On a 3D analogue of the first Hu moment invariant and a family of shape ellipsoidness measures

In this paper we show that a 3D analogue for the first Hu moment invariant, originally introduced for planar shapes, is minimized by a sphere. By exploiting this fact we define a new compactness measure for 3D shapes. The new compactness measure indicates how much a given shape differs from a sphere, which is assumed to be the most compact 3D shape. The new measure is invariant with respect to the similarity transformation and takes the maximum value if and only if the shape considered is a sphere. The methodology, used to derive the theoretical framework necessary for the definition and explanation of the behavior of the new compactness measure, is further extended and a family of 3D ellipsoidness measures is obtained. These ellipsoidness measures are also invariant with respect to similarity transformations and are maximized with certain ellipsoids only (these ellipsoids have the specific ratios between ellipsoid axes). The measures from the family distinguish among ellipsoids whose axes ratios differ—not all the ellipsoids have the same certain ellipsoidness measure. All the new measures are straightforward and fast to compute, e.g. within an asymptotically optimal $$\mathcal{O}(n)$$O(n) time, if n is the number of voxels of the shape considered, or n is the number of triangles in the object surface triangular mesh. Several experiments, on the well-known McGill 3D Benchmark Data Set, are provided to illustrate the behavior of the new measures.