Journal of Mathematical Imaging and Vision

, Volume 55, Issue 3, pp 370–377 | Cite as

Connectivity Preserving Multivalued Functions in Digital Topology

  • Laurence Boxer
  • P. Christopher Staecker


We study connectivity preserving multivalued functions (Kovalevsky in A new concept for digital geometry, shape in picture, 1994) between digital images. This notion generalizes that of continuous multivalued functions (Escribano et al. in Discrete geometry for computer imagery, lecture notes in computer science, 2008; Escribano et al. in J Math Imaging Vis 42:76–91, 2012) studied mostly in the setting of the digital plane \({\mathbb {Z}}^2\). We show that connectivity preserving multivalued functions, like continuous multivalued functions, are appropriate models for digital morphological operations. Connectivity preservation, unlike continuity, is preserved by compositions, and generalizes easily to higher dimensions and arbitrary adjacency relations.


Digital topology Digital image Continuous multivalued function Shy map Morphological operators Retraction Simple point 



We are grateful for the suggestions of the anonymous reviewers.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Computer and Information SciencesNiagara UniversityNYUSA
  2. 2.Department of Computer Science and EngineeringState University of New York at BuffaloBuffaloUSA
  3. 3.Department of MathematicsFairfield UniversityFairfieldUSA

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