On a new basic concept and topological invariant

Abstract

Component is one of the most basic topological concepts for digital images. However, there is still few simple method, to date, permitting to distinguish holes from external borders. Such theoretic difficulty on components leads to quite complex and unnatural algorithms on component analysis. In this paper, the author proves that there exists, in 2D images, a basic topological invariant called total curvature. The total curvature on a border is the sum of local curvature representing the external angle of polygon vertices. We have discovered an important binary feature of this invariant: it equals to 2π for external contours and −2π for hole contours, independent of the orientation of contour tracing. The binary characteristics of total curvature reveals that external contours are global convex despite of local concavities and hole contours are global concave despite of local convexities. Therefore hole and external contours can be unified as total convex and total concave borders. Classifying all the borders of images by total curvature provides directly the number of both components and holes and it can be calculated easily in digital images.

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