ADR shape descriptor - Distance between shape centroids versus shape diameter

Abstract

In this paper we study the ADR shape descriptor ρ(S), where ADR is short for "asymmetries in the distribution of roughness". This descriptor was defined in 1998 as the ratio of the squared distance between two different shape centroids (namely of area and frontier) to the squared shape diameter. After known for more than ten years, the behavior of ρ(S) was not well understood till today, thus hindering its application. Two very basic questions remained unanswered so far:What is the range for ρ(S), if S is any bounded compact shape?How do shapes look like having a large ρ(S) value?This paper answers both questions. We show that ρ(S) ranges over the interval [0, 1). We show that the established upper bound 1 is the best possible by constructing shapes whose ρ(S) values are arbitrary close to 1. In experiments we provide examples to indicate the kind of shapes that have relatively large ρ(S) values.