How to maximize the contact perimeter and contact area of shapes composed of cells in the discrete domain

Copyright c © 2014 Carlos Velarde and Ernesto Bribiesca. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The contact perimeter of a shape composed of pixels is the number of joints between neighboring pixels of the shape, and the contact area of a shape composed of voxels (three dimensional cells) is the number of planar joints between neighboring voxels. In this paper, we present the maximal contact perimeter function for shapes composed of pixels, and the maximal contact area function for voxel-based shapes. Also, we see how each one of these functions induce the definition of the corresponding standard function that extends the concept of contact perimeter (area) to non discrete squares (cubes), and which is used in the definition of the normalized discrete compactness.