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DOI: 10.1007/s00454-007-1315-x Discrete Comput Geom 37:629640 (2007) Discrete & Computational

Summary: DOI: 10.1007/s00454-007-1315-x
Discrete Comput Geom 37:629­640 (2007) Discrete & Computational
Geometry© 2007 Springer Science+Business Media, Inc.
The Genus of a Digital Image Boundary Is Determined
by Its Foreground, Background, and Reeb Graphs
Lowell Abrams1
and Donniell E. Fishkind2
1Department of Mathematics, George Washington University,
Washington, DC 20052, USA
2Department of Applied Mathematics and Statistics, Johns Hopkins University,
Baltimore, MD 21218-2682, USA
Abstract. Weprovethatthegenusoftheboundaryofadigitalimageispreciselyhalfofthe
sum of the cycle ranks of three particular graphs: the "foreground graph" and "background
graph," which capture topological information about the digital image and its complement,
respectively, and the Reeb graph, relative to the natural height function, associated with the
digital image's boundary. We prove several additional results, including a characterization
of when the cycle rank of the Reeb graph fails to equal the genus of the digital image's
boundary (which can happen by virtue of the failure of the natural height function on the


Source: Abrams, Lowell - Department of Mathematics, George Washington University


Collections: Mathematics