A Tunable Measure of 3D Compactness

Abstract

The field of shape description can be applied in domains ranging from medicine to engineering. Defining new metrics may allow to better describe shapes. It is therefore an essential process of development of the field. In this work, a new family of compactness metrics is introduced. It is proven that they range over (0, 1] and are translation, rotation and scaling independent. The sphere is the shape that has the smallest volume for a fixed surface, this is a definition of compactness. Therefore, the metrics of this family are called compactness measures since they all reach 1 if and only if the considered shape is a sphere. The different metrics of the family are obtained by the modification of a parameter β involved in the mathematical definition of the metric. They are proven to be different from each other and a thorough study of their behaviour resulted in the formulation of two interesting conjectures concerning the limit cases of β. Finally several experiments investigate how McGill’s database classes of shapes are represented when using the new family.