Digital Jordan Curve Research Articles

Digital Jordan curves and surfaces with respect to a graph connectedness

We introduce a graph connectedness induced by a given set of paths of the same length. We focus on the 2-adjacency graph on the digital line ℤ with a certain set of paths of length n for every positive integer n. The connectedness in the strong product of two and three copies of the graph is used to define digital Jordan curves and digital Jordan surfaces, respectively. Such definitions build on an edge-to-edge tiling with triangles in the digital plane and a face-to-face tiling by cubes, prisms and pyramids in the (3D) digital space, respectively.

Digital Jordan Curves and Surfaces with Respect to a Closure Operator

In this paper, we propose new definitions of digital Jordan curves and digital Jordan surfaces. We start with introducing and studying closure operators on a given set that are associated with n-ary relations (n > 1 an integer) on this set. Discussed are in particular the closure operators associated with certain n-ary relations on the digital line ℤ. Of these relations, we focus on a ternary one equipping the digital plane ℤ2 and the digital space ℤ3 with the closure operator associated with the direct product of two and three, respectively, copies of this ternary relation. The connectedness provided by the closure operator is shown to be suitable for defining digital curves satisfying a digital Jordan curve theorem and digital surfaces satisfying a digital Jordan surface theorem.

Beyond the digital Jordan curve: Unconstrained simple pixel-based raster relations

Abstract Artificial intelligence and satellites have brought spatial-image interpretation to the forefront. Over the past two decades, the literature has provided three distinct methods for identifying and classifying the boundary of raster objects: the digital Jordan curve, the hyper-raster, and a marginal approach. While each approach has its own merits and drawbacks, there should be methods by which to convert one representation to another. This, however, is currently unattainable as the digital Jordan curve approach and the hyper-raster approach are formulated upon different domains of objects, with the more verbose digital Jordan curve approach serving as a subset of the hyper-raster approach with respect to objects which are simple regions in R 2 . This paper revisits the approach of the digital Jordan curve by removing the digital Jordan curve restriction, replacing it with the set of pixels that neighbor an exterior pixel, allowing for a host of different relations in pixel space. This increase in domain results in a set of 62 available 9-intersection matrices, a nearly four-fold increase from the original 16 matrices and demonstrates the novelty of this approach with respect to the egg-yolk relations, a model in R 2 for relations between regions with broad boundaries.

Relation-induced connectedness in the digital plane

We introduce and discuss a connectedness induced by n-ary relations (\(n>1\) an integer) on their underlying sets. In particular, we focus on certain n-ary relations with the induced connectedness allowing for a definition of digital Jordan curves. For every integer \(n>1\), we introduce one such n-ary relation on the digital plane \({\mathbb {Z}}^2\) and prove a digital analogue of the Jordan curve theorem for the induced connectedness. It follows that these n-ary relations may be used as convenient structures on the digital plane for the study of geometric properties of digital images. For \(n=2\), such a structure coincides with the (specialization order of the) Khalimsky topology and, for \(n>2\), it allows for a variety of Jordan curves richer than that provided by the Khalimsky topology.

A ternary relation for structuring the digital plane

We discuss certain ternary relations, called plain, and show that each of them induces a connectedness on its underlying set. This connectedness allows for definitions of concepts of simple closed and Jordan curves. We introduce a particular plain ternary relation on the digital plane ℤ2 and, as the main result, we prove a digital analogue of the Jordan curve theorem for the connectedness induced by this relation. It follows that the ternary relation introduced may be used as a convenient structure on the digital plane for the study of the geometric properties of digital images that are related to boundaries because boundaries of objects in digital images are represented by digital Jordan curves. An advantage of this structure over the Khalimsky topology is that it allows Jordan curves to turn at the acute angle π/4 at some points.

Alexandroff pretopologies for structuring the digital plane

We explore the possibility of employing Alexandroff pretopologies as structures on the digital plane Z2 convenient for the study of geometric and topological properties of digital images. These pretopologies are known to be in one-to-one correspondence with reflexive binary relations so that graph-theoretic methods may be used when investigating them. We discuss such Alexandroff pretopologies on Z2 that possess a rich variety of digital Jordan curves obtained as circuits in a natural graph with the vertex set Z2. Of these pretopologies, we focus on the minimal ones and study their quotient pretopologies on Z2, which are shown to allow for various digital Jordan curve theorems. We also develop a method for identifying Jordan curves in the minimal pretopological spaces by using Jordan curves in their quotient spaces. Utilizing this method, we conclude the paper with proving a digital Jordan curve theorem for the minimal pretopologies.

A Jordan curve theorem with respect to a pretopology on ℤ2

We study a pretopology on ℤ2 having the property that the Khalimsky topology is one of its quotient pretopologies. Using this fact, we prove an analogue of the Jordan curve theorem for this pretopology, thus showing that such a pretopology provides a large variety of digital Jordan curves. Some consequences of this result are discussed too.

The Jordan curve theorem in the Khalimsky plane

The connectivity in Alexandroff topological spaces is equivalent to the path connectivity. This fact gets some specific properties to Z 2 , equipped with the Khalimsky topology. This allows a sufficiently precise description of the curves in Z 2 and permit to prove a digital Jordan curve theorem in Z 2 .

Digital Jordan curves

A digital Jordan curve theorem is proved for a new topology defined on Z 2 . This topology is compared with the classical Khalimsky and Marcus topologies used in digital topology. We show that the Jordan curves with respect to the topology defined, unlike the Jordan curves with respect to any of the two classical topologies mentioned, may turn at the acute angle π 4 . We also discuss a quotient topology of the new topology.

Generic axiomatized digital surface-structures

In digital topology, Euclidean n-space R n is usually modeled either by the set of points of a discrete grid, or by the set of n-cells in a convex cell complex whose union is R n . For commonly used grids and complexes in the cases n=2 and 3, certain pairs of adjacency relations ( κ, λ) on the grid points or n-cells (such as (4,8) and (8,4) on Z 2 ) are known to be “good pairs.” For these pairs of relations ( κ, λ), many results of digital topology concerning a set of grid points or n-cells and its complement (such as Rosenfeld's digital Jordan curve theorem) have versions in which κ-adjacency is used to define connectedness on the set and λ-adjacency is used to define connectedness on its complement. At present, results of 2D and 3D digital topology are often proved for one good pair of adjacency relations at a time; for each result there are different (but analogous) theorems for different good pairs of adjacency relations. In this paper we take the first steps in developing an alternative approach to digital topology based on very general axiomatic definitions of “well-behaved digital spaces.” This approach gives the possibility of stating and proving results of digital topology as single theorems which apply to all spaces of the appropriate dimensionality that satisfy our axioms. Specifically, this paper introduces the notion of a generic axiomatized digital surface-structure ( GADS)—a general, axiomatically defined, type of discrete structure that models subsets of the Euclidean plane and of other surfaces. Instances of this notion include GADS corresponding to all of the good pairs of adjacency relations that have previously been used (by ourselves or others) in digital topology on planar grids or on boundary surfaces. We define basic concepts for a GADS (such as homotopy of paths and the intersection number of two paths), give a discrete definition of planar GADS (which are GADS that model subsets of the Euclidean plane) and present some fundamental results including a Jordan curve theorem for planar GADS.

Generic Axiomatized Digital Surface-Structures

In digital topology, Euclidean n-space R n is usually modeled either by the set of points of a discrete grid, or by the set of n-cells in a convex cell complex whose union is R n. For commonly used grids and complexes in the cases n = 2 and 3, certain pairs of adjacency relations (κ, λ) on the grid points or n-cells (such as (4,8) and (8,4) on Z 2 are known to be “good pairs”. For these pairs of relations (κ, λ), many results of digital topology concerning a set of grid points or n-cells and its complement (such as Rosenfeld's digital Jordan curve theorem) have versions in which κ-adjacency is used to define connectedness on the set and λ-adjacency is used to define connectedness on its complement. At present, results of 2D and 3D digital topology are usually proved for one good pair of adjacency relations at a time — so for each result there are different (but analogous) theorems for different good pairs of adjacency relations. In this paper we take the first steps in developing an alternative approach to digital topology based on very general axiomatic definitions of “well-behaved digital spaces”. This approach gives the possibility of stating and proving results of digital topology as single theorems which apply to all spaces of the appropriate dimensionality that satisfy our axioms. Specifically, this paper introduces the notion of a generic axiomatized digital surface-structure (GADS) — a general, axiomatically defined, type of discrete structure that models subsets of the Euclidean plane and of other surfaces. Instances of this notion include GADS corresponding to all of the good pairs of adjacency relations that have previously been used (by ourselves or others) in digital topology on planar grids and boundary surfaces. We define basic concepts for a GADS (such as homotopy of paths and the intersection number of two paths), give a discrete definition of planar GADS (which are GADS that model subsets of the Euclidean plane) and present some fundamental results including a Jordan curve theorem for planar GADS.

Digital Jordan curves — a graph-theoretical approach to a topological theorem

We give a proof of a graph-theoretical Jordan curve theorem which generalizes both the topological results of Khalimsky et al. as well as some of the graph-theoretical results of Rosenfeld and Klette and Voss.