On 1-gaps in 3D digital objects

Abstract

In Digital Geometry, a gap is a location of a digital object through which a discrete ray can penetrate with no intersection. More specifically, for a 3D digital object we distinguish between 0-and 1-gaps depending on the relative position of such a ray. Although in some applications it is important to know how many gaps has a set of voxels, it is quite complicated to find an efficient algorithm to directly count them. In this paper, we provide a formula that states the number of 1-gaps of a generic 3D object using the notion of free cell of dimension 1 and 2.

A formula for the number of (n - 2)-gaps in digital n-objects

“…Theorem 27. The formulas g n−2 = (n − 1)c * n−1 − c * n−2 (6) and g n−2 = −2n(n − 1)c n + 2(n − 1)c n−1 − c n−2 + β n−2 (7) are equivalent.…”

“…In [3] and [6], a constructive definition of gap for a digital object D in spaces of dimensions 2 and 3 was proposed, and a relation between the number of such a gaps and the numbers of free cells was found.…”

“…the degree of connectedness of a digital image). Recently (see [6]), we have found a formula for expressing the number of 1-gaps of a digital 3-object by means of the number of its free cells of dimension 1 and 2. During the submission process of that paper, the anonymous referee raised to our attention the existence of another recent and more general formula presented in [7] which gives the number of a generical (n − 2)-gaps of any digital n-object.…”

“…Unfortunately, such formula involves some parameters (the number of (n − 2)-blocks and of n-, (n − 1)-and (n − 2)-cells) that are non-intrinsic or that can not be easily obtained by the geometrical knowledge of the object. For such a reason, in the present paper, we propose a generalization of the formula obtained in [6] that allow us to express the number of (n − 2)-gaps using only two basic parameters, that is the number of free (n − 2)-and (n − 1)-cells of the object itself. Although we prove the equivalence between these two formulas, the latter approach seems simpler and more suitable as a model for an automatic computation.…”

0-Gaps on 3D Digital Curves

“…More recently, in [16] and [17] two formulas which express, respectively the number of 1-gaps of a generic 3D object of dimension α = 1, 2 and the number of (n − 2)-gaps of a generic digital n-object, by means of a few simple intrinsic parameters of the object itself were found. Furtermore, in [18] the relationship existing between the dimension of a 2D digital object equipped with an adjacency relation A α (α ∈ {0, 1}) and the number of its gaps was investigated.…”