The Number of Gaps in Binary Pictures

Brimkov, Valentin E.; Maimone, Angelo; Nordo, Giorgio; Barneva, Reneta P.; Klette, Reinhard

doi:10.1007/11595755_5

Cited by 14 publications

(8 citation statements)

References 10 publications

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“…Moreover, β 0 is the number of connected components of S, while β 1 is the number of its holes. From here we immediately obtain the formula of [15].…”

Section: Proof For Anymentioning

confidence: 96%

“…In particular, we can easily obtain the formula from [15] recalled in the Introduction (its original proof by induction was pretty long). To this end, first observe that for n = 2 the only gaps in S are the 0-gaps.…”

Section: Proof For Anymentioning

confidence: 97%

“…A recent work [15] provides the formula g = v − 2(p + c − h) + b, where g is the number of gaps, v the number of vertices, p the number of pixels, h the number of holes, c the number of connected components, and b the number of 2 × 2 grid squares in a two-dimensional digital picture. For another similar result we refer to [16].…”

Section: Introductionmentioning

confidence: 99%

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Formulas for the number of (n−2)-gaps of binary objects in arbitrary dimension

Brimkov

2009

Discrete Applied Mathematics

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a b s t r a c tIn this paper we define the notion of a gap in an arbitrary digital binary object S in a digital space of arbitrary dimension. Then we obtain an explicit formula for the number of gaps in S of maximal dimension, derive combinatorial relations for digital curves, and discuss possible applications to image analysis of digital surfaces (in particular planes) and curves.Published by Elsevier B.V.

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“…Moreover, β 0 is the number of connected components of S, while β 1 is the number of its holes. From here we immediately obtain the formula of [15].…”

Section: Proof For Anymentioning

confidence: 96%

Section: Proof For Anymentioning

confidence: 97%

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Formulas for the number of (n−2)-gaps of binary objects in arbitrary dimension

Brimkov

2009

Discrete Applied Mathematics

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“…More precisely, if v 1 and v 2 are strictly (n − 1)-adjacent, then they form an (n − 1)-block centered on an (n − 1)-cell that is bounded by e; instead, if they are strictly (n − 2)-adjacent, they form an (n − 2)-gap having e as (n − 2)-hub. If c n (V) = 3, by Definition 8 (2) and Remark 4, the unique possible configuration is given by the L-block centered on e. Finally, if c n (V) = 4, V coincides with the (n − 2)-block centered on e.…”

Section: Definition 14 a Free I-cell Of A Digital N-object That Is Nmentioning

confidence: 99%

“…Since such a notion is strictly connected with some applications in the field of Computer graphics (e.g. the rendering of a 3D image by the ray-tracing technique), many papers (see for example [1], [2], [3], [4], and [5]) concerned the study of 0-and 1-gaps of 3-dimensional objects and of some of their topological invariant such as dimension and genus (i.e. the degree of connectedness of a digital image).…”

Section: Introductionmentioning

confidence: 99%

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

Maimone

Nordo

2013

Filomat

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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.

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