Construction of Convex Sets on Quadrilateral Ordered Tiles or Graphs with Propagation Neighborhood Operations. Dales, Concavity Structures. Application to Gray Image Analysis of Human-Readable Shapes

TL;DR

This paper introduces a novel topological approach to gray image analysis using convex hulls, concavities called dales, and spatial relations on quadrilateral tiles, demonstrated with symbols and human faces.

Contribution

It presents new methods for constructing convex sets and concavities in gray images, including the concept of dales, using parallel operations and topological properties, with applications to symbol and face analysis.

Findings

Convex hulls and concavities are effectively constructed with a single operation.

New topological relations among gray objects are established.

Applications include symbol recognition and analysis of human faces.

Abstract

An effort has been made to show mathematicians some new ideas applied to image analysis. Gray images are presented as tilings. Based on topological properties of the tiling, a number of gray convex hulls: maximal, minimal, and oriented ones are constructed and some are proved. They are constructed with only one operation. Two tilings are used in the Constraint and Allowance types of operations. New type of concavity described: a dale. All operations are parallel, possible to realize clock-less. Convexities define what is the background. They are treated as separate gray objects. There are multiple relations among them and their descendants. Via that, topological size of concavities is proposed. Constructed with the same type of operations, Rays and Angles in a tiling define possible spatial relations. Notions like "strokes" are defined through concavities. Unusual effects on levelized gray objects are shown. It is illustrated how alphabet and complex hieroglyphs can be described through concavities and their relations. A hypothesis of living organisms image analysis is proposed. A number of examples with symbols and a human face are calculated with new Asynchwave C++ software library.