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

Abstract

We provide a formula that expresses the number of (n − 2)-gaps of a generic digital n-object. Such a formula has the advantage to involve only a few simple intrinsic parameters of the object and it is obtained by using a combinatorial technic based on incidence structure and on the notion of free cells. This approach seems suitable as a model for an automatic computation, and also allow us to find some expressions for the maximum number of i-cells that bound or are bounded by a fixed j-cell.By the above expression and the definition of dimension of a cell, we have that the dimension of the dual e ′ of a cell e = (x, θ) coincides with the number of non-null components of the direction θ, that is dim(e ′ ) = n j=1 [θ j 0]. Consequently, the dual e ′ of an i-cell e is an (n − i)-cell.Definition 3. Let D be a digital object. The dual D ′ of D is the set of all dual cells e ′ , with e ∈ D.We say that two n-cells v 1 , v 2 are i-adjacent (i = 0, 1, . . . , n − 1) if v 1 v 2 and there exists at least an i-cell e such that e ⊆ v 1 ∩ v 2 , that is if they are distinct and share at least an i-cell. Two n-cells v 1 , v 2 are strictly i-adjacent, if they are i-adjacent but not j-adjacent, for any jn . The set of all n-cells that are i-adjacent to a given n-voxel v is denoted by A i (v) and called the i-adjacent neighborhoods of v. Two cells v 1 , v 2 ∈ C n are incident each other, and we write e 1 Ie 2 , if e 1 ⊆ e 2 or e 2 ⊆ e 1 . Definition 4. Let e 1 , e 2 ∈ C n . We say that e 1 bounds e 2 (or that e 2 is bounded by e 1 ), and we write e 1 < e 2 , if e 1 Ie 2 and dim(e 1 ) < dim(e 2 ). The relation < is called bounding relation.Definition 5. Let e be an i-cell of a digital n-object D (with i = 0, . . . n − 1). We say that e is simple if e bounds one and only one n-cell.

0-Gaps on 3D Digital Curves

“…The following propositions derive from some general ones proved in [17] for the n-dimensional case. Proposition 7.…”

“…The following three propositions were proved in [17] Proposition 3. For any i, j ∈ N such that 0 ≤ i < j, it is c i→j = 2 j−i j i .…”

“…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.…”