On the Notion of Dimension in Digital Spaces

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

doi:10.1007/11774938_19

Cited by 3 publications

(2 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Another field of investigation could consist in finding a formula, analogous to (4), which express the number of any k-gaps with 0 ≤ k ≤ n − 3, by means of same basic parameters of the digital n-object.…”

Section: Conclusion and Perspectivementioning

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%

See 1 more Smart Citation

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

Maimone

Nordo

2013

Filomat

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Conclusion and Perspectivementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Maimone

Nordo

2013

Filomat

Self Cite

View full text Add to dashboard Cite

show abstract

Genus and Dimension of Digital Images and Their Time- And Space-Efficient Computation

Brimkov¹,

Nordo²,

Barneva³

et al. 2008

Int. J. Shape Model.

View full text Add to dashboard Cite

A note on dimension and gaps in digital geometry

Maimone

Nordo

2017

Filomat

View full text Add to dashboard Cite

The notion of gap is quite important in combinatorial image analysis and it finds several useful applications in fields as CAD and computer graphics. On the other hand, dimension is a fundamental concept in General Topology and it was recently extended to digital objects. In this paper, we show that the dimension of a 2D digital object equipped with an adjacency relation A?(? ? {0,1g} can be determinated by the number of its gaps besides some other parameters like the number of its pixel, vertices and edges.

show abstract

On the Notion of Dimension in Digital Spaces

Cited by 3 publications

References 10 publications

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

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

Genus and Dimension of Digital Images and Their Time- And Space-Efficient Computation

A note on dimension and gaps in digital geometry

Contact Info

Product

Resources

About