A measure of compactness for 3D shapes

A measure of compactness for 3D (three dimensional) shapes composed of voxels, is presented. The work proposed here improves and extends to the measure of discrete compactness [1] from 2D (two dimensional) domain to 3D. The measure of discrete compactness proposed here corresponds to the sum of the contact surface areas of the face-connected voxels of 3D shapes. A relation between the area of the surface enclosing the volume and the contact surface area, is presented. The concept of contact surfaces is extended to 3D shapes composed of different polyhedrons, which divide space generating different 3D lattices. The measure proposed here of discrete compactness is invariant under translation, rotation, and scaling. In this work, the term of compactness does not refer to point-set topology, but is related to intrinsic properties of objects. Finally, in order to prove our measure of compactness, we calculate the measures of discrete compactness of different volcanos (which are compared with their classical measures) from the valley of México using Digital Elevation Model (DEM) data.