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.

I. Introduction

Shape descriptors provide a quantitative representation of a particular characteristic of the shape. Afterwards these values can be used in intelligent image processing applications, for example image classification or image retrieval. Classification is performed by grouping together images which represent similar shapes, for example grouping together oranges and apples (similar because of their roundness), and separating them from pineapples and bananas. Image retrieval consists on searching a database for images which have similar shape to the given query image.