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Borders, frontiers and limits: some computational concepts beyond words

Marges, frontières et limites : formalisation sémantique
Sébastien Gadal et Robert Jeansoulin


D'un point de vue ontologique, l'existence d'un mot est fondamental pour exprimer un concept cognitif. Nous discutons dans cet article plus particulièrement de la notion de connexité à partir de trois termes pouvant la définir. Nous cherchons à rattacher étymologiquement les termes de bord (bordure), de frontière et de limite à des modèles de " séparations ". Le bord qui traduit la limite d’un objet par rapport à un autre appartient-il à un objet, à plusieurs objets, à aucun d'entre eux ? Nous confrontons cette démarche à différentes définitions d’axiomes méréologiques de connexité : connexité faible, frontières de type bona fine ou fiat. Son application s’est faite à partir de l’étude des formes de développement urbaines et de métropolisation par télédétection. Les limites détectées à partir notamment des propriétés texturales des images satellitales donnent lieu à une typologie des frontières dont leurs caractéristiques témoignent du comportement dynamique des différentes composantes d’un territoire (fragilités, résistance, vulnérabilité d’un territoire…).

typologie des frontières, ontologie, télédétection

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Entrées d’index

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Notes de l’auteur

This article is a version extended of the communication presented in 1998 to the AGILE conference (ITC – Netherlands).

Texte intégral


1When considering the geographical space, we all need to differentiate it, to split it into parts to which we attach several different attributes or qualifications, generally (and at least) through the words of our (mother tongue) natural language. But the operation of splitting itself leads to the use of many different words : "borders, bounds, edges, ends, frontiers, limits,…" are used to split the space into "entities, parts, objects, zones, territories, areas, basin, fields, …".

2These words are not inter-exchangeable, we have heuristics, habits or preferences depending on the particular use we have in mind. Our purpose is to excerpt some (at least a few) different underlying concepts, that the words convey and that may be translated into different mathematical concepts, which we can represent and process differently though computers. Hence to implement several usable and computable data structures into GIS.

3This is what we call an "ontological proof" if different representations co-exist into the language (either mother tongue or mathematics) and persist through the ages, then different features must exist into the real world.

Table 1

4Let’s use this array-based representation to list several synonyms for "separation", according to etymological origin, basic meanings and tentative underlying concept ( ? !)

The words in the dictionary : a mere terminological point of view

5Let's try to propose a lexicon about the words that help us to split the world.

Table 2

6These different terms are several expressions for a same general (unstructured) topic. But this rather unstructured use of the words points to several different concepts ; there is at least two approaches : the frontier, limit, border … refer to a "termination" or a "separation" according as the focus is on only one or two objects.

7The specific use of one particular term can be determined by :

  • the thematic of the study (e.g. mathematics versus geography, solid bodies versus competitive entities, permanent versus dynamic aspects …)

  • the technical processing (continuous versus discrete representations, …).

The words in geography : typological definition of the "frontier"

8We can distinguish two thematic aspects of what is a frontier in geography :

political frontiers

9the outer line of a surface or a totality of surfaces on which a political, economic, social system exerts its sovereignty. Its layout is a decision or a totality of decisions that have been taken by this territorial system and is the result of force reports, confrontations or negotiations with neighbour systems. These frontiers can take two forms in space :

  • the staked frontiers are linear ruptures that splits two territories in a brutal manner. They are a priori not deterministic, not geophysical frontiers, and fasten to terminology’s end, frontier and limit and to the notion of fiat boundary [Smith-Varzi] ;

  • the no man's land. frontiers are intervals that mark the separation between two politico -territorial entities.

structural frontiers

10They have a determinist character in that they are the result of adjustments of a society to a territory in the course of time, i.e. : the result of relationships that a society maintains with the space and constraints that exert natural objects. The location, the nature, the form of frontiers are determined by the nature of spatial structures and dynamics of a territory and the geohistorical inheritance. It results from the conjugated action of the society and the ecosystem. This partition of the geographical space can be approached with the help of remote sensing images, which brings two advantages :

  • Satellite images allow to significantly increasing the flow of geographic data, for they cover it in a continuous manner. The information is more synthetic, more connected. The space appears as a finite geographical continuum. It allows apprehending in a global manner the spatial structures and the dynamics that fashion the landscape. The frontiers can be identified with image segmentation methods on the remote sensing images, at several spatial levels.

  • The different radiometric textures model different states of the land, and allow to describe the spatial heterogeneous state and spatial discontinuities. This heterogeneous state, results in a differentiation of the territory into similar texture zones. Their spatial arrangement (defined by their degree of connexity or proximity), by their textural characteristics, determine their degree of interdependence. This degree can differentiate "two spatial" systems or measure the interactions between two subsystems composing a spatial system.

11Different geographical concepts are restricted by a same term (frontier). In the territory they can share a same nature but explain different themes. The remote sensing processing approach permits to model the nature of these frontiers.

The world remotely sensed : a processing approach

12The partition of the geographical space is model through regional homogeneity concepts [Cocquerez-Philipp]. They are based on the nature of attributes competing to develop a mode of organisation of space. They can be developed into two approaches in image analysis :

  • "frontier" approach : the search for intensity variations of the image marks limit of geographical objects and give them forms.

  • "Region" approach aim to identify criteria of similarity between attributes of an image at a given spatial level. The particular arrangement of related pixels to similar attributes determines the form of the geographical objects.

The Processing

13The goal of the processing on remote sensing images, is to detect and display the geographical objects that corresponds to a specific spatial level such as shown in the table below.

Table 3

14There are three steps in the image processing :

Table 4

  • The PCA goal is to synthesise the information within one image : the axis1 band is the one which carries the greatest quantity of information, taken from the original bands.

  • The goal of average filter is to aggregate pixels in order to show the geographical objects which characterise a certain spatial level. Hence we must harmonise the data resolution with this spatial level. The choice of the filter-size is 900 meters with both sensors (45*20m with XS and 30*30m with TM).

  • The goal of the frontier detection is to let emerge geographical objects characterising a spatial level. It requires to eliminate the extrema values of the image, each extrema being the expression of an frontier between two states of the landscape. An averaging operation, repeated six time, reveals homogeneous spatial bodies by the elimination of weak frontiers. The remaining strong frontiers mark the limit between geographical objects characterising a spatial level. Beyond six iterations, one reaches a threshold where it's no longer possible to eliminate the extrema of a certain amplitude. We can palliate this problem by using a wider window, hence working at a higher spatial level. either :
    (a) radiometric thresholding, then edge detection, or :
    (b) direct edge detection Each spatio - temporal level corresponds to an amplitude level of the extrema of the function f (x) of the image. The greater amplitude determinates the stronger spatial discontinuity.

15At this stage, two alternate approaches are possible :

  • If we have an a priori knowledge of the geographical objects which structure the space, let's try to identify the radiometric signatures of these objects, using radiometric thresholds : each spatial object matches one radiometric interval. We can speak of a structural approach of the texture analysis. The extraction of the container, i.e. the frontier of the geographical object, uses golay H morphological operator [Coster-Chermant].

  • Without a priori knowledge of the geographical objects that structure space, the job consists in enlightening inherent spatial discontinuities at a spatial level ; then, to identify frontiers. The method consists in undertake morphological processing with the same morphological operator. It gives place to a topographic map where each segment of the spatial discontinuities of a spatial level is listed.

16The identification of frontiers has been made on a reasoning based on the notion of distance periodicity between discontinuities. Each segment of discontinuity is assimilated, in a vivid manner, to the segment of a wave whose centre of emission is the subsystem. It returns therefore to the notion of hierarchy to three spatial levels that repeat whatever is the spatial dimension of the geographical space studied. Each of these three encased spatial levels, is governed by hierarchical interdependence relationships of the meta -system.

17From the arrangement of frontiers results the next interpretation grid (below). One distinguishes currently twelve form frontiers that can be regrouped in three degrees which describe their space and time permanence, their evolution and the nature of the spatial discontinuity.

Table 5

18The next figures illustrate such frontier types, as computed on the zone of the Comtat of Avignon. The contours, classified into three classes, are follows as :

Table 6

19Differences (between TM and XS) result in the frontier layout of the geographical objects and the attribution of permanence indices to these frontiers, that are differentiated by a specific type of spatial discontinuity. The variability of the automatic layout of limit geographical objects has for origin statistical differences (spatial) and dynamics (spectral) between data. Geographical objects are better aggregated with TM data because of their weakest spatial resolution. XS Image generate more underlined differentiation between geographical objects.

Nature of frontiers between two connex objets of same or different spatial levels

Table 7

20A constant periodicity, is an equal interval between the outline of a frontier. It is the expression of an identical radiometric texture, the last discontinuity showing the limit of the spatial entity. Deformations of arc segments of spatial discontinuities and their interval distinguish a zone of contact : the frontier. The zones of strong periodicity put forward the rupture frontiers linear, of the spaces of confrontation between geographical objects. The transition zone between transition linear rupture frontiers appears as a space of appropriation.

21Frontiers model the nature of relationships between different geographical objects at different levels. They give a description of the spatial discontinuity intensity. They shade light on laws that govern geographical objects relationship at different spatial levels. Relationships of hierarchical order define the notion of proximity/connexity. Two subsystems belonging to two meta - different systems, spatially close, will be differentiated, not by a zone transition frontier, but by a linear rupture frontier. The relative positioning of geographical objects among them, just as their respective membership to one of the three spatial levels, determine the types of interrelations as well as the spatial frontier aspect of a zone.

Table 8 : Theoretical quadtree representation of a portion of geographical space. Comtat venessin, region of Châteaurenard, South of France, 1995

22Transitional frontiers (Gradient frontier) appear as weak constraint spaces, and the linear frontiers as spaces of confrontation where constraints are strong. That depends on the proximity of modes of organisation and their gradient of differentiation. Deformations of a geographical object are the result of the interaction of the former with related geographical objects. The land cover (land state) will be qualified :

  • No geographic, if the periodicity is constant and if the absence of constraints. It shows a spatial uniformity on the totality of the plan. In this case, frontiers do not exist. The space is infinite.

  • Geographic, if there is a spatial heterogeneous state that show multiple land cover. Frontiers have an existence determined by the nature of relationships between geographical objects.

23The two approaches return the same definition of frontier namely : Frontiers are geographical attributes that determine the intervals of differentiation between geographical objects of a same spatio-temporal level.

24Two interpretations of topological notion "open/closed" are possible :

  • If one considers the region as a closed space, the frontier is a geographical object to specific dynamics, independent of its geographical object environments. The form of these last depends of the dynamics of frontiers. Their dynamics and their evolution determine the structure and forms of a territory.

  • If one considers the frontier as an open space, that is a zone of differentiation between geographical objects that have their own dynamics. Frontiers are the spatial dynamics witnesses that animate a territory.

Mereo-Topology : a relational approach of the topology

25The basic notion that underlies the (classical) mathematical topology is the "neighbourhood". That is a tiny, open, environment of a non dimensional object named a "point". Hence we say that the classical topology is point-based. For any part of the two-dimensional space, we can consider its "open part", its "closed part", its "border". Several works about the formalisation of spatial relations between solid bodies [Ascher-Vieu] led to use a "mereological" (or "part-to-whole") approach. Non-dimensional points are not used and the notion of "border" is not so easy to define.

object A IR2 : close(A), open(A), border(A), complement(A)
properties for A not empty : border(A)
close(A), border(A) complement(open(A))... but some relations are very difficult to understand : how can two open parts be "in contact" ?
two closed parts ? and yet this is the intuitive notion for two solid bodies.

26The cognitive difficulties that the natural languages (at least latin-derived ones) show through the existence of several words for the notions of object separation are "ontologically" sound and meaningful. This is our perception of the "ontological proof" : if the word exist, some underlying concept must exist and, through the extensionality principle, wherever two words exist, two different concepts also exist.

27Let's recall the axiomatic foundation of mereo-topology and let's start with the one used by Allen for the one-dimensional space of "intervals" (as figured by non null time duration’s).

28Only three axioms associated to a relation, named "meet" (m), are necessary to provide the full and algebraically complete set of the 13 mutually exclusive relations r(i1,i2) :

{<, m, o, ft, dt, s, eq, d, f, st, ot, mt, >}

29(before, meet, overlaps, finished by, contains, starts, equal, during, finishes, started by, overlapped by, met by, after)

30The two-dimensional regions (as figured by not null surface areas) have been approached in a similar way, the meet relation is re-named "contact" (c) and its axioms are :

31The full and algebraically complete set of the 8 mutually exclusive relations r(r1,r2) :

{dc, ec, po, tpp, ntpp, eq, tppi, ntppi}

32(disconnected, externally connected, partially overlaps, tangential proper part, non tangential proper part, equal, tangential proper part inverse, non tangential proper part inverse)

33In both cases, the definitions for open, close, complement, require some more axioms.

34It has been shown that the classical topology cannot be a model for such theories, but can be considered as a "limit" case of mereo-topological models (the problem comes from that infinite intersection of opens is not an open).

35On the other side, the natural notions of separations may have a more versatile translation.

36The c relation ("contact") provides a somehow difficult intuition, but the ec relation can easily be associated with the solid bodies "touch" relation.

37The works of Varzi et al. also turned to Latin (as geographers do) in order to distinguish several types of boundaries : specifically "fiat" (imposed by an external authority : no one shares the -conflictual- boundary) and "bona fide" (common agreement : shared boundary).

Table 9

From the paper [Asher-Vieu95] three different relations, characterised by different axioms, are related to the contact notion :

38The weak contact model intuitively the contact between an object and the "air" around, this is exactly what we associated to a "border" (the border of the ship belongs to the ship, and there is no "border" for the water around, may be a front, unless if the nature of the water changes when teaching the ship, e.g. when you get out of the water, you're wet : some water "belongs" to you).

39In the paper [Plewe] the separation between objects are classified according to the "crisp" or "fuzzy" character of the objects, hence there are four types of separation points : C-C, C-F, F-F and F-C. The difference between C-F and F-C depends on the way the "membership" functions are computed (object A against object B or object A against the complement, not A)

40It’s rather difficult to expect to obtain a unique table like the following, where lexicon entries can be associated to concepts developed by referenced authors, whatever they are (linguists, geographers, computer scientists).

Table 10


41In conclusion the notion of frontier refers to the notion of end. The end term indicates the type of spatial organisation, more specifically his area limit.

42Generally we propose to use the bound term to express the limit of an objet and the term of frontiers to designate the limit between two objets. Thematically, in the case of geographic, we suggest to use the term limit when we have a staked frontier between two political systems such as administrative limits or countries. The term of frontier is used when a zone separate two geographical objects such as a no man's land. The orée French word defines the degree of interactions and interrelations of the limits or the frontiers between the geographical objects. These types of limits or frontiers characterise the notion of interface or synapse [Brunet-Ferras].

43In the image processing geographical approach using the frontier methods, the edge term is more appropriate (1) designate the separation between two entities. In the region methodological approaches, they are the bound and contour terms (2).

  • In the first case, the computation consist to research the variation of intensity of the image delimiting the objects.

  • In the second case, it is the criteria of similarity which are researched. The proceeding consist to extract the limit of each object.

Table 11

44The linear rupture frontiers defining in the table 5 can be assimilated to the notion of limit, the transition linear rupture frontiers to the border (margin and limit) and the transition frontier too (margin only).

45The frontier marks the separation between the geographical objects, that is the separation zone between two states of the geographical space. These spatial states are the particular view of the structures that compose space and that give a specific form to a territory. In these ways, the frontier is the object defining and identifying the spatial structure and the form of the geographical object. It’s opposed dialectically to the notion of content. The frontier is by excellence the object of measure and qualification of relationships between geographical systems. It’s revealing their rules and laws. It is the potential materialisation of our intelligence and our culture to some apprehension forms of the World by our capacity to separate, analyse, coagulate and to model a structure.

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Sébastien Gadal et Robert Jeansoulin, « Borders, frontiers and limits: some computational concepts beyond words  », Cybergeo: European Journal of Geography [En ligne], Epistémologie, Histoire de la Géographie, Didactique, document 125, mis en ligne le 01 mars 2000, consulté le 15 juin 2024. URL : ; DOI :

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Sébastien Gadal CAMS/LATES, EHESS, 54, bd Raspail, 75270 Paris cedex 06, France

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Robert Jeansoulin CMI/LIM, CNRS, Université de Provence (Aix-Marseille 1,) 39, rue Joliot-Curie, 13453 Marseille cedex13. France

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