Minkowski compactness measure

Many compactness measures are available in the literature. In this paper we present a generalised compactness measure C_q(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 C_q(S) can be understood as a function of q and as such it is possible to calculate a spectum of C_q(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 C_q(S) is applied to various shapes and describes how measure and its spectrum can be used for image processing applications.