A new area-based convexity measure with distance weighted area integration for planar shapes

In this paper we propose a new area-based convexity measure. We assume that convexity evaluation of an arbitrary planar shape is related to the total influence of dents of the shape, and discover that those attributes of the dents, such as the position, area, and depth with respect to the Geometric Center of Convex Hull (GCCH) of the shape, determine the dent influence. We consider that the convex hull of the shape consists of infinitely small patches, to each of which we assign a weight showing the patch influence. We can simply integrate all the patch weights in any regions within the convex hull to calculate their total influence. We define this operation as the Distance Weighted Area Integration, if the weight is associated with the Euclidean distance from the patch to the GCCH. Our new measure is a distance weighted generalization of the most commonly used convexity measure, making this conventional measure fully replaceable for the first time. Experiments demonstrate advantages of the new convexity measure against the existing ones.