{"title":"Holes and Genus of 2D and 3D Digital Images","authors":"Lee C.N., Poston T., Rosenfeld A.","doi":"10.1006/cgip.1993.1002","DOIUrl":null,"url":null,"abstract":"<div><p>\"Hole\" has been a confusing idea in the 3D digital literature. We replace counting holes by the clear geometrical idea of counting non-separating cuts, and show that this gives the Betti number <em>b</em><sub>1</sub>, while <em>b</em><sub>0</sub> counts components and <em>b</em><sub>2</sub> cavities. Connected sets with equal <em>b</em><sub>1</sub> and <em>b</em><sub>2</sub> must match topologically when <em>b</em><sub>1</sub> = 0 (implying simple connectedness). When <em>b</em><sub>1</sub> ≠ 0, contrary to digital folklore, they need not. This paper is a conceptually self-contained introduction for computer scientists to these numbers of 2D and 3D images, and to other topological features such as Euler and linking numbers.</p></div>","PeriodicalId":100349,"journal":{"name":"CVGIP: Graphical Models and Image Processing","volume":"55 1","pages":"Pages 20-47"},"PeriodicalIF":0.0000,"publicationDate":"1993-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1006/cgip.1993.1002","citationCount":"42","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"CVGIP: Graphical Models and Image Processing","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1049965283710023","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 42
Abstract
"Hole" has been a confusing idea in the 3D digital literature. We replace counting holes by the clear geometrical idea of counting non-separating cuts, and show that this gives the Betti number b1, while b0 counts components and b2 cavities. Connected sets with equal b1 and b2 must match topologically when b1 = 0 (implying simple connectedness). When b1 ≠ 0, contrary to digital folklore, they need not. This paper is a conceptually self-contained introduction for computer scientists to these numbers of 2D and 3D images, and to other topological features such as Euler and linking numbers.