in complexes. It proves this definition is equivalent to existing ones and presents an efficient algorithm for determining cellular convexity.

Area of Science:

Computational geometry
Image analysis
Digital topology

Background:

Cellular convexity is a key property in image analysis and computational geometry.
Existing definitions of cellular convexity lack a unified geometric interpretation.
Understanding cellular convexity is crucial for analyzing complex shapes in discrete spaces.

Purpose of the Study:

To introduce a novel geometric definition of cellular convexity for complexes.
To establish the equivalence between the new definition and existing ones (Sklansky, Minsky & Papert).
To develop an efficient algorithm for assessing cellular convexity.

Main Methods:

Geometric characterization of cellular convexity.
Proof of equivalence between different definitions.

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Development of a linear time algorithm (O(n)) for run-length encoded complexes.

Main Results:

A new, geometrically intuitive definition of cellular convexity is proposed.
The new definition is shown to be equivalent to prior definitions for regular complexes.
A regular complex is cellularly convex if its minimum-perimeter polygon avoids the complex boundary.
An efficient O(n) algorithm is presented for determining cellular convexity.

Conclusions:

All discussed definitions of cellular convexity are shown to be virtually equivalent.
The new geometric definition provides a clearer understanding of cellular convexity.
The developed algorithm offers an efficient method for practical applications in image analysis.