Computation of the area in the discrete plane: Green’s theorem revisited

Alain Chalifour; Fathallah Nouboud; Yvon Voisin

doi:10.1117/1.JEI.26.6.063022

5 December 2017 Computation of the area in the discrete plane: Green’s theorem revisited

Alain Chalifour, Fathallah Nouboud, Yvon Voisin

Author Affiliations +

Journal of Electronic Imaging, Vol. 26, Issue 6, 063022 (December 2017). https://doi.org/10.1117/1.JEI.26.6.063022

Abstract

The detection of the contour of a binary object is a common problem; however, the area of a region, and its moments, can be a significant parameter. In several metrology applications, the area of planar objects must be measured. The area is obtained by counting the pixels inside the contour or using a discrete version of Green’s formula. Unfortunately, we obtain the area enclosed by the polygonal line passing through the centers of the pixels along the contour. We present a modified version of Green’s theorem in the discrete plane, which allows for the computation of the exact area of a two-dimensional region in the class of polyominoes. Penalties are introduced and associated with each successive pair of Freeman displacements along the contour in an eight-connectivity system. The proposed equation is shown to be true and properties of the equation related to the topology of the regions are presented. The proposed approach is adapted for faster computation than the combinatorial approach proposed in the literature.

CC BY: © The Authors. Published by SPIE under a Creative Commons Attribution 4.0 Unported License. Distribution or reproduction of this work in whole or in part requires full attribution of the original publication, including its DOI.

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Alain Chalifour, Fathallah Nouboud, and Yvon Voisin "Computation of the area in the discrete plane: Green’s theorem revisited," Journal of Electronic Imaging 26(6), 063022 (5 December 2017). https://doi.org/10.1117/1.JEI.26.6.063022

Received: 14 March 2017; Accepted: 10 November 2017; Published: 5 December 2017

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KEYWORDS

Lead

Binary data

Picosecond phenomena

Calculus

Image segmentation

Metrology

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