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Construction of Persistent Voronoi Diagram on 3D Digital Plane

  • Ranita Biswas
  • Partha Bhowmick
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10256)

Abstract

Different distance metrics produce Voronoi diagrams with different properties. It is a well-known that on the (real) 2D plane or even on any 3D plane, a Voronoi diagram (VD) based on the Euclidean distance metric produces convex Voronoi regions. In this paper, we first show that this metric produces a persistent VD on the 2D digital plane, as it comprises digitally convex Voronoi regions and hence correctly approximates the corresponding VD on the 2D real plane. Next, we show that on a 3D digital plane D, the Euclidean metric spanning over its voxel set does not guarantee a digital VD which is persistent with the real-space VD. As a solution, we introduce a novel concept of functional-plane-convexity, which is ensured by the Euclidean metric spanning over the pedal set of D. Necessary proofs and some visual result have been provided to adjudge the merit and usefulness of the proposed concept.

Keywords

Digital Voronoi diagram 3D digital plane Distance metric Digital convexity Digital geometry 

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringIndian Institute of TechnologyRoorkeeIndia
  2. 2.Department of Computer Science and EngineeringIndian Institute of TechnologyKharagpurIndia

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