A new shape descriptor based on a Q-convexity measure

In this paper we define a new measure for shape descriptor. The measure is based on the concept of convexity by quadrant, called Q-convexity. Mostly studied in Discrete Tomography, this convexity generalizes hv-convexity to any two or more directions, and presents interesting connections with “total” convexity. The new measure generalizes that proposed by Balázs and Brunetti (A measure of Q-convexity, LNCS 9647 (2016) 219–230), and therefore it has the same desirable features: (1) its values range intrinsically from 0 to 1; (2) its values equal 1 if and only if the binary image is Q-convex; (3) its efficient computation can be easily implemented; (4) it is invariant under translation, reflection, and rotation by 90°. We test the new measure for assessing sensitivity using a set of synthetic polygons with rotation and translation of intrusions/protrusions and global skew, and for a ranking task using a variety of shapes. Based on the geometrical properties of Q-convexity, we also provide a characterization of any binary image by the matrix of its “generalized salient points”, and we design a linear-time algorithm for the construction of the binary image from its associated matrix.