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.
Highlights
Let us consider Ω to be a binary four-connected region with or without holes
We suppose that Ω is filled with white pixels and surrounded by black pixels
To correct Eq (2) in its discrete formulation, we introduce a penalty associated with each Freeman displacement from one pixel to its successor along the positively oriented outer contour Γ
Summary
Let us consider Ω to be a binary four-connected region with or without holes. We suppose that Ω is filled with white pixels and surrounded by black pixels. A different version of the Green theorem[9,10] based on the exterior edges of the contour’s pixels (boundary) of a polyomino is used to calculate discrete geometric parameters, such as the center of gravity and the moment of inertia of the polyomino In these studies, the contour is the adjacent pixels of the polyomino that are four-connected, and the boundary is encoded by a four-letter alphabet. Our approach is to extract simultaneously the contour (eight-connected black pixels) and the ordered list of Freman’s directions between each pair of successive pixels These two topics are related in the calculation of the area.
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