Digital Topology

Digital topology deals with the topological properties of digital images: or, more generally, of discrete arrays in two or more dimensions. It provides the theoretical foundations for important image processing operations such as connected component labeling and counting, border following, contour filling, and thinning—and their generalizations to three- (or higher-) dimensional “images.” This paper reviews the fundamental concepts of digital topology and surveys the major theoretical results in the field. A bibliography of almost 140 references is included.

In an earlier paper written for a different readership [Computers and Graphics 13(2) (1989) 159–166] the first author defined a digital fundamental group—an analog, for binary digital pictures, of the fundamental group. In general the definition of the digital fundamental group involves continuous deformation. But an alternative, discrete, definition of the digital fundamental group was proposed for the strongly normal digital picture spaces defined in the same paper. The above-mentioned paper also defined a “continuous analog” C() for each binary digital picture on such a DPS (DPS=digital picture space). C() is a polyhedron constructed by “filling in the gaps” between black points (1's) of the binary digital picture in a specific way. Other kinds of continuous analog had previously been used by the first two authors.In seeking the simplest and most efficient algorithms for performing image processing operations, researchers have considered many different combinations of grids and adjacency relations. Almost all of those combinations are isomorphic to special cases of the concept of a strongly normal DPS.The main contribution of the present paper is a proof that the digital fundamental groups of binary digital pictures on a strongly normal DPS are naturally isomorphic to the fundamental groups of the digital pictures' continuous analogs. We use this result to establish that on a strongly normal DPS the discrete and continuous definitions of the digital fundamental group are equivalent, up to a natural group isomorphism. We also show that many topological results which hold in the Euclidean plane or Euclidean 3-space have analogs that hold in every strongly normal DPS. Our results suggest that a strongly normal DPS is a suitable domain for studying topology-related image processing operations such as thinning, border tracking and contour filling.The definitions of digital fundamental group, strongly normal DPS and continuous analog are included in this paper so as to make it self-contained.

Digital topology is the study of the topological properties of digital images. In most of the literature, a digital image has been endowed with a graph structure; the vertices being the points of the image, and the edges giving the connectivity between the points. This has enabled the use of combinatorial methods to provide theorems and proofs for basic topological results. However, these methods have been shown to be inadequate for a full discussion of object thinning in three dimensions, and also for the development of a topological theory in dimensions higher than three. This has led to the investigation of algebraic topology as a means of providing results in digital topology; this paper surveys the results so far obtained, and shows how they relate to classical algebraic topology, and to digital topology as it has developed over the last two decades.

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 ( 2 f0; 1g) can be determinated by the number of its gaps besides some other parameters like the number of its pixel, vertices and edges.

In digital topology, Euclidean n-space 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 . 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 ) 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 planarGADS (which are GADS that model subsets of the Euclidean plane) and present some fundamental results including a Jordan curve theorem for planar GADS.

The Avalanche Risk Project for the Catalan Pyrenees was started in 1986 by theServei Geològic de Catalunya and theDpt. de Geologia Dinàmica, Geofísica i Paleontologia. One of the aims of this project is to carry out the “Map of Probable Avalanche Paths”, which is a thematic map in where the land areas affected by avalanches are represented. This information is the result of: 1) aerial photointerpretation, 2) field work: observations of the avalanche effects and interviews to the inhabitants and people who work in the mountain areas. All the descriptive complementary information about each avalanche zone has been stored in a database, thus constituting the avalanche Cadastre. A Geographic Information System (G.I.S.) is an informatic system which can acquire, store and manipulate data that describe the land surface. The Cadastre and Map of Probable Avalanche Paths constitutes a complex document. It is also a basic document to perform further analysis, risk maps and defense actions, so it must have the following characteristics: easy to update, possibility to combine with other documents and to exploit as a source of diverse informations. Therefore, a G.I.S. is the most appropriate tool to store, manage, analyze and restore this avalanche data. The area of the Catalan Pyrenees that has been mapped and automated until now is also presented.

Topological invariants are very useful in various areas related to digital images and geometric modelling. In this paper, we study the simplicial homology groups of certain minimal simple closed surfaces, extend an earlier definition of the Euler characteristics of digital image, and show how to compute the Euler characteristic of several digital surfaces.

On one hand the graph isomorphism problem (GI) has received considerable attention due to its unresolved complexity status and its many practical applications. On the other hand a notion of compatible topologies on graphs has emerged from digital topology (see [5, 7, 17, 9, 14, 16]). In this article we study GI from the topological point of view. Firstly we explore the poset of compatible topologies on graphs and in particular on bipartite graphs. Then, from a graph we construct a particular compatible Alexandroff topological space said homeomorphic-equivalent to the graph. Conversely from any Alexandroff topology we construct an isomorphic-equivalent graph on which the topology is compatible. Finally using these constructions, we show that GI is polynomial-time equivalent to the topological homeomorphism problem (T opHomeo). Hence GI and T opHomeo are in the same class of complexity. Key words: Graph isomorphism problem, compatible topology on graph, homeomorphism

Abstract— Given a 2D binary image it is sometimes of interest to determine invariant topological properties, such as Euler number and holes which can reflect directly the complexity of a binary image. Counting the number of holes of binary images is an important problem arising in medical image processing, phototypesetting, petroleum exploration, earthquake prediction, and pattern recognition. In this paper, we propose and prove a new formula to detect and count the number of holes of binary images. A local labeling algorithm is proposed to make detecting holes ...

Recently, the general topology has become the appropriated frame for every collection related to relations because topology is required not only for mathematics and physics but also for biology, rough set theory, biochemistry, and dynamics. In this paper, we have introduced another concept of the closure operator. In so doing, the idempotent condition, which has never been realized, is achieved.

Digital topology is the study of the topological properties of digital images. In most of the literature, a digital image has been endowed with a graph structure; the vertices being the points of the image, and the edges giving the connectivity between the points. This has enabled the use of combinatorial methods to provide theorems and proofs for basic topological results. However, these methods have been shown to be inadequate for a full discussion of object thinning in three dimensions, and also for the development of a topological theory in dimensions higher than three. This has led to the investigation of algebraic topology as a means of providing results in digital topology; this paper surveys the results so far obtained, and shows how they relate to classical algebraic topology, and to digital topology as it has developed over the last two decades.

The goal of this paper is to generalize the notion of lighting function given in [3] in order to integrate strong 26-surfaces [5] into our framework for digital topology. In particular, the continuous analogue for strong 26-surfaces introduced in [10] is extended for arbitrary ...

Recently, the general topology has become the appropriated frame for every collection related to relations because topology is required not only for mathematics and physics but also for biology, rough set theory, biochemistry, and dynamics. In this paper, we have introduced another concept of the closure operator. In so doing, the idempotent condition, which has never been realized, is achieved.

Membrane Computing is a paradigm inspired from biological cellular communication. Membrane computing devices are called P systems. In this paper we calculate some algebraic-topological information of 2D and 3D images in a general and parallel manner using P systems. First, we present a new way to obtain the homology groups of 2D digital images in time logarithmic with respect to the input data involving an improvement with respect to the algorithms development by S. Peltier et al. Second, we obtain an edge-segmentation of 2D and 3D digital images in constant time with respect to the input data.

We present here a new randomized algorithm for repairing the topology of objects represented by 3D binary digital images. By “repairing the topology”, we mean a systematic way of modifying a given binary image in order to produce a similar binary image which is guaranteed to be well-composed. A 3D binary digital image is said to be well-composed if, and only if, the square faces shared by background and foreground voxels form a 2D manifold. Well-composed images enjoy some special properties which can make such images very desirable in practical applications. For instance, well-known algorithms for extracting surfaces from and thinning binary images can be simplified and optimized for speed if the input image is assumed to be well-composed. Furthermore, some algorithms for computing surface curvature and extracting adaptive triangulated surfaces, directly from the binary data, can only be applied to well-composed images. Finally, we introduce an extension of the aforementioned algorithm to repairing 3D digital multivalued images. Such an algorithm finds application in repairing segmented images resulting from multi-object segmentations of other 3D digital multivalued images.

This paper proposes a concise coding of the cells of n-dimensional finite regular grids. It induces a simple, generic and efficient framework for implementing classical digital topology data structures and algorithms. Discrete subsets of multidimensional images (e.g. regions, digital surfaces, cubical cell complexes) have then a common and compact representation. Moreover, algorithms have a straightforward and efficient implementation, which is independent from the dimension or sizes of digital images. We illustrate that point with generic hypersurface boundary extraction algorithms by scanning or tracking. This framework has been implemented and basic operations as well as the presented applications have been benchmarked.

A 3D binary digital image is said to be well-composed if and only if the set of points in the faces shared by the voxels of foreground and background points of the image is a surface in R3. Well-composed images enjoy important topological and geometric properties; in particular, there is only one type of connected component in any well-composed image, as 6-, 14-, 18-, and 26-connected components are equal. This implies that several algorithms used in computer vision, computer graphics, and image processing become simpler. For example, thinning algorithms do not suffer from the irreducible thickness problem if the image is well-composed. In this report, we introduce a new randomized algorithm for making 3D binary digital images that are not well-composed into well-composed ones, analyze its time complexity, and present experimental evidence of its effectiveness when faced with practical medical imaging data. We also introduce a new approach to extract simplicial surfaces from 3D bina...

The theoretical foundations and application availability of topological spatial relationship research for three dimensional Geo-entities in raster space are discussed. The digital topology theory is applied to define the interior, boundary and k-order six neighborhood of a 3D geo-entity which is represented with regular hexahedron tessellation. A new 9-intersection model, which replaces the original interior I, the boundary B and the exterior E of an entity in 9-I model with I6, B6 and E respectively, is presented. A computable method based on relation database query language SQL (Structured Query Language) is adopted for the model computation, topological query and analysis. The experimental analytic results show that the topology analysis model can be properly used for topological relations query and analysis of 3D Geo-entities represented in raster space, and it can also be applied for the reasoning of metric relation, direction relation and other spatial relationships. k 6