Minkowski compactness measure

Abstract

Many compactness measures are available in the literature. In this paper we present a generalised compactness measure Cq(S) which unifies previously existing definitions of compactness. The new measure is based on Minkowski distances and incorporates a parameter q which modifies the behaviour of the compactness measure. Different shapes are considered to be most compact depending on the value of q: for q = 2, the most compact shape in 2D (3D) is a circle (a sphere); for q → ∞, the most compact shape is a square (a cube); and for q = 1, the most compact shape is a square (a octahedron). For a given shape S, measure Cq(S) can be understood as a function of q and as such it is possible to calculate a spectum of Cq(S) for a range of q. This produces a particular compactness signature for the shape S, which provides additional shape information. The experiments section of this paper provides illustrative examples where measure Cq(S) is applied to various shapes and describes how measure and its spectrum can be used for image processing applications.