Digital Sphere Functionality

Ranita Biswas and Partha Bhowmick,
On Functionality of Quadraginta Octants of Naive Sphere
with Application to Circle Drawing
DGCI 2016, LNCS 9647, pp. 256-267, 2016.


Although the concept of functional plane for naive (i.e., 2-minimal) digital plane is studied and reported in the literature in great detail, no similar study is yet found for the naive digital sphere. This article exposes the first study in this line, opening up further prospects of analyzing the topological properties of sphere in the discrete space. Moreover, it indicates the immense possibility to make out symmetry groups and their topological characterization for various other 3D objects, a few of which evidently being hypersphere, ellipsoid, and hyper-ellipsoid. We show that each quadraginta octant Q of a naive sphere forms a bijection with its projected pixel set on a unique coordinate plane, which thereby serves as the functional plane of Q, and hence gives rise to merely mono-jumps during back projection. The other two coordinate planes serve as para-functional and dia-functional planes for Q, as the former is 'mono-jumping' but not bijective, whereas the latter holds neither of the two. Owing to this, the quadraginta octants form symmetry groups and subgroups with equivalent jump conditions. We also show a potential application in generating a special class of discrete 3D circles based on back projection and jump bridging by Steiner voxels. A circle in this class possesses 4-symmetry, uniqueness, and bounded distance from the underlying real sphere and real plane.