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 the most cases such measures were defined by exploiting some of shape properties. An illustrative example might be the shape circularity measure derived by exploiting the well-know result that the circle has the largest area among all the shapes with the same perimeter. In the approach applied here, no desired property is needed and no optimizing shape has to be found. We start from a desired/selected convex polygon, and develop the related shape measure. 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, used to develop the a particular measure. All the measures are invariant with respect to translations, rotations, and scaling transformations. The method used has an straightforward extension to a wider family of shape measures, dependent on a tuning parameter involved. Another extension leads to a family of the new shape convexity measures.