Digital Topology Ulrich Eckhardt Department of Applied Mathematics University of Hamburg Bundesstraße 55 D–20 146 Hamburg, Germany and Longin Latecki Department of Computer Science University of Hamburg Vogt–K¨olln–Straße 30 D–22 527 Hamburg February 26, 2008 To appear in Trends in Pattern Recognition, Council of Scientific Information, Vilayil Gardens, Trivandrum, India. Hamburger Beitr¨agezur Angewandten Mathematik Reihe A, Preprint 89 October 1994 Contents 1 Introduction 1 1.1 Motivation and Scope . 1 1.2 Historical Remarks . 3 2 The Digital Plane 3 2.1 Basic Definitions . 3 2.2 Jordan’s Curve Theorem . 4 2.3 The graphs of 4– and 8–topologies . 6 3 Embedding the Digital Plane 7 3.1 Line Complexes . 7 3.2 Cellular Topology . 11 4 Axiomatic Digital Topology 12 4.1 Definition and Simple Properties . 12 4.2 Connectedness . 14 4.3 Alexandroff Topologies for the Digital Plane . 17 5 Semi–Topology 19 5.1 Motivation . 19 5.2 The Associated Topological Space . 20 5.3 Related Concepts . 21 5.4 Connectedness . 22 5.5 Ordered Sets . 23 6 Applications to Image Processing 24 6.1 Models for Discretization . 24 6.2 Continuity . 24 6.3 Homotopy . 25 6.4 Fuzzy Topology . 25 Abstract to avoid wrong conclusions. For example, if an au- tomatic reasoning system should interpret correctly The aim of this paper is to give an introduction into the sentence “The policemen encircled the house”, it the field of digital topology. This topic of research must be able to understand that it is not possible to arose in connection with image processing. It is im- leave the house without meeting a policeman. Math- portant also in all applications of artificial intelligence ematically speaking, the set of policemen has some dealing with spatial structures. sort of ‘Jordan curve property’ which enables them In this article, a simple but representative model to separate the house from the remainder of the world of the Euclidean plane, called the digital plane, is unless one is able to act in three dimensions. studied. The problem is to introduce a satisfactory It is clear, that no discrete model can exhibit all rel- ‘topology’ for this essentially discrete structure. It evant features of a continuous original. Therefore, one turns out, that all known approaches, which come has to accept compromises. The compromise chosen from different directions of applications and theory, depends on the specific application and the questions converge to virtually one concept of 4/8– or 8/4– which are typical for the application. Digital geome- connectedness. try is an attempt to evaluate the price one has to pay A very natural approach to problems of discrete for discretization. The most simple part of geometry topology is the concept of semi–topological spaces. is topology. The digital model topology necessarily re- flects only certain facets of the underlying continuous structure. It is therefore necessary to apply different 1 Introduction approaches to digital topology. Our visual system seems to be well adapted to cope 1.1 Motivation and Scope with topological properties of the world. For exam- The only sets which can be handled on computers are ple, letters in a document can be classified in a first discrete or digital sets, wich means sets that contain step according to their topological homotopy types. at most a denumerable number of elements. There The layout of documents frequently is based on topo- are two sources for discrete sets logical predicates (e.g., a specific item must appear within a preassigned box). Optical checking of wiring • The data structures of computer science are enu- of chips amounts in finding out whether the wiring merable by definition. So, only discrete objects has the homotopy type wanted. can be represented. This covers a great number There are mainly three approaches for defining a of practical situations. For example in most ap- digital analog of the well–known ‘natural’ topology plications of Artificial Intelligence the universe of the Euclidean space: of discourse is a finite set. Another example is discrete classification e.g. of plants and animals Graph–Theoretic Approach A very elementary or stars into spectral classes etc. structure which can be handled easily on com- puters is a graph. A graph is obtained when a • Continuous objects are discretized, i.e. they are neighborhood relation is introduced into the dig- approximated by discrete objects. This is done ital set. Such a structure allows to investigate e.g. in finite element models in engineering but connectivity of sets. also in image processing where the intrinsically continuous image is represented by a discrete set Imbedding Approach The discrete structure is of ‘pixels’. imbedded into a known continuous structure, usually into an Euclidean space. Topological In some cases the objects under consideration are properties of discrete objects are then defined taken from a space with certain geometric charac- by means of their continuous images. teristics. Any useful discrete model of the situa- tion should model the geometry faithfully in order Axiomatic Approach Certain subsets of the un- 1 derlying digital structure are declared to be • General topology deals mainly with continuous ‘open sets’ and are required to fulfill certain ax- functions. Such functions have no counterparts ioms. These axioms have to be chosen in such in discrete structures. However, it would be use- a way that the digital structure gets properties ful for many applications to have such a concept which are as close as possible to the properties as ‘discrete continuity’. of usual topology. • By means of a suitable definition of continuity All three approaches have advantages and disad- one is able to compare topological structures vantages. The axiomatic approach is mathematically with each other. A very powerful concept in very elegant but it does not directly provide the topology is homeomorphy. Homeomorphic topo- language which is wanted in applications. The ob- logical structures are topologically the same. So, jects which are practically investigated are not open if a topological problem is investigated, one has sets but rather connected sets, sets which are con- to look for a homeomorphic structure in which tained in other sets etc. The graph theoretic approach the problem can be stated and solved as easily yields directly connectedness but is beomes very diffi- as possible. cult to handle more complicated concepts of topology • Homotopy theory deals with properties of sets such as continuity, homotopy etc. The embedding ap- which are invariant under continuous deforma- proach is of course only adequate for structures which tions. Translating homotopy from general topol- can be related to an Euclidean space. The problem is ogy to discrete structures raises a number of that one has to find for each question an appropriate questions which are not yet solved in a satis- embedding. factory way. On the other hand, one of the most There are many proposals in the literature which often used preprocessing methods in image pro- deal with topological questions. Of course, these pro- cessing is thinning which is the numerical real- posals cannot always be characterized by only one of ization of a ‘deformation retract’ from homotopy the approaches given above. theory. We list some of the questions which are typically attacked by digital topology: There are several practical situations other than image processing or spatial reasoning in artificial • Definition of connectedness and connected com- intelligence where topological spaces were used to ponents of a set. model discrete situations, starting from Alexandrov’s • Classification of points in a digital set as interior work in 1935 (see [1, 2]). In these publications, and boundary points, definition of boundaries of Alexandroff provided a theoretical basis for the topol- digital sets. ogy of ‘cell complexes’. • Formulation of Jordan’s Curve Theorem (and • In his book on the lambda calculus, Barendregt its higher dimensional analogs). This theorem uses the Scott topology [10, p. 9 f.] for partially states that a set can be represented by its bound- ordered sets [55], [56]. By means of this topol- ary which leads to a reduction of dimensionality ogy, concepts such as continuity of functions can in the representation. This Theorem is, as men- be introduced. There exist numerous publica- tioned above, essential for understanding spatial tions on application of discrete topologies to data relations and is used for grouping items in spatial structures, abstract computing models, combi- data structures. natorial algorithms, complexity theory and log- ics. • One very important topological ‘invariant’ of a set is its Euler number. This number is in a cer- • By using a suitable topology over the attribute tain sense all what we can get by parallel working set, Baik and Miller attempted to test equiva- machines. lence of heterogeneous relational databases [9]. 2 • Brissaud [12, 13, 14] defined a so–called ‘pre– similar theory was provided in the same time by G. topology’ for modelling preference structures in Herman [23]. economics. (for other applications in economics In 1979 A. Rosenfeld published a paper which had see [4], [5], [6], [7]). the title “Digital Topology” [53]. This paper was very These pre–topological spaces were later used by influential since for the first time some very difficult Arnaud et al. [8] in image processing. problems of digital topology were treated rigorously. The paper was based on results of Duda, Hart and In this paper, only topologies for the 2–dimensional Munson [17]. rectangular grid Z2 are considered. This grid can be understood to be the set of all points of the Euclidean plane R2 having integer coordinates. A more general theory is possible [52] but for easier presentation this is not attempted here. The reader should be familiar with elementary topology of the Euclidean plane.