Local convexity properties of digital curves

The paper studies local convexity properties of digital curves. We locally define convex and concave parts from the slope of maximal digital straight segments and arithmetically characterize the smallest digital pattern required for checking convexity. Moreover, we introduce the concepts of digital edge and digital hull, a digital hull being a sequence of increasing or decreasing digital edges. We show that any strictly convex or concave part has a unique digital hull and arithmetically characterize the smallest digital pattern that contains a point incident to two consecutive digital edges. These theoretical results lead to online and linear-time algorithms that are useful for polygonal representations: convex hull computation of convex digital curves, computation of a reversible polygon that respects the convex and concave parts of a digital curve and computation of the well-known minimum-perimeter polygon.