The number of gaps in binary pictures

Authors:
Valentin E. Brimkov;Angelo Maimone;Giorgio Nordo;Reneta P. Barneva;Reinhard Klette
Affiliations:
Mathematics Department, SUNY Buffalo State College, Buffalo, NY;Dipartimento di Matematica, Università di Messina, Messina, Italy;Dipartimento di Matematica, Università di Messina, Messina, Italy;Department of Computer Science, SUNY Fredonia, Fredonia, NY;CITR Tamaki, University of Auckland, Auckland, New Zealand
Venue:
ISVC'05 Proceedings of the First international conference on Advances in Visual Computing
Year:
2005

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Abstract

This paper identifies the total number of gaps of object pixels in a binary picture, which solves an open problem in 2D digital geometry (or combinatorial topology of binary pictures). We obtain a formula for the total number of gaps as a function of the number of object pixels (grid squares), vertices (corners of grid squares), holes, connected components, and 2 × 2 squares of pixels. It can be used to test a binary picture (or just one region: e.g., a digital curve) for gap-freeness.