An easy measure of compactness for 2D and 3D shapes

An easy measure of compactness for 2D (two dimensional) and 3D (three dimensional) shapes composed of pixels and voxels, respectively, is presented. The work proposed here is based on the two previous works of the measure of discrete compactness [E. Bribiesca, Measuring 2-D shape compactness using the contact perimeter, Comput. Math. Appl. 33 (1997) 1–9; E. Bribiesca, A measure of compactness for 3D shapes, Comput. Math. Appl. 40 (2000) 1275–1284]. The measure of compactness proposed here improves and simplifies the previous measure of discrete compactness. Now, using this proposed measure of compactness, it is possible to compute measures for any kind of object including porous and fragmented objects. Also, the computation of the measures is very simple by means of the use of only one equation. The measure of compactness proposed here depends in large part on the sum of the contact perimeters of the side-connected pixels for 2D shapes or on the sum of the contact surface areas of the face-connected voxels for 3D shapes. Relations between the perimeter and the contact perimeter for 2D shapes and between the area of the surface enclosing the volume and the contact surface area, are presented.The measure presented here of 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 objects. Also, we present an important application for brain structures quantification by means of the use of the new proposed measure of discrete compactness.