Counting Gaps in Binary Pictures

Brimkov, Valentin E.; Maimone, Angelo; Nordo, Giorgio

doi:10.1007/11774938_2

Cited by 14 publications

(10 citation statements)

References 15 publications

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“…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 digital picture. For another similar result we refer to [10].…”

Section: Introductionmentioning

confidence: 77%

Combinatorial Relations for Digital Pictures

Brimkov

Moroni

Barneva

2006

Discrete Geometry for Computer Imagery

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Section: Introductionmentioning

confidence: 77%

Combinatorial Relations for Digital Pictures

Brimkov

Moroni

Barneva

2006

Discrete Geometry for Computer Imagery

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“…If a 3 < a 1 + a 2 , then L is thicker than naive and thinner than standard. (6,7,16) restricted to a (10, 14)-cube. All level lines have identical shape since gcd(6, 7) = 1.…”

Section: Fact 7 Let a Naive Planementioning

confidence: 99%

“…There are 9 cracks in the cube. (b) Level line code for P (6,9,16) restricted to a (10, 14)-cube. Distinct level lines may be different in shape since gcd (6,9) = 3 = 1 (see [10]).…”

Section: Fact 7 Let a Naive Planementioning

confidence: 99%

“…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]. The notion of a gap has been used in higher dimensions, too [4].…”

Section: Introductionmentioning

confidence: 95%

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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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“…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.…”

Section: Theoretical Backgroundsmentioning

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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Counting Gaps in Binary Pictures

Cited by 14 publications

References 15 publications

Combinatorial Relations for Digital Pictures

Combinatorial Relations for Digital Pictures

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

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

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