Tunable cubeness measures for 3D shapes

In this paper we introduce cubeness measure C(S)C(S): a shape similarity measure between a given 3D shape and a cube. The cubeness measure has several desirable properties: it ranges over (0, 1] and reaches 1 only when the given shape is a cube, it is invariant with respect to rotation, translation and scaling, and is also robust with respect to noise. The measure is compared with discrete 3D compactness measure from the existing literature.A modification of the basic definition of cubeness is also given. This modification enables the creation of a family of descriptors Cγ,δ(S)Cγ,δ(S), which vary their behaviour depending on the choice of parameters (γ, δ). Several examples are given, which illustrates the behaviour of these measures. Also some shape retrieval experiments are presented which illustrate the suitability of cubeness measures for such applications. The experimental results are in accordance with theoretical considerations and with our perception.

► We introduce a shape similarity measure between a given 3D   shape and a cube.
► This measure C(S)C(S) ranges over (0, 1], is rotation, translation and scale invariant.
► This measure is defined in continuous space.
► We also introduce a similarity measure between a 3D shape and a cuboid with edge ratio 1:γ:δ.