Measuring Shapes with Desired Convex Polygons

In this paper we have developed a family of shape measures. All the measures from the family evaluate the degree to which a shape looks like a predefined convex polygon. A quite new approach in designing object shape based measures has been applied. In most cases such measures were defined by exploiting some shape properties. Such properties are optimized (e.g., maximized or minimized) by certain shapes and based on this, the new shape measures were defined. An illustrative example might be the shape circularity measure derived by exploiting the well-known result that the circle has the largest area among all the shapes with the same perimeter. Of course, there are many more such examples (e.g., ellipticity, linearity, elongation, and squareness measures are some of them). There are different approaches as well. In the approach applied here, no desired property is needed and no optimizing shape has to be found. We start from a desired convex polygon, and develop the related shape measure. The method also allows a tuning parameter. Thus, there is a new 2-fold family of shape measures, dependent on a predefined convex polygon, and a tuning parameter, that controls the measure's behavior. The measures obtained range over the interval (0,1] and pick the maximal possible value, equal to 1, if and only if the measured shape coincides with the selected convex polygon that was used to develop the particular measure. All the measures are invariant with respect to translations, rotations, and scaling transformations. An extension of the method leads to a family of new shape convexity measures.