A topology for discrete spaces and its application in scratch detection for surface inspections

Abstract

AbstractIn conventional digital image processing, topological properties have been studied only for selected types of neighborhoods such as a 4‐ or 8‐pixels connection. This paper analyzes properties of a finite topological space by defining it as a topological space with no restriction on the shape of a neighborhood. This leads to the identification of topological properties which are independent of the shape of a neighborhood and can be applied to image processing using neighborhoods other than a 4‐ or 8‐connection. Since a finite topological space can treat only a single neighborhood this cannot be applied to image processing which uses multiple neighborhoods simultaneously. A finite topological space has been extended to a formal topological space having multiple neighborhoods, and its properties are analyzed in this paper.This theory is then applied to image processing. It has been difficult to detect scratches on a surface with patterns, such as a hard disk or a hairline‐finished metal, by using a conventional digital topology because the scratches often consist of many small blocks and their images are stained by noise. The properties obtained through the formal topology have successfully been applied to these problems, and its effectiveness has been confirmed.