Methods for Handling Imperfect Spatial Information

Robert Jeansoulin, Odile Papini, Henri Prade, and Steven Schockaert (Eds.) Methods for Handling Imperfect Spatial Information

Studies in Fuzziness and Soft Computing, Volume 256 Editor-in-Chief Prof. Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences ul. Newelska 6 01-447 Warsaw Poland E-mail: [email protected] Further volumes of this series can be found on our homepage: springer.com Vol. 241. Jacek Kluska Analytical Methods in Fuzzy Modeling and Control, 2009 ISBN 978-3-540-89926-6 Vol. 242. Yaochu Jin, Lipo Wang Fuzzy Systems in Bioinformatics and Computational Biology, 2009 ISBN 978-3-540-89967-9 Vol. 243. Rudolf Seising (Ed.) Views on Fuzzy Sets and Systems from Different Perspectives, 2009 ISBN 978-3-540-93801-9 Vol. 244. Xiaodong Liu and Witold Pedrycz Axiomatic Fuzzy Set Theory and Its Applications, 2009 ISBN 978-3-642-00401-8 Vol. 245. Xuzhu Wang, Da Ruan, Etienne E. Kerre Mathematics of Fuzziness – Basic Issues, 2009 ISBN 978-3-540-78310-7 Vol. 246. Piedad Brox, Iluminada Castillo, Santiago Sánchez Solano Fuzzy Logic-Based Algorithms for Video De-Interlacing, 2010 ISBN 978-3-642-10694-1 Vol. 247. Michael Glykas Fuzzy Cognitive Maps, 2010 ISBN 978-3-642-03219-6 Vol. 248. Bing-Yuan Cao Optimal Models and Methods with Fuzzy Quantities, 2010 ISBN 978-3-642-10710-8

Vol. 249. Bernadette Bouchon-Meunier, Luis Magdalena, Manuel Ojeda-Aciego, José-Luis Verdegay, Ronald R. Yager (Eds.) Foundations of Reasoning under Uncertainty, 2010 ISBN 978-3-642-10726-9 Vol. 250. Xiaoxia Huang Portfolio Analysis, 2010 ISBN 978-3-642-11213-3 Vol. 251. George A. Anastassiou Fuzzy Mathematics: Approximation Theory, 2010 ISBN 978-3-642-11219-5 Vol. 252. Cengiz Kahraman, Mesut Yavuz (Eds.) Production Engineering and Management under Fuzziness, 2010 ISBN 978-3-642-12051-0 Vol. 253. Badredine Arfi Linguistic Fuzzy Logic Methods in Social Sciences, 2010 ISBN 978-3-642-13342-8 Vol. 254. Weldon A. Lodwick, Janusz Kacprzyk (Eds.) Fuzzy Optimization, 2010 ISBN 978-3-642-13934-5 Vol. 255. Zongmin Ma, Li Yan (Eds.) Soft Computing in XML Data Management, 2010 ISBN 978-3-642-14009-9 Vol. 256. Robert Jeansoulin, Odile Papini, Henri Prade, and Steven Schockaert (Eds.) Methods for Handling Imperfect Spatial Information, 2010 ISBN 978-3-642-14754-8

Robert Jeansoulin, Odile Papini, Henri Prade, and Steven Schockaert (Eds.)

Methods for Handling Imperfect Spatial Information

ABC

Editors Dr. Robert Jeansoulin CNRS UMR 8049 LabInfo IGM, Université Paris-Est Marne-la-Vallée 77454 Marne-la-Vallée Cedex 2 – France E-mail: [email protected]

Dr. Henri Prade CNRS UMR 5505 IRIT, Université Paul Sabatier, 118 route de Narbonne 31062 Toulouse Cedex 9 – France E-mail: [email protected]

Prof. Odile Papini CNRS UMR 6168 LSIS, ESIL, Université de la Méditerranée, 163, Avenue de Luminy, 13288 Marseille Cedex 9 – France E-mail: [email protected]

Dr. Steven Schockaert Department of Applied Mathematics and Computer Science, Krijgslaan 281, Universiteit Gent 9000 Gent – Belgium E-mail: [email protected]

ISBN 978-3-642-14754-8

e-ISBN 978-3-642-14755-5

DOI 10.1007/978-3-642-14755-5 Studies in Fuzziness and Soft Computing

ISSN 1434-9922

Library of Congress Control Number: 2010934931 c 2010 Springer-Verlag Berlin Heidelberg  This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typeset & Cover Design: Scientific Publishing Services Pvt. Ltd., Chennai, India. Printed on acid-free paper 987654321 springer.com

Acknowledgements

This book originated from a small workshop on Soft Methods for Statistical and Fuzzy Spatial Information Processing, which was co-located with the fourth International Symposium on Soft Methods in Probability and Statistics, and took place on September 11th 2008 in Toulouse, France. The contributions of this book include revised and extended versions of six of the eight papers that were presented at the workshop, together with seven invited contributions in order to improve its overall coverage. We are grateful to all the authors for their willingness to participate to this book, and for the time and effort they have spent to prepare their contribution. The chapters have been mainly reviewed by ourselves, and we also acknowledge the help from Tom M´elange. We thank John Grant for his careful proofreading of the introductory chapter. We have furthermore been helped in the preparation of this book by Florence Bou´e, who has in particular maintained a webpage which was accessible to the contributors, during the preparation of the book. We are grateful to Janusz Kacprzyk for his invitation to submit this volume in the Studies in Fuzziness and Soft Computing series. Lastly, this book may be viewed as a result of the inter-regional action project n◦ 05013992 “GEOFUSE: Fusion d’informations g´eographiques incertaines”, jointly supported by the Conseils R´egionaux of Midi-Pyr´en´ees and of Provence-Alpes-Cˆ ote d’Azur, where three of the editors were involved in. The last editor has benefited from two travel grants of the Research Foundation – Flanders (FWO), giving him the opportunity of staying in Toulouse and working on the preparation of the book. Toulouse, June 2010

The editors

Contents

Introduction: Uncertainty Issues in Spatial Information . . . . . . Robert Jeansoulin, Odile Papini, Henri Prade, Steven Schockaert

1

Part 1: Describing Spatial Configurations Spatial Vagueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brandon Bennett

15

A General Approach to the Fuzzy Modeling of Spatial Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pascal Matsakis, Laurent Wendling, Jing Bo Ni

49

Bipolar Fuzzy Spatial Information: Geometry, Morphology, Spatial Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Isabelle Bloch

75

Fuzzy and Rough Set Approaches for Uncertainty in Spatial Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Theresa Beaubouef, Frederick E. Petry Part 2: Symbolic Reasoning and Information Merging An Exploratory Survey of Logic-Based Formalisms for Spatial Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Florence Dupin de Saint-Cyr, Odile Papini, Henri Prade Revising Geographical Knowledge: A Model for Local Belief Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Omar Doukari, Robert Jeansoulin, Eric W¨ urbel

VIII

Contents

Merging Expressive Spatial Ontologies Using Formal Concept Analysis with Uncertainty Considerations . . . . . . . . . . 189 Olivier Cur´e Generating Fuzzy Regions from Conflicting Spatial Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Steven Schockaert, Philip D. Smart Part 3: Prediction and Interpolation Fuzzy Methods in Image Mining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Alfred Stein Kriging and Epistemic Uncertainty: A Critical Discussion . . . 269 Kevin Loquin, Didier Dubois Scaling Cautious Selection in Spatial Probabilistic Temporal Databases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 Francesco Parisi, Austin Parker, John Grant, V.S. Subrahmanian Imperfect Spatiotemporal Information Analysis in a GIS: Application to Archæological Information Completion Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 Cyril de Runz, Eric Desjardin Uncertainty in Interaction Modelling: Prospecting the Evolution of Urban Networks in South-Eastern France . . . . . . . 357 Giovanni Fusco Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

List of Authors

Theresa Beaubouef Department of Computer Science and Industrial Technology, Southeastern Louisiana University, Hammond, LA 70402 [email protected] Brandon Bennett Division of Artificial Intelligence, School of Computing, University of Leeds, Leeds , LS2 9JT, UK. [email protected] Isabelle Bloch D´epartement Traitement du Signal et des Images, T´el´ecom ParisTech, CNRS LTCI, 46 rue Barrault, F-75634 Paris Cedex 13 [email protected] Olivier Cur´ e Equipe Terre Digitale, Universit´e Paris-Est, 5, bd Descartes, Marne la Vall´ee 77454 France, [email protected] Cyril de Runz CReSTIC-SIC, IUT de Reims-Chˆ alons-Charleville,

Rue des Cray`eres, BP 1035, 51687 Reims, Cedex 2, France [email protected] Eric Desjardin CReSTIC-SIC, IUT de Reims-Chˆ alons-Charleville, Rue des Cray`eres, BP 1035, 51687 Reims, Cedex 2, France [email protected] Omar Doukari CNRS UMR 6168 LSIS, Campus de Saint-J´erˆome, Avenue Escadrille Normandie-Niemen, 13397 Marseille Cedex - France [email protected] Didier Dubois IRIT, Universit´e Paul Sabatier, 118 Route de Narbonne, F-31062 Toulouse Cedex 9 [email protected] Florence Dupin de Saint-Cyr IRIT, Universit´e Paul Sabatier, 118 Route de Narbonne, F-31062 Toulouse Cedex 9 [email protected]

X

List of Authors

Giovanni Fusco UMR 6012 ESPACE, Universit´e de Nice-Sophia Antipolis, 98 Bd Herriot BP 3209, 06204 Nice cedex 3, France [email protected]

Odile Papini LSIS UMR-CNRS 6168, ESIL, Universit´e de la M´editerran´ee, Avenue de luniversit´e BP 132, 83957 La Garde cedex, France [email protected]

John Grant Towson University, Towson, Maryland, USA and University of Maryland, College Park, MD 20742, USA [email protected]

Francesco Parisi Universit´ a della Calabria, Via P.Bucci, cubo 41/C, 87036 Rende (CS), Italy [email protected]

Robert Jeansoulin CNRS UMR 4089 LabInfo IGM, Universit´e Paris-Est Marne-la-Vall´ee, 77454 Marne-la-Vall´ee Cedex - France [email protected]

Austin Parker University of Maryland, College Park, MD 20742, USA [email protected]

Frederick E. Petry Mapping Charting and Geodesy Branch, Naval Research Laboratory, Stennis Space Center, MS 39529 Kevin Loquin D´epartement Traitement du Signal [email protected] et des Images, T´el´ecom ParisTech, CNRS LTCI, Henri Prade 46 rue Barrault, IRIT, Universit´e Paul Sabatier, F-75634 Paris Cedex 13 118 Route de Narbonne, [email protected] F-31062 Toulouse Cedex 9 [email protected] Pascal Matsakis Deptartment of Computing and Steven Schockaert Information Science, Department of Applied Mathematics University of Guelph, and Computer Science, Guelph, ON N1G 2W, Canada Ghent University, [email protected] Krijgslaan 281, 9000 Gent, Belgium [email protected] JingBo Ni Deptartment of Computing and Information Science, University of Guelph, Guelph, ON N1G 2W, Canada [email protected]

Philip Smart School of Computer Science, Cardiff University, 5 The Parade, Roath, Cardiff, UK [email protected]

List of Authors

XI

Alfred Stein ITC, University of Twente, PO Box 6, 7500 AA Enschede, The Netherlands [email protected]

Laurent Wendling Universit´e Paris Descartes, 45, rue des Saints-Pres, 75270 Paris Cedex 06 France [email protected]

V.S. Subrahmanian University of Maryland, College Park, MD 20742, USA [email protected]

Eric W¨ urbel CNRS UMR 6168 LSIS, Universit´e du Sud Toulon-Var, 83957 La Garde Cedex, France [email protected]

Introduction: Uncertainty Issues in Spatial Information Robert Jeansoulin, Odile Papini, Henri Prade, and Steven Schockaert

Abstract. This introductory chapter serves two purposes. First, it provides a brief overview of research trends in different areas of information processing for the handling of uncertain spatial information. The discussion focuses on the diversity of spatial information, and the different challenges that may arise. Second, an overview of the contents of this edited volume is presented. We also point out the novelty of the book, which goes beyond geographical information systems and considers different forms of quantitative and qualitative uncertainty.

1 The Nature of Spatial Information Variety of Spatial Information The term spatial information refers to pieces of information that are associated with locations, which typically refer to points or regions in some two- or threedimensional space. Many applications deal with geographic information [8, 35], in which case the space under consideration is the surface of the Earth. Other Robert Jeansoulin CNRS UMR 8049 LabInfo IGM, Universit´e Paris-Est Marne-la-Vall´ee 77454 Marne-la-Vall´ee Cedex 2, France e-mail: [email protected] Odile Papini CNRS UMR 6168 LSIS, 163, Avenue de Luminy, ESIL, Universit´e de la M´editerran´ee 13288 Marseille Cedex 9, France e-mail: [email protected] Henri Prade CNRS UMR 5505 IRIT, Universit´e Paul Sabatier, 118 route de Narbonne 31062 Toulouse Cedex 9, France e-mail: [email protected] Steven Schockaert Department of Applied Mathematics and Computer Science, Krijgslaan 281, Universiteit Gent 9000 Gent, Belgium e-mail: [email protected] R. Jeansoulin et al. (Eds.): Methods for Handling Imperfect Spatial Info., STUDFUZZ 256, pp. 1–11. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

2

R. Jeansoulin et al.

applications, however, deal with spatial information of a quite different kind, ranging from medical images (e.g. MRI scans [11]) to industrial product specifications (e.g. computer aided design and manufacturing [56]), or the layout of buildings or campuses [31, 32, 3]. In addition to describing aspects of the real world, spatial information may also describe virtual environments [26]. Beyond virtual environments, we may even consider space in a metaphorical way for describing or reasoning about the meaning of concepts, viewed as regions in a multi-dimensional space (e.g. the conceptual spaces of Gardenf¨ors [25]). Spatial information may have various origins. Geographic information may be derived from satellite images or other types of remote sensing, it may be collected using various types of sensors, it may result from the collection of census data, or from textual descriptions, among others. Other types of spatial information may be explicitly created by a human user (e.g. in the case of computer aided design), although it may also be derived e.g. from scanning devices, from a robot mapping its environment, or again from textual descriptions. Given this diversity, it should not come as a surprise that spatial information has been studied from different angles in different research communities, as described in more detail in Section 2. Dealing with Spatial Uncertainty Appropriate abstractions are needed to deal with the complexity of spatial configurations. A first observation is that some information is naturally associated with points, e.g. the altitude of a location, while other types of information pertain only to regions, e.g. the population of a city. A common approach to deal with the former type is to discretize a bounded fragment of space into a finite number of cells. This allows us to quantify the value of the parameters associated with each cell, leading to the so-called field-based models, which are also referred to as grids or matrices. To deal with spatial information that is not tied to particular points, we may introduce a mechanism to refer to spatial entities of interest, e.g. using names or coordinates, and to describe relations between them. Both the qualitative and quantitative spatial information available to applications usually contains uncertainty. Common to other applications that rely on measurement of observed phenomena, sensor data may be only imprecisely known (e.g. in the form of an interval); see e.g. [28]. Similarly, for instance, the classification of land coverage based on remote sensing images may be given in the form of a probability distribution [47]. In addition to these problems related to data acquisition, the way applications use spatial information may, by itself, introduce uncertainty. For instance, as it is impossible to make measurements at every point on Earth, interpolation or extrapolation techniques are required to estimate parameters for points where measured values are missing [9]. Information fusion is another process which may introduce uncertainty [16]. Indeed, in the face of conflicting sources, inconsistency may cast doubts on previously held beliefs, thus introducing uncertainty about them. A final cause for uncertainty is related to the use of labels to refer to places, their properties, and the relations between them. For instance, in fusion problems,

Introduction: Uncertainty Issues in Spatial Information

3

apart from the uncertainty affecting the pieces of information, the fact that different sources may use different partitions of space (e.g. electoral wards vs. parishes) is another reason for imprecision or uncertainty. Also, the labels used to describe properties of regions may belong to different ontologies (e.g. agricultural vs. botanical terminology). While the use of labels is paramount for interactions between a system and human users, they give rise to many practical problems. For instance, how do you say Gen`eve in Italian (Genova, Geneva)? Ghent in French? Lille in Dutch? Even worse, the same label is often used to refer to different places (e.g. Paris, France vs. Paris, Texas [33]), and even when a label unambiguously refers to one place, the spatial extent of this place may be ill-defined (e.g. downtown Toulouse [40]). Similar considerations apply to spatial relations (e.g. near, North-West of, etc.) and properties of places (e.g. densely populated). Note that labels may be used both for stating information, and for expressing queries. An example of the former case may be spatial representations that are derived from textual information (e.g. the web). In the case of queries, the uncertainty in the meaning of the labels may suggest some flexibility regarding what solutions are acceptable. Spatial Information Processing In practice, information processing is often divided into several subproblems. At a high level, we may consider three main steps [15], although they may not all be present in every application. The first one aims at clarifying raw information, e.g. by cleaning sensor information, or by synthesizing and structuring it. In the spatial domain, this may involve removing noise from remote sensing images, as well as analyzing and interpreting them. When collecting census information, for instance, this step also involves the assessment of the confidence levels in different sources. The second step obtains the information needed to address the problem at hand. This involves retrieving and combining information from one or more data repositories, and reasoning about it. Apart from querying geographic information systems (GIS) in general, this step may refer to the aforementioned information fusion and interpolation/extrapolation problems, or to a diagnosis based on the interpretation of medical images. The third and last step uses the retrieved information for decision making, or solving design or optimization problems: finding the best location for building a nuclear power plant, managing air traffic control, or deriving a strategy for avoiding the future flooding of a river. The three steps outlined above are usually studied in different research communities, as briefly described in the next section, but emphasizing only the issue of handling uncertainty. As we shall see in the last section, providing an overview of the book, most of the chapters address problems related to the second step.

2 Research Areas in Spatial Information The causes and remedies of spatial uncertainty have been studied for a long time. Spatial uncertainty is related mostly to spatial reasoning, to decision making

4

R. Jeansoulin et al.

involving space, and to the difficulty, sometimes the impossibility, of building a deterministic approach. The research about spatial uncertainty in the last decade has focused on the following domains: • uncertainty in spatial cognition, artificial intelligence, and vision, which is coupled to spatial reasoning; • uncertainty in geographic information systems, which is coupled to spatial data quality and spatial statistics. Uncertainty in Spatial Cognition, Artificial Intelligence, and Vision Spatial uncertainty has been extensively studied in the domain of cognitive science, from psychology to artificial intelligence. What does it mean to say “I’ve got no sense of direction”? What kind of information must a robot keep in memory to find its path? Choosing the right representation of space for a particular application is always an issue: every time you have to explain a route by phone, you must face a new problem. Is it relevant to use East-West indications? Or rather use “towards downtown”, “towards the river”, etc.? It always depends on what is the easiest to grasp, the most secure information that you will not mishandle, and not mismatch with a similar but wrong reference. Names are not the only source of spatial uncertainty; visual perception is one too. Optical illusions have been noticed and investigated for centuries. They are sometimes used in architecture, in the Greek Parthenon, in art works — noticeably in the Italian and French Renaissance — and more recently by the artist M.C. Escher. Interestingly enough, artificial intelligence constraint satisfaction approaches may be successfully applied for deciding if a line drawing of a three-dimensional object is actually realizable in physical space [12]. Optical illusions have been related to unconscious inferences, an idea first suggested in the mid 19th century by H. von Helmholtz, and to inhibition-influenced vision, by E. Mach. Experimental psychology has also addressed this problem, for instance, focusing on the question of the possible existence of a specific spatial working memory. Experiments made with 3D rotated figures seem to demonstrate that the difficulty of recognizing shapes depends on the rotation angle [38]. Based on such experiments D. Marr developed his computational theory in the late 1970s, collected in his posthumous book “Vision” [37]. At about the same time, the book “Mental Models” by P. Johnson-Laird [36] was published. It makes intensive use of relational reasoning, including spatial and temporal reasoning, as well as defeasible reasoning which allows for a change in one’s beliefs in the face of new observations. When communicating such a relational description, one of the arguments is used as a reference object. The proper choice of the reference objects is important when communicating a series of relational statements, as it may make the overall message easier to understand [20]. In artificial intelligence, leaving aside the early work on computer vision [4], the issue of reasoning about time, space, and uncertainty were considered independently at first. But only once temporal reasoning had been sufficiently mastered, after the introduction of Allen’s temporal interval relations [1], research on spatial reasoning has blossomed [19, 45, 6, 46, 10]. In parallel, studies in uncertainty representation

Introduction: Uncertainty Issues in Spatial Information

5

have led to the development of various settings beyond probability theory, namely Zadeh’s fuzzy sets [57, 30] and possibility theory [14], Shafer’s theory of belief functions [51], and imprecise probability [54]. The most prominent artificial intelligence domain where uncertainty and space have been jointly addressed — already since the late 1980s — is robotics, for automated planning purposes [31]. In particular, simultaneous localization and mapping (SLAM) [53, 41, 17] is a technique used in (mobile) robotics to build up a map within an unknown environment. Estimating uncertain spatial relationships is then one of the key issues. In spite of preliminary early attempts [27, 22] there have been few applications of fuzzy set based methods in robot navigation, with [58, 43, 49] being some of the exceptions. However, outside robotics, some works have focused on representing and reasoning about fuzzy spatial relations [18, 21, 34, 7, 50]. Uncertainty in Geographic Information Systems From artificial intelligence and robotics, we go to another domain where spatial uncertainty is definitely a big issue: cartographic mapping. Cartographic representations, land surveys, remote sensing, geographic information systems, and global positioning are providing data for the two main subdomains of mapping: the mapping of administrative and man-made spatial features, on the one hand, and the mapping of natural resources on the other. It has been a long time before men were able to use maps in the way we know them today. Cadastres are as old as tax collection, invented by the first Babylonian monarchs or Egyptian pharaohs. Areas and relative positions were all they needed to compute taxes, but errors and quarrels fed the courts, as they continue to feed them today. Geometers gained importance as land surveyors, as “arpenteurs”, and often, turning around spatial uncertainty, their statements became the reality. But this does not erase uncertainty, it merely adds other constraints to the system, which sometimes helps, and sometimes hampers the decision making. Spatial uncertainty analysis, in this context, mostly relates to geographic data quality, in the sense that the role of data quality management is to reduce the uncertainty in making decisions. Spatial uncertainty analysis has been, since the 1990s, an acknowledged discipline that integrates expertise from geographical sciences, remote sensing, spatial statistics and many others. Several international organizations have working groups on these issues: ISPRS (quality of spatio-temporal data and models), the European AGILE (spatial data usability), the International Cartographic Association (spatial data uncertainty and map quality). Standardisation bodies are concerned too: ISO 9000, which addresses the production and distribution of goods and services, and those in the field of geographic information, e.g. FGDC, OGC, CEN. There is a community of engineers in spatial uncertainty in the various national mapping agencies and in large private companies such as Microsoft and Google, as indeed the quality of geographic information is clearly an issue for them. In addition, several books have been published on this topic. In particular, [29] presents reflections by members of the International Cartographic Association, while [28, 52, 13] present research

6

R. Jeansoulin et al.

breakthroughs on issues related to the quality and the uncertainty of geographic information, while [59, 2] focus on uncertainty in geographic data. Cartographic mapping is not the only area related to GIS in which uncertainty appears. In the early years of image processing [48], following the release of the first digital cameras on Earth and above (with the release of Landsat Imagery in the early 1970s [55]), there has been a big boost in the automated acquisition of geographical information. Indeed, the automated processing of remote sensing images has proven to be an invaluable tool for feeding field-based models. Clearly, handling uncertainty is a central issue when interpreting remote sensing images in this way. Besides, spatial uncertainty is also prevalent in natural resources assessment [42], and must be estimated as it propagates in ecological models [8]. Geostatistics is a branch of statistics developed originally to predict distributions for mining operations [39]. It is currently applied in diverse branches of hydrology, meteorology, landscape ecology, soil science and agriculture (especially in precision farming), and branches of geography, such as epidemiology, and planning (logistics).

3 Overview Generations of children, reading “Hop o’ My Thumb” by Charles Perrault, also known under its original French title “Le Petit Poucet”, have been scared by spatial uncertainty. They were afraid to lose their path back, if being forced to decide their way: turn right or left? The intent of this book is not to scare children, nor to scare scholars. Still, we may give the reader some small white pebbles to keep him or her from getting lost in the forest of the literature on spatial uncertainty handling, or in the bush of the chapters that follow. To the best of our knowledge, this is the first book which is devoted to spatial uncertainty handling outside the traditional GIS setting. Uncertainty issues are especially addressed from a representation and reasoning point of view. In this sense, we only consider the second step of the information processing chain as described at the end of the first section. As such, the book does not discuss uncertainty issues in the interpretation of remote sensing images at the acquisition stage, and does not exclusively focus on geographical applications. Similarly, the book does not encompass planning and reasoning about actions in spatial environments, which is why uncertainty issues in robotics are not considered. The concept of uncertainty by itself is understood in a broad sense, including both quantitative and more qualitative approaches, dealing with variability, epistemic uncertainty, as well as with vagueness of terms. The contributions of this book are clustered around three general issues: i) description of spatial configurations, ii) symbolic reasoning and information merging, and iii) prediction and interpolation. Part 1 especially focuses on modeling uncertainty in the meaning (vagueness) of linguistic constructs that are used to describe spatial relations and predicates. The first chapter in this part, by B. Bennett, presents a tutorial overview of diverse methods to deal with the problem of spatial vagueness, before focusing in more detail on a new method called standpoint semantics. This

Introduction: Uncertainty Issues in Spatial Information

7

latter approach refines the supervaluation semantics of Fine [23], by adding structure on the set of possible precisifications. In particular, in the spatial domain, precisifications of a vague predicate or relation may depend on the standpoint one takes regarding the most appropriate thresholding of some parameter. The second chapter, by P. Matsakis, L. Wendling and J. Ni, is concerned with describing relative positions between objects, and spatial relations to reference objects, where an object is a crisp or fuzzy region, in raster or vector form, of 2D or 3D space. The models presented may be used, e.g. to identify the most salient spatial relation between two given objects, or to identify the object that best satisfies a given relation to a reference object. Subsequently, the chapter by I. Bloch explores the idea of bipolarity in the modeling of spatial information. The idea is to distinguish between locations considered as being really possible for a given object, and locations which are only not impossible. This bipolar view is then embedded in the framework of fuzzy mathematical morphology, and finally illustrated on a medical application. The last chapter of the first part, by T. Beaubouef and F. E. Petry, provides a broad overview of the possible uses of fuzzy and rough sets [44] in geographical information systems. Rough sets naturally allow for a granular view of space and of the description of land coverage, while the use of fuzzy sets and relations applies to the modeling of linguistic terms. Special attention is paid to the modeling of rough spatial relations, to the use of spatial indexing techniques, such as R-trees, for fuzzy regions, and to rough object-oriented spatial databases. Part 2 of the book deals with applications in artificial intelligence, and in particular with the problem of reasoning about spatial relations, and dealing with inconsistency in information merging. The first chapter by F. Dupin de Saint-Cyr, O. Papini and H. Prade provides an extensive survey of propositional and modal logics for describing mereo-topological or geometrical relations between regions, and for handling properties associated with regions. The handling of uncertainty in such frameworks is discussed. In particular, one may be uncertain whether a property holds in a region, and if it does, whether it holds everywhere in the region, or only in some part(s) of it. The next chapter, by O. Doukari, R. Jeansoulin and E. W¨urbel, presents a particular approach to the revision of propositional knowledge bases when receiving new information, which is well-suited for geographical information. This approach is centered around an assumption of locality, where conflicts related to one region of space do not affect what is known about regions that are far away. The third chapter of the second part, by O. Cur´e, addresses the problem of merging spatial ontologies that are used to describe properties of regions. To deal with the problem of heterogeneous vocabulary usage, an approach based on formal concept analysis [24, 5] is proposed which enables the creation of concepts that are not encountered in any of the given ontologies, and manages the resulting uncertainty. The last chapter, by S. Schockaert and P. D. Smart, deals with generating spatial scenarios which are compatible with available, and possibly conflicting, spatial constraints, using a genetic algorithm. To handle potential conflicts in a more flexible way, the approach may result in fuzzy regions, which are represented as a finite collection of nested polygons.

8

R. Jeansoulin et al.

Part 3 gathers chapters about interpolation and prediction of spatial phenomena. The first three chapters are methodologically oriented, while the two others are directly motivated by real-world case studies. The chapter by A. Stein presents a decision making approach based on the use of remote sensing images. In an image mining approach it discusses how such images are obtained and interpreted, how the resulting information may be used to identify objects, how these objects are tracked over time, and how this may lead to meaningful predictions for the future. Special consideration is given to issues of data quality, and solutions are provided based on fuzzy set theory and spatial statistics. The approach is illustrated with a case study on the flooding of Tongle Sap Lake in Cambodia. Next, the chapter by K. Loquin and D. Dubois deals with the kriging methodology used in geostatistics for the interpolation and extrapolation of parameters which are known only at a finite number of points in space. The kriging methodology is supposed to account for the uncertainty induced by the variability of the considered parameter over space. However, the chapter emphasizes the fact that the epistemic uncertainty appearing both in data specification and random function estimation is not properly taken into account by the standard approach. Subsequently, the merits and limitations of fuzzy and interval based extensions of kriging for handling epistemic uncertainty are discussed. The third chapter, by F. Parisi, A. Parker, J. Grant and V. S. Subrahmanian is concerned with spatial probabilistic temporal databases. The goal is to provide efficient support for queries asking for all pairs of objects and time points such that the object is in a specified region at that time, with a probability that is within a given interval. Solutions to such queries may be represented as convex polytopes in some high-dimensional space, and methods are provided for approximating such polytopes in an efficient way. The approach is evaluated on both synthetic data and real-world data about ship locations. The next chapter, by C. de Runz and E. Desjardin, presents a way to deal with scarce pieces of evidence, obtained from archaeological data. The goal is to reconstruct plausible spatial configurations (e.g. the layout of the streets in an ancient city), and to visualize them. The proposed method takes advantage of a fuzzy extension of the Hough transform from image processing, which may be applied to fuzzy pieces of information. Finally, the chapter by G. Fusco uses a Bayesian network methodology for predicting the evolution of spatial networks derived from data about flows through a dominant flows approach. Such networks may be as diverse as representing people commuting between their home and work, money transfers between different areas, or the migration of people between different suburbs. The presented case study is based on commuter trips in a region of Southeastern France. As a whole, the book intends to illustrate the different circumstances where spatial uncertainty may be encountered, and the different approaches that may be considered to cope with it.

Introduction: Uncertainty Issues in Spatial Information

9

References 1. Allen, J.: Maintaining knowledge about temporal intervals. Communications of the ACM 26(11), 832–843 (1983) 2. Atkinson, P., Foody, G.: Uncertainty in Remote Sensing and GIS: Fundamentals. Wiley & Sons, Chichester (2002) 3. Bahl, P., Padmanabhan, V.N.: Radar: An in-building RF-based user location and tracking system. In: Proceedings of IEEE Infocom, vol. 2, pp. 775–784 (2000) 4. Ballard, D., Brown, C.: Computer Vision. Prentice Hall, Englewood Cliffs (1982) 5. Barbut, M., Monjardet, B.: Ordre et Classification, Alg`ebre et Combinatoire, Hachette, vol. 2 (1970) 6. Bennett, B.: Determining consistency of topological relations. Constraints 3(2-3), 213–225 (1998) 7. Bloch, I.: Spatial reasoning under imprecision using fuzzy set theory, formal logics and mathematical morphology. International Journal of Approximate Reasoning 41(2), 77–95 (2006) 8. Burrough, P.A., McDonnell, R.A.: Principles of Geographical Information Systems. Oxford University Press, Oxford (1998) 9. Chil`es, J.-P., Delfiner, P.: Geostatistics: Modeling Spatial Uncertainty. Wiley, Chichester (1999) 10. Cohn, A., Renz, J.: Qualitative spatial representation and reasoning. In: van Hermelen, F., Lifschitz, V., Porter, B. (eds.) Handbook of Knowledge Representation, pp. 551–596. Elsevier, Amsterdam (2008) 11. Colliot, O., Camara, O., Bloch, I.: Integration of fuzzy spatial relations in deformable models–Application to brain MRI segmentation. Pattern Recognition 39(8), 1401–1414 (2006) 12. Cooper, M.: Line Drawing Interpretation. Springer, Heidelberg (2008) 13. Devillers, R., Jeansoulin, R.: Fundamentals of Spatial Data Quality. Geographical Information Systems. ISTE (2006) 14. Dubois, D., Prade, H.: Possibility Theory: An Approach to Computerized Processing of Uncertainty. Plenum Press, New York (1988) 15. Dubois, D., Prade, H., Yager, R.R.: Information engineering and fuzzy logic. In: Proceedings of the Fifth IEEE International Conference on Fuzzy Systems, pp. 1525–1531 (1996) 16. Dupin de Saint-Cyr, F., Jeansoulin, R., Prade, H.: Fusing uncertain structured spatial information. In: Greco, S., Lukasiewicz, T. (eds.) SUM 2008. LNCS (LNAI), vol. 5291, pp. 174–188. Springer, Heidelberg (2008) 17. Durrant-Whyte, H., Bailey, T.: Simultaneous localisation and mapping (SLAM): Part I the essential algorithms. Robotics and Automation Magazine 13(2), 99–110 (2006) 18. Dutta, S.: Approximate spatial reasoning: integrating qualitative and quantitative constraints. International Journal of Approximate Reasoning 5(3), 307–330 (1991) 19. Egenhofer, M., Franzosa, R.: Point-set topological spatial relations. International Journal of Geographical Information Systems 5(2), 161–174 (1991) 20. Ehrlich, K., Johnson-Laird, P.: Spatial descriptions and referential continuity. Journal of Verbal Learning and Verbal Behavior 21(3), 296–306 (1982) 21. Esterline, A., Dozier, G., Homaifar, A.: Fuzzy spatial reasoning. In: Proceedings of the 1997 International Fuzzy Systems Association Conference, pp. 162–167 (1997) 22. Farreny, H., Prade, H.: Tackling uncertainty and imprecision in robotics. In: Proceedings of the 3rd International Symposium on Robotics Research, pp. 85–91 (1986) 23. Fine, K.: Vagueness, truth and logic. Synthese 30, 265–300 (1975)

10

R. Jeansoulin et al.

24. Ganter, B., Wille, R.: Formal Concept Analysis. Springer, Heidelberg (1999) 25. G¨ardenfors, P.: Conceptual Spaces: The Geometry of Thought. MIT Press, Cambridge (2000) 26. Gillner, S., Mallot, H.: Navigation and acquisition of spatial knowledge in a virtual maze. Journal of Cognitive Neuroscience 10(4), 445–463 (1998) 27. Goguen, J.: On fuzzy robot planning. In: Zadeh, L.A., Fu, K.-S., Tanaka, K., Shimura, M. (eds.) Fuzzy Sets and their Applications to Cognitive and Decision Processes, pp. 429–447. Academic Press, London (1975) 28. Goodchild, M., Jeansoulin, R.: Data Quality in Geographic Information: From Error to Uncertainty. Hermes (1998) 29. Guptill, S., Morrison, J.: Elements of Spatial Data Quality. Pergamon, Oxford (1995) 30. Klir, G., Bo, Y. (eds.): Fuzzy Sets, Fuzzy Logic, and Fuzzy Systems: Selected Papers by L.A. Zadeh. World Scientific, Singapore (1996) 31. Latombe, J.-C.: Robot Motion Planning. Kluwer Academic Publishers, Norwell (1991) 32. Laumond, J.-P.P.: Robot Motion Planning and Control. Springer, New York (1998) 33. Leidner, J.L.: Toponym Resolution in Text. CRC Press, Boca Raton (2008) 34. Li, Y., Li, S.: A fuzzy sets theoretic approach to approximate spatial reasoning. IEEE Transactions on Fuzzy Systems 12(6), 745–754 (2004) 35. Longley, P., Goodchild, M., Maguire, D., Rhind, D.: Geographical Information Systems and Science. John Wiley & Sons, Chichester (2005) 36. Mani, K., Johnson-Laird, P.: The mental representation of spatial descriptions. Memory and Cognition 10(2), 181–187 (1982) 37. Marr, D.: Vision: A Computational Investigation into the Human Representation and Processing of Visual Information. Henry Holt and Co. (1982) 38. Marr, D., Nishihara, H.: Representation and recognition of the spatial organization of threedimensional shapes. Proceedings of the Royal Society of London. Series B, Biological Sciences 200(1140), 269–294 (1978) 39. Matheron, G.: Trait´e de G´eostatistique Appliqu´ee. Editions Technip (1962) 40. Montello, D., Goodchild, M., Gottsegen, J., Fohl, P.: Where’s downtown?: Behavioral methods for determining referents of vague spatial queries. Spatial Cognition & Computation 3(2), 185–204 (2003) 41. Montemerlo, M., Thrun, S., Koller, D., Wegbreit, B.: FastSLAM: A factored solution to the simultaneous localization and mapping problem. In: Proceedings of the 18th National Conference on Artificial Intelligence, pp. 593–598 (2002) 42. Mowrer, H., Congalton, R.: Quantifying Spatial Uncertainty in Natural Resources: Theory and Applications for GIS and Remote Sensing. CRC Press, Boca Raton (2000) 43. Oriolo, G., Ulivi, G., Vendittelli, M.: Fuzzy maps: a new tool for mobile robot perception and planning. Journal of Robotic Systems 14(3), 179–197 (1997) 44. Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning about Data. Springer, Heidelberg (1991) 45. Randell, D., Cui, Z., Cohn, A.: A spatial logic based on regions and connection. In: Proceedings of the 3rd International Conference on Knowledge Representation and Reasoning, pp. 165–176 (1992) 46. Renz, J., Nebel, B.: On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the Region Connection Calculus. Artificial Intelligence 108(1-2), 69–123 (1999) 47. Richards, J., Jia, X.: Remote Sensing Digital Image Analysis: An Introduction. Springer, Heidelberg (2006) 48. Rosenfeld, A., Kak, A.: Digital Picture Processing. Academic Press, London (1982)

Introduction: Uncertainty Issues in Spatial Information

11

49. Saffiotti, A.: The uses of fuzzy logic in autonomous robot navigation. Soft Computing 1(4), 180–197 (1997) 50. Schockaert, S., De Cock, M., Kerre, E.E.: Spatial reasoning in a fuzzy region connection calculus. Artificial Intelligence 173(258-298) (2009) 51. Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976) 52. Shi, W., Fisher, P., Goodchild, M.: Spatial Data Quality. Taylor & Francis, Abington (2002) 53. Smith, R.C., Cheeseman, P.: On the Representation and Estimation of Spatial Uncertainty. The International Journal of Robotics Research 5(4), 56–68 (1986) 54. Walley, P.: Statistical Reasoning with Imprecise Probabilities. Chapman & Hall, Boca Raton (1991) 55. Williams, R., Carter, W.: ERTS-1, a new window on our planet. U.S. Geological Survey (1976) 56. Wilson, R., Latombe, J.: Geometric reasoning about mechanical assembly. Artificial Intelligence 71(2), 371–396 (1994) 57. Yager, R., Ovchinnikov, S., Tong, R., Nguyen, H. (eds.): Fuzzy Sets and Applications: Selected Papers by L.A. Zadeh. Wiley, Chichester (1987) 58. Yen, J., Pfluger, N.: A fuzzy logic based extension to Payton and Rosenblatt’s commandfusion method for mobile robot navigation. IEEE Transactions on Systems, Man and Cybernetics 25(6), 971–978 (1995) 59. Zhang, J., Goodchild, M.: Uncertainty in Geographical Information. Taylor & Francis, Abington (2002)

Part 1: Describing Spatial Configurations

Spatial Vagueness Brandon Bennett

Abstract. This chapter explores the phenomenon of vagueness as it relates to spatial information. It will be seen that many semantic subtleties and representational difficulties arise when spatial information is affected by vagueness. Moreover, since vagueness is particularly pervasive in spatial terminology, these problems have a significant bearing on the development of computational systems to provide functionality involving high-level manipulation of spatial data. The paper begins by considering various foundational issues regarding the nature and semantics of vagueness. Overviews are then given of several approaches to spatial vagueness that have been proposed in the literature. Following this, a more detailed presentation is given of the relatively recently developed standpoint theory of vagueness and how it can be applied to spatial concepts and relations. This theory is based on the identification of parameters of variability in the meaning of vague concepts. A standpoint is a choice of threshold values determining the range of variation over which a vague predicate is judged to be applicable. The chapter concludes with an examination of a number of particularly significant vague spatial properties and relations and how they can be represented.

1 Introduction Suppose I am asked whether the Leeds City Art Gallery is close to Leeds University. Perhaps am not sure what to say. Suppose I then take an accurate map and determine the distance between the university main entrance and the entrance to the art gallery is 965m. Now, I have a pretty accurate measure of the distance but still I may be undecided as to whether to describe the gallery as ‘close to’ the university. This is because ‘close to’ is a vague relation. The example, illustrates a key property of vagueness, which is that it is distinct from uncertainty. Vagueness is not a result of Brandon Bennett School of Computing, University of Leeds e-mail: [email protected] R. Jeansoulin et al. (Eds.): Methods for Handling Imperfect Spatial Info., STUDFUZZ 256, pp. 15–47. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

16

B. Bennett

lack or imprecision of our knowledge about the world, rather it arises from lack of definite criteria for the applicability of certain linguistic terms. Even with complete and accurate knowledge of all relevant objective facts, one may be unsure as to whether a vague description is appropriate to a given situation. In this chapter, I shall examine the phenomenon of vagueness as it relates to spatial information. We shall see that vagueness is particularly pervasive in spatial terminology, and also that, where a description involves both vagueness and spatiality, semantic subtleties are encountered that do not arise when vagueness operates in a non-spatial context. I shall begin by examining some fundamental issues relating to our understanding of vagueness, how it should be represented, and how it interacts with spatial information and representations. Following this, I shall give overviews of some approaches to spatial vagueness that have been proposed in the literature. I shall then present a particular analysis of spatial vagueness, which is based on the identification of parameters of variability in the meaning of vague predicates and on the notion of a standpoint. A standpoint is a choice of threshold values determining the range of variation over which a vague predicate is judged to be applicable. Finally, I shall look in more detail at certain vague spatial predicates that I consider to be of particular significance and examine how these might be modelled in the context of a formal representation system.

2 The Nature of Vagueness 2.1 Distinguishing Vagueness from Generality and Uncertainty ‘Vagueness’ as I use the term is distinct from both generality and uncertainty. A predicate or proposition may be more or less general according to the range of individuals to which it applies or the range of circumstances under which it holds. However, as long as the boundary between correct and incorrect applications is definite, I shall not consider generality to be a form of vagueness. For example, the sentence ‘Tom’s house is within 5 miles of the Tower of London’ is general but not vague, since its truth can be determined by measuring the distance between the two buildings. By contrast ‘Tom lives near to the Tower of London’ is vague, since even when I know the exact distance, the truth of this sentence is indeterminate. A proposition is uncertain if we do not know whether it is true or not. In most circumstances we describe a proposition as uncertain when the reason we do not know whether it is true is that we do not possess complete and accurate knowledge about the state of the world. In so far as the term ‘uncertainty’ is taken to apply only in such cases, vagueness is distinct from uncertainty, since it may occur even where we do have complete and accurate knowledge of the world. However, some have argued (notably Williamson (1992)) that vagueness does indeed arise from lack of knowledge: lack of knowledge about the meanings of terms; and is thus a kind of uncertainty. This position is known as the epistemic view of vagueness. Although it has a devoted following, the epistemic view is not widely accepted. The

Spatial Vagueness

17

main objection is that, in the case of a vague linguistic term, there is no fact of the matter as to what is its true precise meaning. Hence, vagueness does not consist in ignorance of any objective fact and is thus not epistemic in nature. Proponents of the epistemic view may counter this objection by claiming that there is a fact of the matter but it is unknowable.1 My opinion is that vagueness is distinct from uncertainty and that any strong form of the epistemic view is misleading. Nevertheless, vagueness and uncertainty do share significant logical properties and one would expect similar kinds of formal representation to apply to the two phenomena. So, if vagueness is not a kind of generality or uncertainty, what is it? As one might expect, many different theories have been proposed. In fact, even once we have distinguished vagueness from generality and uncertainty, there may well be more than one semantic phenomena operating under the name of ‘vagueness’.

2.2 Vagueness in Different Linguistic Categories Whether or not there are multiple kinds of vagueness, it is certainly true that vagueness affects different types of linguistic expression, different syntactic categories, and these must be treated in, at least superficially, and perhaps fundamentally, distinct ways. I need only briefly consider the category of propositions. Obviously, propositions can be and often are vague. However, it is, I believe, uncontroversial that propositions are only vague in so far as they contain constituent parts that are vague. Hence, I shall proceed to examine sub-propositional expressions. But in considering these I shall of course be concerned with the central semantic issue of how they contribute to the truth conditions of propositions in which they occur. Perhaps the most obvious and most studied type of vague expression is adjectives, prime examples being ‘bald’, ‘rich’ and ‘red’. This category also includes many spatially oriented adjectives, such as ‘tall’, ‘short’, ‘large’, ‘small’, ‘elongated’, ‘steep’, ‘undulating’ etc.. For such concepts, their vagueness seems to lie in the lack of clearly defined criteria for their applicability. Thus, when a vague adjective is predicated of an individual, the resulting proposition will be true in some cases and false in others, but for some individuals it will not be clear whether the predication should be considered true or false — such individuals are borderline cases. For count noun predicates such as ‘table’, ‘mountain’, ‘lake’, ‘village’, the manifestation of vagueness is similar but not exactly the same as in adjectives. As Kamp (1975) observed, the vagueness of count nouns differs from that of adjectives as to the number of parameters of variation that are usually involved. The applicability of an adjective typically depends on a relatively small number of factors (tallness 1

Lawry (2008) has proposed a weakened version of the epistemic view, which he calls the epistemic stance. Lawry suggests that while there may be no objective fact of the matter underlying the meaning of vague terms, people tend to use vague language as if there were.

18

B. Bennett

depends on height, redness on colour).2 By contrast, the conditions for applicability of a count noun usually involve a large number of factors. For instance, a table should be made of suitably solid material; it should have a flat surface, which is supported by legs; its height and other dimensions should be within certain appropriate ranges; and various other constraints should apply to its shape. The reason these diverse characteristics are gathered together within the meaning of ‘table’ is that objects with such a combination of properties are useful to people, and hence are frequently encountered and referred to by language users.3 This correlation of relevant attributes also means that for many count nouns it is difficult to decide which of them are essential and which are merely typical features (c.f. Waismann (1965)). Relational expressions may also be vague, and in the spatial domain the prototypical examples are the relations ‘... is near to ...’ and ‘... is far from ...’. The vagueness of such relations appears to be similar to that of adjectives. That is, their applicability is dependent on one or two parameters (for instance distance) but the values of the parameter(s) for which the relation is applicable are not precisely determined. As well as affecting predictive expressions, vagueness may also be associated with referential terms, i.e. names and definite descriptions. In the first category we have nominal expressions such as ‘Mount Everest’ and ‘The Sahara Desert’. In the second we have more complex constructs such as ‘the foothills of Mount Everest’ and ‘the area around the church’. There has been considerable debate about whether the vagueness of nominals is parasitic upon that of predicative expressions, or whether it is a separate form of vagueness. One way that vagueness of proper names might be analysed in terms of vagueness of predicates is via the following logical re-writing of a nominal into a definite description, which is then represented using Russell’s analysis of definite descriptions:4

Φ (‘Mount Everest’) ≡ ∃!x[Mountain(x) ∧ Named(x, ‘Mount Everest’) ∧ Φ (x)] Under this analysis, the vagueness of ‘Mount Everest’ is completely explained by the vagueness of ‘mountain’. A similar approach could be applied to definite descriptions, although these expressions can take many different forms and may require a variety of different analyses involving constituent vague count nouns, adjectives or relations, or some combination of these. In opposition to this analysis, is the view that the vagueness of nominal expressions cannot or should not be reduced to vagueness of predicative expressions, but has its own mode of operation. Here again two contrasting explanations can be given: one is that vague nominals indeterminately refer to one of many possible referents; the other is that the referents of vague nominals are intrinsically vague 2

3 4

There are some adjectives that might be considered exceptions to this. For example, ‘intelligent’ depends on intelligence, but this quality is manifest in many different attributes. Nevertheless, we often do treat intelligence as if it were a single measurable quantity. The phenomenon of clustering of exemplars, where logically distinct properties tend in the real world to occur in combination, is discussed in (Bennett, 2005). Following Russell, the syntax ∃![Φ (x)] is used to abbreviate ∃x[Φ (x) ∧ ∀y[Φ (y) → y = x]].

Spatial Vagueness

19

entities. According to the first view, the semantics for vague nominals would involve an indeterminate denotation function, perhaps modelled by a one-to-many function or by a collection of possible functions (such an account is usually called supervaluationist and will be considered in detail is section 6). According to the second view, the semantics must provide an explicit model of vague entities and incorporate a domain of such entities as the range of the denotation function for nominals. I shall further consider these alternatives in the next section which deals with the issue of whether vagueness is intrinsic or linguistic. Our consideration of vagueness in different categories of linguistic expression may be summarised as follows. Vagueness is present in its most obvious form in adjectives and relations, where conditions of applicability are indeterminate but are typically dependent on a small number of relevant properties. The vagueness of count nouns is similar to that of adjectives, but is typically dependent on more factors and it is often unclear which factors are relevant. The nature of the vagueness of proper nouns and definite descriptions is controversial. In relation to spatial information, the difference between vagueness located in a relation and vagueness located in a nominal expression is illustrated by the following example sentences: 1. 2. 3. 4.

The treasure is 20 km from the summit of Mount Everest. The treasure is 20 km from of Mount Everest. The treasure is near to the summit of Mount Everest. The treasure is near to Mount Everest.

In sentence 1 both the relation and the reference object are precise. In 2 the relation is precise but the reference object vague, whereas in 3 the relation is vague but the reference object precise. Finally, in 4 both relation and reference object are vague.

2.3 Is Vagueness Intrinsic or Linguistic? A fundamental issue in the ontology of vagueness concerns whether vagueness is an intrinsic property of certain kinds of real-world object (often called de re or ontic vagueness; or whether it is an entirely linguistic phenomenon (de dicto vagueness). For instance Tye (1990) supports the view that there are actual objects in the world that are vague (as well as vague linguistic expressions); whereas Varzi (2001a)) argues that all vagueness is essentially linguistic.5 It is not clear whether the handling of vagueness within a computational information system requires one to take a philosophically defensible position on ontological status of vagueness. However, the debate does have some bearing on the choice of a suitable representational and semantic framework. If one takes the view that vagueness is purely linguistic, then the phenomenon will be modelled in terms of indeterminate interpretation of predicates and/or naming terms, but the objects to which the predicates and names are applied will be modelled as precisely 5

A much discussed paper by Evans (1978) contains a formal proof that appears to show that some fundamental logical problems may arise from taking an ontic view of vagueness. However, the significance and implications of Evans’ proof are unclear and controversial.

20

B. Bennett

determined entities. For instance, the denotation of “Mount Everest” would be considered to be indeterminate, but each possible denotation would correspond to a precisely bounded volume of matter. By contrast, if one adopts an ontic view of vagueness, then the term “Mount Everest” would denote an entity that is in itself vague. For instance it could be identified with a fuzzy set of points, or with a cluster of possible extensions. Tye (1990) suggests that vagueness can be present both in predicates and also in objects. He argues that the vagueness of objects cannot simply be explained by saying that they are instances of vague predicates. In the case of material objects, Tye proposes that a vague material object is one for which the set of parts of the object is not fully determinate.6 This condition is of particular relevance to the investigation of spatial vagueness because it specifically identifies ontic vagueness with indeterminacy of spatio-temporal parts. I shall not take a rigid position on whether vagueness is primarily linguistic or ontic. In fact, it seems to me that it may not be possible to make a sharp distinction between ontic and linguistic properties. This is because the entities to which we refer are not given prior to our linguistic conventions; rather the ways that we use language play a significant role in determining the domain of objects. So the entities to which we might ascribe ‘ontic’ properties are themselves, at least partially, determined by linguistic convention and stipulation. This is especially evident in the case of predicates that are both vague and spatial. As we shall see in the next section, when considering the truth conditions for a predicate that vaguely characterises a spatial entity, one cannot assume that there is a pre-determined and well-defined object to which the predicate is applied. Moreover, from the point of view of establishing formal representations and semantics, the issue of whether vagueness should be modelled in terms of vague entities or in terms of indeterminate linguistic reference may be seen as more a matter of technical convenience than philosophical significance.

3 Vagueness and Spatiality Circumstances where vagueness interacts with spatiality present issues that do not arise where vagueness operates in a non-spatial context. In this section I describe some of the main issues that arise and present some illustrative examples.

3.1 The ‘Sorites’ Paradox The issue that has dominated much of the debate about vagueness over the last couple of millenia is the logical anomaly known as the sorites paradox. I will not be looking into this in detail here since, although it affects many spatial predicates, the 6

In fact Tye also suggests the further criterion that “there is no determinate fact of the matter about whether there are objects that are neither parts, borderline parts, nor non-parts of o”. This condition relates to the issue of second-order vagueness (i.e. the vagueness of the borderline between clear and indeterminate cases), which will not be considered in this chapter.

Spatial Vagueness

21

paradox is not essentially spatial in nature. Hence, considering it from a specifically spatial angle would not add much to the a great deal has already been written about this topic elsewhere (see e.g. Williamson (1994); Keefe and Smith (1996); Beall (2003)). The classic form of the sorites paradox involves the predicate ‘... is a heap’,7 and can be stated as follows: 1. This pile of 1,000,000 grains of sand is a heap. 2. If one removes one grain of sand from a heap, the remainder will still be a heap. ∴ 3. Even if we removed all the grains from this heap, it would still be a heap. Here, the conclusion, 3, appears to follow from the premisses by a kind of induction, with premiss 1 being the base case and premiss 2 being used as a principle of induction. Similar ‘slippery slope’ arguments can be formulated using pretty much any vague predicate — indeed, susceptibility to sorites arguments is often considered to be a necessary requirement for a predicate to be vague. Many such predicates are spatial in nature, obvious examples being ‘tall’, ‘large’, ‘near’ etc.. For example: 1. A man whose height is 2m is tall. 2. If a man of a given height is tall, then a man whose hight is 1mm less is also tall. ∴ 3. A man whose height is 1m is tall. A characteristic feature of sorites susceptible predicates is that they are associated with some mode of variation, often a measurable property, that is either continuous or fine grained. In attempting to explain the sorites, it is usually the inductive premiss that comes under most scrutiny. A typical diagnosis is that this premiss must be false, although we are for some reason compelled to believe that it ought to be true. The problem then is explaining why we have a tendency to think it should be true. Accounts of this often refer to the supposition that vague predicates cannot be used to make precise distinctions because their limits of applicability are not well-defined. Hence, if two samples are very similar in all attributes that are relevant to the application of a vague predicate V(x), then it must be that either both of them or neither of them are instances of V. Such reasoning can be used to justify the inductive premiss of a sorites argument. In practice it seems that the sorites paradox has not had a direct impact on existing computational spatial information systems. The types of computer system most in danger of being affected by sorites paradoxes are those concerned with representing commonsense knowledge or implementing commonsense reasoning — for instance, systems such as CYC (Guha and Lenat, 1990). It would be inadvisable for such a system to directly encode any proposition or rule similar to a sorites induction step, since it would be clear that it could quickly lead to problems. 7

‘Sorites’ is the Greek word for heap. More accurately, it is an adjective meaning ‘heaped up’.

22

B. Bennett

Current implemented systems generally avoid the paradox simply by not taking vagueness into account. However, as we shall see later in the chapter, a number of computationally-oriented approaches to representing vagueness have been proposed. In fact, these tend to tackle the phenomenon using modelling techniques that are rather different from the axiomatic analysis that leads to the paradox. Hence, they might be regarded as skirting round the problem rather than solving it. It is not yet clear whether such circumvention will be ultimately satisfactory. As more sophisticated applications and representations are developed, it may turn out that the sorites paradox will need to be confronted more directly.

3.2 The Problem of Individuation That vagueness results in a lack of well-defined criteria for the application of predicates is widely recognised. However, there is a further consequence of vagueness that becomes especially important when considering cases where the vagueness of a predicate affects the determination of the spatial extension of entities satisfying that predicate. The problem is that of individuation — i.e. the determination of the class of entities to which the predicate might be applied. The issue of individuation is illustrated in Fig. 1, which shows a section of an extended water body. Suppose we now wish to find instances of the count noun predicates ‘river’ and ‘lake’ in relation to this water region. One interpretation is that the region is simply a river section that is rather irregular in width and includes a number of bulges. But another interpretation is that the water body consists of three lakes connected by short river channels.8 The possibility of these different interpretations arises from the vagueness of the terms ‘river’ and ‘lake’. However, what makes this case especially problematic is that there is no pre-existing division of the water region into segments that we choose to describe as ‘river’, ‘lake’ or whatever. Rather, the acceptable ways in which water can be segmented into features is dependent upon the meanings of these feature terms. And since these meanings are affected by vagueness, it is indeterminate what is the most appropriate segmentation.

Fig. 1 Is this just an irregular river, or is it three lakes joined by a river?

Fig. 2 illustrates a similar case concerning the demarcation of forest regions based on the density of trees. If we vary the minimum tree density threshold 8

One might comment that the distinction between lake and river also significantly depends on water flow. This is certainly true but much the same segmentation issue would still arise, and it is much easier to illustrate in terms of shape rather than flow.

Spatial Vagueness

23

Fig. 2 Possible forest demarcations for a given tree distribution. Inner contours are based on a high threshold on the tree density. Outer contours are based on lower thresholds.

required for a region to be classified as a forest, the extension of forest regions will clearly vary. Furthermore, the number of forest regions may alter according to the threshold.

3.3 Consequences of Indeterminate Spatial Extension A well as illustrating the problem of individuation, the examples given in the previous section also show how in association with spatial concepts vagueness not only affects classification but also results in indeterminacy of spatial extension. Consequently vagueness in spatially related terms such as geographic feature types often leads to an indeterminacy in spatial predicates and relations, even where the predicates and relations themselves are apparently precise. Suppose we want to compare the size of a particular expanse of desert at two different time points on the basis of precipitation data collected at two different time points. (Of course the classification of desert may depend on features other than precipitation, but whatever measure of aridity is employed the issue I am about to describe will occur.) In order to demarcate the extension of desert at each time point, we need to choose some threshold for the amount of precipitation below which we will classify a region as ‘desert’. But if we do this we may find that whether the demarcated desert region expands or contracts between the two time point depends on the particular threshold value that we choose. This problem is illustrated in Fig. 3. The sub-figures on the left of the diagram show precipitation contour maps for the year 1965. In the upper map a threshold of 30mm per month is taken to bound the shaded desert region, whereas in the lower one a threshold of 20mm per month is used to demarcate the desert. On the right of the diagram we have precipitation contours for the year 2009 and again precipitation thresholds of 30 and 20 mm per month respectively are used to bound the desert region in the upper and lower maps. We see that if we choose the higher threshold

24

B. Bennett

(30mm) the demarcated desert region is seen to expand from 1965 to 2009, whereas if we choose the lower threshold (20mm), over the same period the demarcated desert appears to contract. What has happened is that although the driest region has shrunk the region that is dry but somewhat less so has expanded. We could of course avoid this issue by asking about the change in overall precipitation calculated by integrating the precipitation values over the whole map. But this is a different question from that of whether the desert expands or contracts. Moreover, describing the world in terms of average or cumulative measures over a large area can be misleading. For example, if a land region consists of one part that is bitterly cold and another that is intolerably hot, it would be uninformative to classify the whole region as temperate.

Fig. 3 Desert demarcation according to different precipitation thresholds

4 A Theory of Crisp and Blurred Regions In this section I give an overview of the theory of vague regions presented by Cohn and Gotts (1996b). This paper is best known for the so-called ‘egg-yolk’

Spatial Vagueness

25

model of vague spatial regions, which had originally been proposed by Lehmann and Cohn (1994).9 However, the Cohn and Gotts paper starts by developing a purely axiomatic theory of vague regions, for which the egg-yolk model is given as only one possible interpretation. The style of presentation of this theory suggests a de re ontology of vagueness — i.e. vagueness lies in the regions themselves, rather than in the linguistic expressions that refer to these regions. Moreover the egg-yolk semantics can be regarded as a providing a simple model of a spatially vague object. Although presented primarily as a theory of ‘vague’ spatial regions, the theory is more or less neutral about the type of indeterminacy that it models. Thus, it can equally well be used to represent spatial indeterminacy arising from uncertainty, and indeed the terms ‘vague’ and ‘uncertain’ are used more or less interchangeably in the original papers that proposed the theory. The theory is described by means of the following terminology. The term ‘crisp’ is used to mean that a region is precisely determined, and ‘blurred’ to mean that it is indeterminate (either vague or uncertain). More generally, a region x may be described as being a crisper version of another region y, or more succinctly, one may say that ‘x is a crisping of y’. These propositions both assert that the region x is a more precise version of y. This relation can be understood as meaning that every precise boundary that could be considered as a possible boundary of x could also be considered as a possible boundary of y.

4.1 An Axiomatic Theory of the ‘Crisping’ Relation A formal theory of crisp and blurred regions is developed in similar fashion to axiomatic theories of mereology (Simons, 1987) or the topological Region Connection Calculus (Randell et al., 1992). The theory is based on the primitive relation x ≺ y, read as ‘x is a crisping of y’. This relation is defined to be a strict partial order: A1) ∀xy[x ≺ y → ¬(y ≺ x)] A2) ∀xyz[(x ≺ y ∧ y ≺ z) → x ≺ z)] Before stating the further axioms satisfied by ≺, it will be helpful to introduce some definitions: D1) x  y ≡def (x ≺ y ∨ x = y) D2) MA(x, y) ≡def ∃z[z  x ∧ z  y] D3) Crisp(x) ≡def ¬∃y[y ≺ x] D1 is just a convenient abbreviation, giving the reflexive counterpart of the crispness ordering. D2 defines the key relation of ‘mutual approximation’. MA(x, y) holds just in case there is some third region z that is a crisping of both x and y. Thus x and y could potentially approximate the same region. D3 defines a Crisp region to be one that has no crispings.

9

Cohn and Gotts (1994, 1996a) are also precursors of Cohn and Gotts (1996b).

26

B. Bennett

Further axioms of the theory can now be stated as follows: A3) A4) A5) A6) A7) A8) A9) A10)

∀xy[x ≺ y → ∃z[z ≺ y ∧ ¬MA(x, z)]] ∀xy[MA(x, y) → ∃z∀w[w  z ↔ (w  x ∧ w  y)]] ∀xy[∃z[x  z ∧ y  z ∧ ∀w[(x  w ∧ y  w) → z  w]]] ∀xy[MA(x, y) ↔ ∃z[x  z ∧ y  z ∧ ∀w[(x  w ∧ y  w) ↔ z  w]]] ∃x∀y[y  x] ∀xy[∀z[Crisp(z) → ((z  x) ↔ (z  y))] → (x = y)] ∀x∃y[y  x ∧ Crisp(y)] ∀xy[x ≺ y → ∃z[x ≺ z ∧ z ≺ y]]

Axiom A3 says that, if a region can be made more precise in one way, it can also be made precise in another way that is incompatible with the first. Axiom A4 says that, if two regions are mutually approximate, then there is not only a region that is a crisping of both but, more specifically, there is a region that is the least crisp crisping of both. Axiom A5 says that any two regions have a ‘crispest common blurring’. (In terms of the egg-yolk model the crispest common blurring of two vague region is the region whose white is the sum of the two whites and yolk is the intersection of the two yolks. This assumes that we may have regions with an empty yolk.) Axiom A6 states that if two regions have a mutual approximation, they must have a ‘maximally blurred’ mutual approximation. Axiom A7 states that there is a maximally blurred region (‘the complete blur’). Cohn and Gotts remark that one may wish to omit this axiom or even assert its negation, which says that for any region one can always find a more blurred region. Axiom A8 specifies a condition for identity. It says that if two regions have the same complete crispings they must be equal. (Cohn and Gotts (1996b) give a slightly more complex but equivalent formula, and they also consider some alternative identity axioms.) Axiom A9 states that every region can be crispened to a completely crisp region. Axiom A10 states that whenever one region is crisper than another, there is a third intermediate region, more blurred than the first but crisper than the second. (This gives makes the crisping ordering dense). Exactly what kinds of relation structure satisfy this axiom set is not known. However, one possible model is considered in the next section.

4.2 The ‘Egg-Yolk’ Model The axiomatic theory of ‘crisping’ can of course be given a semantics based on the general purpose model theory of first-order logic. However, the theory is normally understood in terms of the more specific semantics originally proposed in Lehmann and Cohn (1994). The so-called ‘Egg-Yolk’ model Lehmann and Cohn (1994) interprets a vague region in terms of a pair of nested crisp regions representing its maximal and minimal possible extensions. The maximal extension is called the ‘egg’ and the minimal is the ‘yolk’, which is required to be a part of the egg (see Fig. 4). (The case where the yolk is equal to the egg is allowed, such cases

Spatial Vagueness

27

Fig. 4 A typical ‘egg-yolk’ interpretation of a vague region

corresponding to ‘crisp’ regions.) When an egg-yolk pair is given as the spatial extension of a vague region this means that the region is definitely included in the egg and definitely includes the yolk. In terms of the Egg-Yolk model, the crisping relation x ≺ y can be understood as holding whenever the egg associated with x is contained within the egg associated with y and the yolk of x contains the yolk of y, and furthermore x and y do not have identical eggs and yolks (if this last condition does not hold, we have x  y but not x ≺ y). In order to tie the theory of the crisping relation explicitly to the Egg-Yolk model, Cohn and Gotts (1996b) introduce two functions to the vocabulary: egg-of(x) and yolk-of(x), denoting respectively the egg and yolk regions associated with x. The mereological relation P(x, y), meaning ‘x is part of y’, is also introduced. The following additional axioms are then specified: A11) ∀x[P(yolk-of(x), egg-of(x))] A12) ∀xy[x ≺ y → (P(egg-of(x), egg-of(y)) ∧ P(yolk-of(y), yolk-of(x)) ∧ ¬(P(egg-of(y), egg-of(x)) ∧ P(yolk-of(x), yolk-of(y))))] A11 ensures that the yolk of every region is contained within its egg. A12 states that if x is crisper than y then x’s egg must be part y’s egg and y’s yolk must be part of x’s yolk and at least one of these relations must be a proper part relation (i.e. the eggs and yolks can’t both be equal). Note that axiom A12 is stated as an implication rather than an equivalence. The explanation given for this is that the egg-yolk condition is proposed as a necessary but not sufficient condition for one vague region being a legitimate crisping of another. The rationale behind this is that there might be a vague region that satisfies the relatively weak spatial constraints required to be a crisping of another region and yet would not be considered as a reasonable crisping for other reasons, such as the shape of its egg and yolk. Fig. 5 illustrates two potential candidates for the crisping of a given region. The egg and yolk of the initially given region are shown as bounded by solid lines, and the egg and yolk of the candidate crispings are outlined with dashed lines. In case

28

B. Bennett

Fig. 5 a) Reasonable, and b) anomalous crispings in the ‘egg-yolk’ interpretation

a) the egg and yolk of the candidate crisping are similar in shape to the initial region, so the vague region that they define may be regarded as a legitimate crisping of the original. But in case b) the jagged outline of the egg and yolk of the candidate mean that it is implausible that the original could be crispened in this way. Although, the egg-yolk model has become well-known, the papers of Cohn and Gotts (1994, 1996a,b) suggest that this is just one of a range of possible interpretations that could be given to the axiomatic theory. The egg-yolk model cannot of itself, account for any kind of constraint on a region’s plausible extensions between its maxima and minima. However, when dealing with real phenomena, such as vague geographic features, we would expect the range of possible extensions to be structured in accordance with the underlying conditions relative to which a vague region is individuated. For instance, one might expect these extensions to exhibit a contour-like structure, such as we saw in Fig. 2 and Fig. 3 above. Another strand of research on spatial vagueness, somewhat related to the work of Cohn and Gotts and the egg-yolk theory, is work based on rough sets and granular partitions. See for example Bittner and Stell (2002). Consideration of this approach is beyond the scope of this chapter.

5 Fuzzy Logic Approaches The most popular approach to modelling vagueness used in AI is that of fuzzy logic and the theory of fuzzy sets. These theories originated with the works of Goguen (1969) and Zadeh (1975) and have given rise to a huge field of research (see (Dubois and Prade, 1988; Zimmermann, 1996) for surveys of fuzzy logics and their applications). Within the context of the current collected work, I presume that a general introduction to fuzzy logic will not be required. The coverage of fuzzy approaches to vagueness here will be very limited, since these are considered in detail in other chapters of this collection.

Spatial Vagueness

29

5.1 Spatial Interpretation of Fuzzy Sets Fuzzy sets can be given an intuitive spatial interpretation. In the same way that a precise spatial region can be identified by a set of spatial points, a vague spatial region may be associated with a fuzzy set of points. Here, the degree of membership of the point in the fuzzy set corresponds to the degree to which that point is considered as belonging to the vague region. Fuzzy representations have been popular with many geographers as they provide a natural way of representing features with ill-defined boundaries (Wang and Brent Hall, 1996). The particular way that fuzzy sets have been employed varies greatly according to the type of feature being analysed, the kinds of data available, and the particular aims of each analysis. Kronenfeld (2003) proposed a fuzzy approach to classification and partitioning of continuously varying land cover types and applies this to the classification of forest types. Arrell et al. (2007) use a fuzzy classification of elevation derivatives to identify natural landforms such as peaks and ridges. Evans and Waters (2007) use fuzzy sets to model the regions referred to by vernacular place names.

5.2 Fuzzy Region Connection Calculus A fuzzy version of the well-known Region Connection Calculus (Randell et al., 1992) of topological relations has been developed by Schockaert et al. (2008, 2009), by treating the primitive connection relation, C, of the RCC theory as a fuzzy relation, and replacing the definitions of other spatial relations by analogous definitions formulated using fuzzy logic operators. Let T (φ , ψ ) be the value of a T -norm (fuzzy conjunction) function on the truth values of propositions φ and ψ ; and let IT (φ , ψ ) be the residual implicator function relative to T , given by IT (φ , ψ ) =def sup{λ | λ ∈ [0, 1] and T (φ , λ ) ≤ ψ }. Then a fuzzy parthood relation P is defined in terms of the connection relation as follows: P(a, b) ≡def (∀x ∈ U)[C(x, a) → C(x, b)] =⇒ infx∈U {IT (C(x, a), C(x, b))} So, following a standard approach to translating first-order logic into fuzzy logic, the universal quantification in the classical definition is replaced by the infimum of the truth values for all regions in the domain, and implication is replaced by the residual implicator. Similar mappings from first-order logic to fuzzy relations can be given for all the topological relations of the original RCC theory. This fuzzification method is certainly well-principled, and various strong results can be proved regarding the structure and reasoning capabilities of the resulting fuzzy region connection calculus. However, some properties of the relations defined in this way appear to be somewhat unexpected. In particular, the use of the infimum (and supremum) as corresponding to universal (and existential) quantification leads to results that may seem counter-intuitive. For example, consider the case illustrated in Fig. 6. Here we have one region b which seems to be very much part of region a, except from a thin protruding spike, whereas c lies on the edge of a but has no part that is strongly within a. In fuzzy RCC one may find that in such a situation c

30

B. Bennett

Fig. 6 a, b and c are three fuzzy regions. Under Schockaert’s definition of fuzzy parthood, b will be evaluated as being part of a to a higher degree than is c

will be evaluated as being part of a to a higher degree than is b. This is because a region near the tip of b may have a high degree of connection with b but a very low degree of connection with a. Hence, even though regions connected to most parts of b will be highly connected to a, the infimum in the fuzzy P definition means that the evaluation of the overall degree of truth of the P relation is determined by that part of b that is least connected to a. By contrast, the whole of c is on the periphery of a but no part of c is very far from a. Since the furthest part of c is nearer to a than is the furthest part of b, P(c, a) is likely to be evaluated as having a higher degree of truth than P(b, a). Of course, this depends on the particular details of the situation and the distribution of the fuzzy C relation. To avoid cases such as that just described, one could define C in such a way that contact with a protruding spike of a region always counts as a low degree of connection. However, it is evident that very natural measures of the degree of connection can give rise to measures of parthood that may seem unnatural. The reason for this is that, when we hold the view that a topological relation between spatial regions is almost true, we often mean that it would hold if we disregard some comparatively insignificant part of one or both of the regions. But the use of the infimum in the fuzzy part definition means that we cannot disregard any part of a region (except by reducing the degree to which it is considered a part of the region).

6 Supervaluationist Approaches 6.1 Origins and Motivations of Supervaluation Semantics The fundamental idea of the supervaluationist account of vagueness, is that a language containing vague predicates can be interpreted in many different ways, each of which can be modelled in terms of a more precise version of the language, which is known as a precisification. In specifying a formal supervaluation semantics each

Spatial Vagueness

31

precisification is associated with a valuation of the symbols (i.e. predicates and constants) of the language. In some accounts each precisification is simply a classical assignment, corresponding to a completely precise interpretation; but it is common to allow precisifications that are not completely precise, and are associated with partial assignments. The interpretation of the vague language itself is determined by a supervaluation, which is the collection of assignments at all precisifications. The view that vagueness can be analysed in terms of multiple senses was proposed by Mehlberg (1958), and a formal semantics based on a multiplicity of classical interpretations was used by van Fraassen (1969) to explain ‘the logic of presupposition’. It was subsequently applied to the analysis of vagueness by Fine (1975) and independently by Kamp (1975). Thereafter, it has been one of the more popular approaches to the semantics of vagueness adopted by philosophers and linguists, and to a lesser extent by logicians. Only a brief overview of supervaluation can be given here. A more thorough account and extensive discussion can be found in works such as Williamson (1994) and Smith (2008). A major strength of the supervaluation approach is that it enables the expressive and inferential power of classical logic to be retained (albeit within the context of a somewhat more elaborate semantics) despite the presence of vagueness. In particular, necessary logical relationships among vague concepts can be specified using classical axioms and definitions. These analytic interdependencies will be preserved, even though the criteria of correspondence between concepts and the world are ill-defined and fluid. For example, in the sentence ‘Tom is tall and Simon is short’, both ‘tall’ and ‘short’ are vague. However, their meaning is coordinated in that, when applied in the same context relative to a given comparison class, it must be that in any admissible precisification the minimal height at which an object is ascribed the property ‘tall’ is higher than the maximal height at which an object is ascribed the property ‘short’. Thus if it were to turn out that Tom’s height is less than that of Simon, the claim would be false in any admissible precisification. Investigation of supervaluation semantics in the philosophical literature tends, as one might expect, to be drawn towards subtle foundational questions (such as those concerning the sorites paradox and second-order vagueness). Consequently, there has been relatively little development towards practical use of supervaluation semantics in formal representations designed for computational information processing applications. Supervaluation semantics is often regarded as fundamentally opposed to multivalued and fuzzy approaches to vagueness. Whereas the latter approaches modify the notion of truth and the logical operators, supervaluation sememantics locates vagueness in the interpretation of terms and accounts for this within an extended model theory, whilst retaining essentially classical notions of truth and deduction.

6.2 Admissible Precisifications and ‘Supertruth’ In standard supervaluation semantics the concept of an admissible precisification plays a key role. It is used to elaborate the simple classical concept of truth by

32

B. Bennett

defining the concept of supertruth, which is applicable to propositions that include vague terminology. In a semantics where all precisifications are associated with complete classical models, the following definition is may be given:

φ is supertrue just in case φ is true according to every admissible precisification. In many accounts (e.g. Fine (1975)) the notion of admissible precisification is taken as primitive. It is normally assumed that, in addition to satisfying conditions of logical consistency (in virtue of being associated with classical models), an admissible precisification also satisfies an appropriate theory specifying analytic properties and relationships between vocabulary terms (e.g. nothing can be an instance of both the predicates ‘tall’ and ‘short’ — here we are assuming the predicates are applied in the same context). Stipulating such semantic conditions may be complex in general, but poses no particular theoretical problems. But if the only restrictions on admissible precisifications are that they must satisfy logical and analytical axioms, then the only propositions that will count as supertrue will be those that are analytically true. This is usually regarded as too strong a requirement. What we would like is a notion of supertruth such that a proposition is true if it comes out as true on any reasonable interpretation of its terms not on every possible interpretation. Consequently, the set of admissible precisifications is normally taken to be those that are in some sense reasonable. For instance if a man is 6’6” in hight, one would expect him to be tall according to all admissible precisifications. However, this begs the question of what precisifications should be counted as admissible. The problem of determining the set of admissible precisifications is often sidestepped in presentations of supervaluation semantics. However, it causes difficulties both from a theoretical and practical point of view. From a theoretical perspective, any attempt to stipulate a set of admissible precisifications brings to the fore the problem of second-order vagueness. To explain briefly: the admissible precisifications are intended to correspond to possible ways of deciding the truth of predications applied to borderline cases. That is, every complete precisification corresponds to a set of decisions as to whether each borderline case is an instance of a given vague predicate. By contrast, a non-admissible precisification would one one that assigns a predicate as true of an object to which it is clearly not applicable (or as not true of something to which it is clearly applicable). The problem is that any sharp distinction between admissible and non-admissible precisifications assumes a correspondingly sharp distinction between borderline instances and clear instances (or clear non-instances). But given that we are considering vague predicates, it seems untenable that there should be a precise boundary between borderline and clear-cut cases. The problem with admissibility from a practical point of view, is that if we actually wanted to reason with or implement a formal system in which the set of admissible precisifications plays a key role, we would have to explicitly specify all the

Spatial Vagueness

33

conditions required of an admissible precisification. This would not only be complex but would also seem to require one to make stipulations without any obvious means of justification.

6.3 Computational Applications of Supervaluationism The uptake of supervaluation-style approaches in computational applications has been relatively limited. One reason for this is probably the difficulty in specifying the set of admissible precisifications, as mentioned above. Nevertheless it is worth mentioning some works which have taken preliminary steps in the application of supervaluation semantics. Bennett (1998) proposed a two-dimensional model theory and a corresponding modal logic, in which the interpretations of propositions are indexed both by precisifications and possible worlds. Relative to this semantics a spectrum of entailment relations were defined corresponding to more or less strict requirements on how the vague senses of premisses and conclusion are allowed to vary. In Bennett (2006) the semantics of vague adjectives was characterised in terms of their dependence on relevant objective observables (e.g. ‘tall’ is dependent on ‘height’). This may be seen as a precursor of the standpoint semantics, which will be presented in the next section. An example of the use of a supervaluationist approach in an implemented computer system for processing geographic information can be found in Bennett et al. (2008). Applications of supervaluation or similar theories to geographic information have been proposed by Smith and Mark (2003), Varzi (2001a) and Kulik (2003). Halpern (2004) analyses vagueness in terms of the subjective reports of multiple agents, but these play a similar role in his semantics to precisifications in the semantics proposed in this paper.

7 Standpoint Semantics Standpoint Semantics is both a refinement and an extension of supervaluation semantics whose purpose is to make more explicit the modes of variability of vague concepts and to support a definition of truth that is relative to a particular attitude to the meanings of terms in a vague language. Whereas supervaluation semantics provides a very general framework within which vagueness can be analysed formally, standpoints semantics is more geared towards detailed modelling of specific vague concepts within some particular application domain.

7.1 What Is a Standpoint? In making an assertion or a coherent series of assertions, one is taking a standpoint regarding the applicability of linguistic expressions to describing the world. Such a standpoint depends partly on one’s beliefs about the world and partly on one’s linguistic judgements about the criteria of applicability of words to a particular situation. This is especially so when some of the words involved are vague. For instance,

34

B. Bennett

one might take the standpoint that a certain body of water should be described as a ‘lake’, whereas another smaller water-body should be described as a ‘pond’. It is not suggested that each person/agent has fixed standpoint, which they stick to in all situations. Rather an agent adopts a given standpoint at a particular time as a basis for describing certain features of the world. In a different situation the agent might find that adopting a different standpoint is more convenient for describing salient features of the world. This is somewhat misleading since even a person thinking privately may be aware that an attribution is not clear cut. Hence a person may change their standpoint. Moreover this is not necessarily because they think they were mistaken. It can just be that they come to the view that a different standpoint might be more useful for communication purposes. Different standpoints may be appropriate in different circumstances. The core of standpoint semantics does not explain why a person may hold a particular standpoint or the reasons for differences or changes of standpoint, although a more elaborate theory dealing with these issues could be built upon the basic formalism. In taking a standpoint, one is making somewhat arbitrary choices relating to the limits of applicability of natural language terminology. But a key feature of the theory is that all assertions made in the context of a given standpoint must be mutually consistent in their use of terminology. Hence, if I take a standpoint in which I consider Tom to be tall, then if Jim is greater in height than Tom then (under the assumption that height is the only attribute relevant to tallness) I must also agree with the claim that Jim is tall.

7.2 Parameterised Precisification Spaces By itself, supervaluation semantics simply models vagueness in terms of an abstract set of possible interpretations, but gives no analysis of the particular modes of semantic variability that occur in the meanings of natural language vocabulary. A key idea of standpoint semantics is that the range of possible precisifications of a vague language can be described by a (finite) number of relevant parameters relating to objectively observable properties; and the limitations on applicability of vocabulary according to a particular standpoint can be modelled by a set of threshold values, that are assigned to these parameters. To take a simple example, if the language contains a predicate Tall (as applicable to humans), then a relevant observable is ‘height’. And to determine a precisification of Tall we would have to assign a particular threshold value to a parameter, which could be called tall human min height. One issue that complicates this analysis is that vague adjectives tend to be context sensitive in that an appropriate threshold value depends on the category of things to which the adjective is applied. This is an important aspect of the semantics of vague terminology but is a side issue in relation to our main concerns in this chapter. Here we shall assume that vague properties are applied uniformly over the set of things to which they can be applied. To make this explicit we could always use separate properties such as Tall-Human and Tall-Giraffe, although we won’t actually need

Spatial Vagueness

35

to do this for present purposes. A formal treatment of category dependent vague adjectives is given in Bennett (2006). In general a predicate can be dependent on threshold valuations of several different parameters (e.g. Lake might depend on both its area and some parameter constraining its shape.) Thus, rather than trying to identify a single measure by which the applicability of a predicate may be judged, we allow multiple vague criteria to be considered independently. In the initial development of the standpoint approach Santos et al. (2005a); Mallenby and Bennett (2007); Third et al. (2007); Bennett et al. (2008)), it was assumed that standpoints can be given a model theoretic semantics by associating each standpoint with a unique threshold valuation characterising a complete precisification. In so far as standpoints may be identified with an aspect of a cognitive state, this idea is perhaps simplistic. It is implausible that an agent would ever be committed to any completely precise value for a threshold determining the range of applicability of a vague predicate. Cognitive standpoints are more plausibly associated with constraints on a range of possible threshold values rather than exact valuations of thresholds. For instance, if I call someone tall, then my claim implies an upper bound on what I consider to be a suitable threshold for tallness — the threshold cannot be higher than the height of that person. This elaboration of the status of standpoints in relation to thresholds is being developed in ongoing research. But, in the context of implementing cartographic displays showing the spatial extensions of instances of vague terms, modelling a standpoint as a fully determinate parameterised precisification has been found to be useful and informative. It has the advantage that the regions displayed in accordance with a standpoint always corresponds to some precise definition. This is desirable if one wants to compare different instances of vague predicates. Moreover, it is relatively easy to design an interface such that a user can easily change their standpoint by altering the thresholds assigned to one or more of the parameters that define the standpoint. To summarise, the key ideas of standpoint semantics are: a) to identify precisifications with threshold valuations — i.e. assignments of threshold values to a set of parameters that model the variability in meaning of the vague concepts of a language; and b) to always evaluate information relative to a standpoint, which in the simplest case corresponds to a single precisification, but could correspond to a set of precisifications compatible with an agent’s current attitudes to language use. A threshold valuation appropriate for specifying a standpoint in relation to the domain of hydrographic geography might be represented by something like the following: V = [ pond vs lake area threshold = 200 (m2 ), desert max precipitation = 20 (mm per month), elongated region min elongation ratio = 2.2, ...] Here, the first parameter determines a cut-off between ponds and lakes in terms of their surface area and the second sets the maximum precipitation at which a region

36

B. Bennett

could be considered to be a desert. The last parameter might be used to specify conditions under which a region is considered to be ‘elongated’ (how this property might be defined will be discussed in section 9.2 below).

7.3 Defining and Interpreting Vague Concepts Using Parameters As well as providing a formal structure that defines the semantic choices associated with a particular precisification, the parameters of semantic variation and the threshold assignments to these parameters play further key roles in standpoint semantics. As well as their role in the semantics, the parameters may also be referred to explicitly in the formal object language in which we both axiomatise or define vague concepts and in which we also represent information expressed in terms of these concepts. We here assume that the object language is first-order logic — or rather it is firstorder logic with a small syntactic innovation (in fact a similar extension is likely to be possible for other formal languages, but we will not examine this possibility here). The innovation is that, for each vague predicate, we allow additional arguments to be attached to it corresponding to semantic variation parameters, relating to the variability in the meaning of that concept. Specifically, where a vague n-ary predicate V depends on m parameters we write it in the form: V[p1 , . . . , pm ](x1 , . . . , xn ) The following examples illustrate the use of this language augmented with precisification parameters to define some vague spatial concepts: 1. Tall[tall thresh](x) ≡def height(x) > tall thresh 2. Forested[ forest max tree dist ](r) ≡def ∀p[In(p, r) → ∃t[Tree(t) ∧ (dist(p,t) < forest max tree dist)]] 3. Forest[ forest max tree dist, forest min area ](r) ≡def Forested[forest max tree dist](r) ∧ (area(r) > forest min area) ∧ ¬∃r [Forested[forest max tree dist](r ) ∧ PP(r, r )] Example 1 is a simple definition of ‘tall’ as a predicate that applies to anything whose height is greater than a particular threshold. Definition 2 specifies that a region is forested just in case every point in that region is less than a certain threshold distance from a tree. Finally, example 3 defines a forest as being a forested region whose area is greater than a given minimum and that is not contained within some large forested region (here PP(x, y) means that x is a proper part of y). Actually, the additional parameter syntax ‘[p1 , . . . , pm ]’ is not really essential since we could either just treat the variability parameters as ordinary additional arguments or we could simply omit them from the predicate arguments altogether and just have them as constants embedded within the definitions. However, the extended syntax seems to be both convenient and informative as it ensures that the parameters of variability of each vague predicate are clearly indicated and highlights the

Spatial Vagueness

37

conceptual difference between the objects to which a predicate is applied and the parameters used to precisify the predicate’s meaning. We can now understand how a threshold valuation (associated with a standpoint) is used to interpret each vague predicate in a precise way. All we need do is to substitute the values given by the threshold valuation in place of the corresponding threshold parameters given in the definition. If we then remove the ‘[p1 , . . . , pm ]’ argument lists we end up with ordinary first-order formulae, defining precise versions of the vague predicates, in accordance the given threshold valuation.

8 Comparison between Approaches It may be useful to compare how the variable extension of a vague spatial region is modelled according to the different approaches we have considered. We assume that we wish to model the spatial extension of an instance r of some vague spatial predicate V(x). Fig. 7 illustrates the different models that arise. In Fig. 7.a we see the egg-yolk model with its inner yolk corresponding to the region that is definitely part of the extension of r, and its outer egg boundary within which the extension of r must lie. The dotted lines indicate the boundaries of crisp regions that are possible crispings of r. We note that by itself, the egg-yolk model does not place any further restrictions on the shape of these possible crispings. Fig. 7.b depicts region r represented as a fuzzy set of spatial points. The shading represents the degree to which each point is considered to lie within r — the darker the shade the higher the degree of membership. Fig. 7.c is supposed to indicate possible extensions of a feature under a general supervaluation semantics. This is rather misleading as it would depend very much on what semantic conditions were specified in the theory determining the set of admissible precisifications. Most likely the set of extensions according to different precisifications would be much more structured. What this diagram is intended to indicate is just that supervaluation semantics by itself does not impose particular conditions on the range of possible extensions of a region characterised by a vague predicate. Fig. 7d shows possible extensions according to a standpoint semantics. Here we assume that a single parameter has been determined that is relevant to application of a predicate that is used individuate the region. According to different choices of a threshold of applicability, we get different extensions. These will typically be structured like a contour map. If a strict threshold is set (which may be high or low depending on whether the parameter is a positive or negative indicator of the feature under consideration) then a small region is identified, and with less strict thresholds monotonically larger regions are demarcated. We note that models a, b and d have similar structure. The Egg-Yolk boundaries of a can be regarded as representing two distinguished contours chosen from the continuum of contours resulting from threshold choices made in relation to the standpoint model d. Similarly, the fuzzy membership function depicted in b could be chosen so that its α -cuts also correspond to contours in d. Moreover, although

38

B. Bennett

Fig. 7 Comparison of extensions of a region instantiating a vague predicate, as modelled by: a) the Egg-Yolk model, b) a fuzzy set of spatial points, c) the precisifications of a supervaluation semantics, and d) the standpoints of a standpoint semantics

the supervaluation model c shows extensions corresponding to arbitrary precisifications, suitable conditions on the set of admissible precisifications could make this too correspond with the contours of d. A further connection is that the egg and yolk of a (or indeed the contours of d) could be associated with limitations on the extensions of r that are possible in admissible precisifications of a supervaluation semantics. c would then look more like a (or d). Nevertheless, despite the similarity between the resulting models. There are deep conceptual differences. In the fuzzy model, b, the meaning of the vague predicate V is considered to be static and the points are associated with V to a more or less strong degree. But with the standpoint semantics it is the meaning of V that varies, and a point may be part of the extension of an instance of V for some but not others of these meanings. The differences between these approaches would become more evident if we considered examples involving a combination of inter-related classifications — for

Spatial Vagueness

39

instance the extensions of two or more regions that are instances of different but related vague predicates. In fuzzy logic the conjunction of multiple vague predications is modelled by a function of the truth degrees associated with each extensions. This mode of combination makes it difficult to capture the significance of dependencies between vague concepts. But in the standpoint approach, interdependent concepts will share parameters of variability, so that a choice of threshold value may affect the meaning of several different vague concepts. Consequently, for any given standpoint the extensions of related features are coordinated. For example, suppose that the terms ‘forest’ and ‘heath-land’ are defined so that areas with a tree density above a chosen threshold are classified as ‘forest’ but those with tree density less than this threshold (but above some lower threshold) are classified as ‘heath-land’. According to the standpoint semantics the border between forest and heath-land would move according to the adopted standpoint; and for any particular standpoint no region would be both forest and heath-land. By contrast, in the borderline between forest and heath-land we would have an area that is forest to some degree and is also heath-land to some degree. The ways in which separate items of vague information can be combined and collectively interpreted within each of the various formalisms is a very significant and interesting topic. However, further consideration of this aspect of vagueness is beyond the scope of this chapter.

9 Some Significant Vague Spatial Predicates In this section we shall examine certain vague spatial predicates in more detail. Several significant kinds of spatial concept and relation will be covered. But, since there are a large number of ways in which vagueness affects spatial predication, this is not an exhaustive analysis.

9.1 Vague Distance Relations: Near and Far Arguably the most fundamental spatial relations are those relating to distance. Indeed Tarski (1959) showed that all precise geometric predicates can be defined starting from the basic relation of the equidistance of two pairs of points. In natural language we describe distances in a variety of ways. Sometimes we use numerical distance measurements, typically with a very high degree of approximation. For instance, if I say that Leeds is 200 miles from London, I do not imply an exact measurement — if the distance were 10 or even 20 miles more or less, this would still be regarded as a reasonable claim (the semantics of numerical approximations has been studied by a number of authors — e.g. (Corver et al., 2007; Krifka, 2007)). Although these measurement approximations are closely related to vagueness they are essentially numerical rather than spatial in character and will not be considered further here. Another equally, if not more, common way of describing distances is by means of terms referring to vague distance relationships — i.e. words such as ‘near’, ‘far’,

40

B. Bennett

‘close’, ‘distant’. Examination of the informal concept of nearness in geography goes back at least to the work of Lundberg and Eckman (1973). More recent studies of how people use the words ‘near’ and ‘far’ include (Fisher and Orf, 1991), (Frank, 1992), (Gahegan, 1995) and (Worboys, 2001) Application of fuzzy logic techniques to representing spatial relations such as near are investigated by Robinson (1990, 2000), who describe a system that uses question answering by human subjects to learn a fuzzy representation of the concept ‘near’ by constructing a fuzzy set of the places near the reference place. Fisher and Orf (1991) in their survey of subjects ascription of ‘near’ (in a university campus setting) found that, rather than judgements of nearness being clustered within a single distance range, several clusters were found (three in fact), which seemed to indicate that different semantic interpretations of ‘near’ were being employed by different subjects. What is clear (as indicated by the experimental results of Gahegan (1995)) is that many contextual factors, apart from the actual distances involved, have a strong influence on how subjects apply the description ‘near’. These include the relative sizes of objects or regions involved, connection paths between places, scale, and the perceived significance of objects.

9.2 Elongation vs. Expansiveness A qualitative distinction that appears to be of general importance in our description of the world, and has particular significance in relation to many geographic features, is that between elongated and expansive regions. A typical example is the distinction between a river, which is elongated, and a lake, which is expansive. The term ‘elongated’ is used here to refer to a region that is long an thin, but not necessarily straight. Thus, a river may curve and wind but is still, in this sense, elongated. Given the infinite possible shapes that a region may take, it is not obvious how to measure the degree to which an arbitrary region is elongated. One relatively simple idea is to consider the ratio of the radius of a region’s minimal bounding circle to the radius of its maximal inscribed circle. For the case of a 2-dimensional region, this measure is illustrated in Fig. 8. A similar measure can be defined for 3-dimensional regions, using spheres instead of circles. Such a measure can easily be utilised within a standpoint semantics formalism. For example, one might define the vague property Elongated, using the standpoint semantics notation, as follows: • Elongated[elong ratio](x) ≡def (min bound rad(x)/max insc rad(x)) ≤ elong ratio This formulation is fine as long as we have already demarcated the boundaries of all regions that we wish to classify. In this case we can directly evaluate the truth of the Elongated predicate, relative to a given value of the threshold τ (elong ratio), for any given region. However, in many domains — certainly in geography and biology — we often encounter cases where we are looking for an elongated part of a larger system. For example we may want to individuate a river that is part of a complex hydrological system.

Spatial Vagueness

41

Fig. 8 Calculation of elongation ratio. The elongation ratio is obtained by dividing the radius of the minimum bounding circle by the radius of the maximal inscribed circle — i.e. L = R/r.

Fig. 9 Medial axis skeleton of the Humber estuary and its tributaries

A method for identifying elongated parts of a larger region (and hence partitioning it into elongated and expansive segments) was proposed by Santos et al. (2005b) and further developed in (Mallenby and Bennett, 2007). Initial geometrical processing is carried out to find the medial axis skeleton of the region under analysis.10 This is the locus of all points that are equidistant from two or more boundary points of the region. Fig. 9 shows the medial axis skeleton computed for the Humber estuary in the UK. It can be seen that the skeleton includes line segments of two somewhat different kinds. Some of the segments run along what we might naturally think of as the 10

This was carried out using the software VRONI (Held, 2001).

42

B. Bennett

Fig. 10 Determination of elongated segments of a larger region

middle of the channels of the water system, whereas others run from the middle to the edge of the water region. The latter type of segment arise from relatively small indentations in the river boundary and are not particularly relevant to the shape of the water system as a whole. In order to remove these unwanted segments, the skeleton is then pruned by removing all segments whose rate of approach towards the boundary is greater than a certain threshold. Once this pruning has been carried out, the remaining, more globally significant, part of the medial axis skeleton is used to identify elongated parts of the region. The idea is that these are associated with segments of the medial axis along which the width remains approximately constant. As shown in Fig. 10, the width variance is calculated for each point along a (pruned) medial axis section. The calculation is based on the highest and lowest widths evaluated along a sample segment of the medial axis, extending a certain distance either side of the point under consideration. In order to make the measure of width variation scale invariant, the length of medial axis over which it is calculated is taken to be equal to the width of the region at that point. The width variation is then computed as the ratio of the highest to lowest width (i.e. distance from a point on the medial segment to the nearest boundary point) along the sample segment.11 Fig. 10 illustrates the width variation measure at two points p and q along the medial axis of a region. At p the width variation is low since all widths at medial points within the maximal inscribed circle centred at p are similar, whereas at q the width variation is high. Once this variation measure has been computed for each point on the (pruned) medial axis, the elongated parts of the region are determined as those parts of region such that every point on the segment of medial axis running through that region has a width variation below a given threshold value. Points with width variation above the threshold are considered to lie in expansive sections of the region (as are points that lie on branching points of the medial axis, for which the measure is not well defined). The application of this method to the Humber estuary region is shown below in Fig. 11. Here we see two possible demarcations of the elongated sections of the estuary and the river Hull from which it opens. The upper map shows the 11

A number of variants of this simple calculation also give reasonable results — it is as yet unclear which is the most appropriate.

Spatial Vagueness

43

demarcation obtained by using a strict threshold of 1.09 on the maximum width variance, whereas the lower map was produced using the much more tolerant threshold of 1.32. This provides a good example of how a standpoint semantics approach can be used to visualise a geographic feature according to different interpretations of a vague concept.

Max width variance = 1.09

Max width variance = 1.32

Fig. 11 River stretches identified according to different standpoints

9.3 Geographic Feature Types and Terminology Geographers, and more especially surveyors and cartographers, have long been aware of the difficulties of giving precise definitions of spatial features (see for example (Maling, 1989, chapters 5 and 12)). The prevalence of cartographic maps as the prime medium for geographic information may have hidden the true extent of indeterminism in geographic features and their boundaries. Constructing a map involves the use of complex procedures and conventions for converting observations measurements into cartographic regions and entities. Moreover, many stages of these procedures require a certain amount of subjective judgement in order to transform the multifarious characteristics of the world into precise cartographic objects. Thus the resulting map representation gives an impression that the world is far more neatly organised and compartmentalised than is really the case.

44

B. Bennett

As we have seen from many of the illustrative examples given above, many of the representations proposed by researchers in AI, formal logic and ontology have been developed with geographic applications in mind. Relevant works include that of Varzi (2001a) and the collection of papers in Varzi (2001b). Certain geographic feature types have received particular attention. I now briefly summarise some contributions in this area. The nature of forests and how they should be defined and identified has been examined by Bennett (2001) and Lund (2002). The case of mountains has been considered by Smith and Mark (2003). As mentioned above a number of papers have tackled the definition and individuation of hydrographic features. As well as being interested in specific types of geographic feature, geographers are also concerned with general aspects of the way humans describe the world. Such descriptions are of course greatly affected by the vagueness of our natural language terminology. The question of how vernacular terms related to geographic regions has been examined by Evans and Waters (2007) and the semantics of the notion of ‘place’ has been explored by Bennett and Agarwal (2007).

10 Conclusion This chapter has examined the issue of spatial vagueness from a variety of perspectives and has explored a number of different approaches that have been taken to representing and processing vague spatial information. The reader will no doubt have become aware that this is a subtle and complex area, which is very much open to further exploration. Although the field is still very much in its infancy, it is clear that representation and processing of vague spatial information can potentially play a crucial role in many computational applications ranging from geographic and biological information systems to natural language understanding and robotics. However, it is also evident that in order to support such applications in a general and flexible way, many theoretical and practical obstacles remain to be overcome. Acknowledgements. I would like to thank David Mallenby and Paulo Santos for their collaboration with myself in developing and implementing a standpoint based approach to geographic feature identification. Their work has provided a significant contribution to the material reported in this chapter.

References Arrell, K., Fisher, P., Tate, N.: A fuzzy c-means classification of elevation derivatives to extract the natural landforms in Snowdonia, Wales. Computers and Geoscience 33(10), 1366–1381 (2007) Beall, J.: Liars and Heaps. Clarendon Press, Oxford (2003) Bennett, B.: Modal semantics for knowledge bases dealing with vague concepts. In: Cohn, A.G., Schubert, L., Shapiro, S. (eds.) Principles of Knowledge Representation and Reasoning: Proceedings of the 6th International Conference (KR 1998), pp. 234–244. Morgan Kaufmann, San Francisco (1998)

Spatial Vagueness

45

Bennett, B.: What is a forest? on the vagueness of certain geographic concepts. Topoi 20(2), 189–201 (2001) Bennett, B.: Modes of concept definition and varieties of vagueness. Applied Ontology 1(1), 17–26 (2005) Bennett, B.: A theory of vague adjectives grounded in relevant observables. In: Doherty, P., Mylopoulos, J., Welty, C.A. (eds.) Proceedings of the Tenth International Conference on Principles of Knowledge Representation and Reasoning, pp. 36–45. AAAI Press, Menlo Park (2006) Bennett, B., Agarwal, P.: Semantic categories underlying the meaning of ‘place’. In: Winter, S., Duckham, M., Kulik, L., Kuipers, B. (eds.) COSIT 2007. LNCS, vol. 4736, pp. 78–95. Springer, Heidelberg (2007) Bennett, B., Mallenby, D., Third, A.: An ontology for grounding vague geographic terms. In: Eschenbach, C., Gruninger, M. (eds.) Proceedings of the 5th International Conference on Formal Ontology in Information Systems (FOIS 2008), IOS Press, Amsterdam (2008) Bittner,T., Stell, J.G.: Vagueness and rough location. Geoinformatica 6(2), 99–121 (2002) Cohn, A.G., Gotts, N.M.: A theory of spatial regions with indeterminate boundaries. In: Eschenbach, C., Habel, C., Smith, B. (eds.) Topological Foundations of Cognitive Science (1994) Cohn, A.G., Gotts, N.M.: The ‘egg-yolk’ representation of regions with indeterminate boundaries. In: Burrough, P., Frank, A.M. (eds.) Proceedings, GISDATA Specialist Meeting on Geographical Objects with Undetermined Boundaries, Francis Taylor, pp. 171–187 (1996a) Cohn, A.G., Gotts, N.M.: Representing spatial vagueness: a mereological approach. In: Aiello, J.D.L.C., Shapiro, S. (eds.) Proceedings of the 5th conference on principles of knowledge representation and reasoning (KR 1996), pp. 230–241. Morgan Kaufmann, San Francisco (1996b) Corver, N., Doetjes, J., Zwarts, J.: Linguistic perspectives on numerical expressions: Introduction. Lingua, special issue on linguistic perspectives on numerical expressions 117(5), 751–775 (2007) Dubois, D., Prade, H.: An introduction to possibilistic and fuzzy logics. In: Smets, P., Mamdani, E.H., Dubois, D., Prade, H. (eds.) Non-Standard Logics for Automated Reasoning, Academic Press, London (1988) Evans, A.J., Waters, T.: Mapping vernacular geography: web-based GIS tools for capturing ‘fuzzy’ or ‘vague’ entities. International Journal of Technology, Policy and Management 7(2), 134–150 (2007) Evans, M.: Can there be vague objects? Analysis 38, 208 (1978); reprinted in his Collected Papers, Oxford, Clarendon Press (1985) Fine, K.: Vagueness, truth and logic. Synth`ese 30, 263–300 (1975) Fisher, P., Orf, T.: An investigation of the meaning of near and close on a university campus. Computers, Environment and Urban Systems (1991) Frank, A.: Qualitative spatial reasoning about distances and directions in geographic space. Journal of Visual Languages and Computing 3, 343–371 (1992) Gahegan, M.: Proximity operators for qualitative spatial reasoning. In: Kuhn, W., Frank, A.U. (eds.) COSIT 1995. LNCS, vol. 988, pp. 31–44. Springer, Heidelberg (1995) Goguen, J.: The logic of inexact concepts. Synthese 19, 325–373 (1969) Guha, R., Lenat, D.: CYC: a mid-term report. AI Magazine 11(3), 32–59 (1990) Halpern, J.Y.: Intransitivity and vagueness. In: Principles of Knowledge Representation: Proceedings of the Ninth International Conference (KR 2004), pp. 121–129 (2004)

46

B. Bennett

Held, M.: VRONI: An engineering approach to the reliable and efficient computation of voronoi diagrams of points and line segments. Computational Geometry: Theory and Applications 18(2), 95–123 (2001) Kamp, H.: Two theories about adjectives. In: Keenan, E. (ed.) Formal Semantics of Natural Language, Cambridge University Press, Cambridge (1975) Keefe, R., Smith, P.: Vagueness: a Reader. MIT Press, Cambridge (1996) Krifka, M.: Approximate interpretation of number words: a case for strategic communication. In: Bouma, G., Kraer, I., Zwarts, J. (eds.) Cognitive foundations of interpretation, Koninklijke Nederlandse Akademie van Wetenschapen, pp. 111–126 (2007) Kronenfeld, B.J.: mplications of a data reduction framework to assignment of fuzzy membership values in continuous class maps Special issue on Spatial Vagueness, Uncertainty and Granularity. In: Bennett, B., Cristani, M. (eds.) Spatial Cognition and Computation, vol. 3(2/3), pp. 221–238 (2003) Kulik, L.: Spatial vagueness and second-order vagueness. Spatial Cognition and Computation 3(2&3), 157–183 (2003) Lawry, J.: Appropriateness measures: an uncertainty model for vague concepts. Synthese 161, 255–269 (2008) Lehmann, F., Cohn, A.G.: The EGG/YOLK reliability hierarchy: Semantic data integration using sorts with prototypes. In: Proc. Conf. on Information Knowledge Management, pp. 272–279. ACM Press, New York (1994) Lund, H.G.: When is a forest not a forest? Journal of Forestry 100(8), 21–28 (2002) Lundberg, U., Eckman, G.: Subjective geographic distance: A multidimensional comparison. Psychometrika 38, 113–122 (1973) Maling, D.: Measurements from Maps: principles and methods of cartometry. Pergamon Press, Oxford (1989) Mallenby, D., Bennett, B.: Applying spatial reasoning to topographical data with a grounded ontology. In: Fonseca, F., Rodr´ıgues, M.A., Levashkin, S. (eds.) GeoS 2007. LNCS, vol. 4853, pp. 210–227. Springer, Heidelberg (2007) Mehlberg, H.: The Reach of Science. Extract on Truth and Vagueness, pp. 427–455. University of Toronto Press (1958); reprinted in Keefe, Smith (1996) Randell, D.A., Cui, Z., Cohn, A.G.: A spatial logic based on regions and connection. In: Proc. 3rd Int. Conf. on Knowledge Representation and Reasoning, pp. 165–176. Morgan Kaufmann, San Mateo (1992) Robinson, V.: Interactive machine acquisition of a fuzzy spatial relation. Computers and Geosciences 16, 857–872 (1990) Robinson, V.: Individual and multipersonal fuzzy spatial relations acquired using humanmachine interaction. Fuzzy Sets and Systems 113, 133–145 (2000) Santos, P., Bennett, B., Sakellariou, G.: Supervaluation semantics for an inland water feature ontology. In: Kaelbling, L., Saffiotti, A. (eds.) Proceedings of the 19th International Joint Conference on Artificial Intelligence (IJCAI 2005), Edinburgh, pp. 564–569 (2005a) Santos, P., Bennett, B., Sakellariou, G.: Supervaluation semantics for an inland water feature ontology. In: Kaelbling, L.P., Saffiotti, A. (eds.) Proceedings of the 19th International Joint Conference on Artificial Intelligence (IJCAI 2005), pp. 564–569. Professional Book Center, Edinburgh (2005b) Schockaert, S., De Cock, M., Kerre, E.E.: Spatial reasoning in a fuzzy region connection calculus. Artificial Intelligence 173(2), 258–298 (2009) Schockaert, S., De Cock, M., Cornelis, C., Kerre, E.E.: Fuzzy region connection calculus: An interpretation based on closeness. International Journal of Approximate Reasoning 48(1), 332–347 (2008)

Spatial Vagueness

47

Simons, P.: Parts: A Study In Ontology. Clarendon Press, Oxford (1987) Smith, B., Mark, D.M.: Do mountains exist? towards an ontology of landforms. Environment and Planning B: Planning and Design 30(3), 411–427 (2003) Smith, N.J.: Vagueness and Degrees of Truth. Oxford University Press, Oxford (2008) Spatial Cognition and Computation: special issue on spatial vagueness, uncertainty and granularity (2003) Tarski, A.: What is elementary geometry? In: Henkin, L., Suppes, P., Tarski, A. (eds.) The Axiomatic Method (with special reference to geometry and physics), North-Holland, Amsterdam (1959) Third, A., Bennett, B., Mallenby, D.: Architecture for a grounded ontology of geographic information. In: Fonseca, F., Rodr´ıguez, M.A., Levashkin, S. (eds.) GeoS 2007. LNCS, vol. 4853, pp. 36–50. Springer, Heidelberg (2007) Tye, M.: Vague objects. Mind 99, 535–557 (1990) van Fraassen, B.C.: Presupposition, supervaluations and free logic. In: Lambert, K. (ed.) The Logical Way of Doing Things, ch. 4, pp. 67–91. Yale University Press, New Haven (1969) Varzi, A.C.: Vagueness in geography. Philosophy and Geography 4, 49–65 (2001a) Varzi, A. (ed.): Topoi: special issue on the philosophy of geography, vol. 20(2). Kluwer, Dordrecht (2001b) Waismann, F.: Verifiability. In: Flew, A. (ed.) Logic and Language, Doubleday, New York, pp. 122–151 (1965), Wang, F., Brent Hall, G.: Fuzzy representation of geographical boundaries in GIS. International Journal of GIS 10(5), 573–590 (1996) Williamson, T.: Vagueness and ignorance. In: Proceedings of the Aristotelian Society, vol. 66, pp. 145–162 (1992) Williamson, T.: Vagueness. The problems of philosophy. Routledge, London (1994) Worboys, M.F.: Nearness relations in environmental space. International Journal of Geographical Information Science 15(7), 633–651 (2001) Zadeh, L.A.: Fuzzy logic and approximate reasoning. Synthese 30, 407–428 (1975) Zimmermann, H.-J.: Fuzzy set theory—and its applications, 3rd edn. Kluwer Academic Publishers, Norwell (1996)

A General Approach to the Fuzzy Modeling of Spatial Relationships* Pascal Matsakis, Laurent Wendling, and JingBo Ni

Abstract. How to satisfactorily model spatial relationships between 2D or 3D objects? If the objects are far enough from each other, they can be approximated by their centers. If they are not too far, not too close, they can be approximated by their minimum bounding rectangles or boxes. If they are close, no such simplifying approximation should be made. Two concepts are at the core of the approach described in this paper: the concept of the F-histogram and that of the F-template. The basis of the former was laid a decade ago; since then, it has naturally evolved and matured. The latter is much newer, and has dual characteristics. Our aim here is to present a snapshot of these concepts and of their applications. It is to highlight (and reflect on) their duality⎯a duality that calls for a clear distinction between the terms spatial relationship, relationship to a reference object, and relative position. Finally, it is to identify directions for future research.

1 Introduction Philosophers, physicists and mathematicians have been debating about space for centuries. Here, space is considered Euclidean and independent of time (our apology to Einstein). It is not, however, a mere abstract void (and Leibniz would rejoice): talking about space implies talking about (spatial) objects and relationships. Indeed, space is viewed as “the structure defined by the set of spatial relationships between objects” [58]. In the present paper as in related literature, space is usually two- or three-dimensional, with a Cartesian coordinate system. A physical object is of no interest in itself; the focus is on the part of space it occupies. Objects, therefore, are seen as subsets of the Euclidean space. A point, a line segment, a disk, a toroid, the union of these, are examples of objects. An object may be bounded or unbounded, convex or concave, open or closed, connected or disconnected, etc. Practically, it is either in raster or vector form. A Pascal Matsakis · JingBo Ni University of Guelph, Ontario, Canada e-mail: {pmatsaki,jni}@uoguelph.ca Laurent Wendling Université Paris Descartes, France e-mail: [email protected] R. Jeansoulin et al. (Eds.): Methods for Handling Imperfect Spatial Info., STUDFUZZ 256, pp. 49–74. springerlink.com © Springer-Verlag Berlin Heidelberg 2010

50

P. Matsakis, L. Wendling, and J. Ni

raster object in 2D space, for example, is sometimes seen as the union of unit squares (pixels), and others as a cloud of points (pixel centers). Finally, note that fuzzy subsets of the Euclidean space may also be considered. Fuzzy sets make it possible to encapsulate information regarding the imprecision or the uncertainty in the spatial extent of some physical objects. There is, in the end, a variety of spatial objects. So there is a variety of spatial relationships. Some are language-based, in the sense that they are naturally referred to using everyday terms, e.g., the relationships “right” (is to the right of), “far” (is far from), “touch” (touches). Others are math-based, and may or may not be named (e.g., the 512 relationships defined by the 9-intersection model). Some are binary; they involve two objects only (e.g., object A is to the right of object B). Others are not (e.g., object A is between objects B and C). In this paper, we limit our discussion to binary relationships, which are by far the most common subject of studies. They are usually categorized into directional (e.g., “right”), distance (e.g., “far”), and topological (e.g., “touch”) relationships. This is not surprising, since angles and distances are at the core of Euclidean geometry, and Euclidean spaces are, above all, topological spaces. Other categories, however, are sometimes considered (e.g., “intersect” is set-theoretical before being topological). Spatial relationships are often modeled by fuzzy relations on the set of all objects. Consider, for example, the statement “A is far from B”. In many everyday situations, one would find it neither completely true nor completely false (even if A and B are very simple crisp objects). A fuzzy model of “far” attaches a numerical value to the pair (A, B), and this value is seen as the degree of truth of the statement above. Not only the use of fuzzy relations seems more natural than the use of standard, all-ornothing relations, but it also allows two fundamental questions to be answered. How to identify the most salient relationship between two given objects in a scene? How to identify the object that best satisfies a given relationship to a reference object? Answering these questions comes down to calculating and comparing the degrees of truth of several statements. See Figs. 1 and 2. These statements, however, are not independent from each other. Part of the calculation of each degree of truth might therefore be common to all degrees of truth and yield an intermediate result, interesting if only for efficiency purposes. This result can be seen as a quantitative representation of either the relative position between the two objects (first question, Fig. 1) or the relationship to the reference object (second question, Fig. 2). What we argue here is that a clear distinction should be made between the terms spatial relationship (a binary relationship), relationship to a reference object (a unary relationship), and relative position. True, the position of an object with respect to another may be described in terms of spatial relationships. However, it may also have a representation of its own, as mentioned above. Ideally, such a representation should allow any relationship between the two objects to be assessed. Practically, this is never the case. The information relative to a given relationship might not have been captured by the representation, or might have been encapsulated in an unfathomable way. One representation may be better suited than another to the assessment of some relationships, and vice versa. At any rate, we may be interested in relative positions for what they are, and not in any particular relationship (e.g., when carrying out a scene matching task).

A General Approach to the Fuzzy Modeling of Spatial Relationships

51

Section 2 illustrates the discussion above. Its aim is to clarify, through examples, the differences between the terms spatial relationship, relationship to a reference object, and relative position. Sections 3 and 4 introduce the two concepts at the core of the general approach described in this paper, while pointing out dual characteristics. Sections 5 and 6 show how these concepts may rely on two others, also with dual characteristics. Section 7 deals with algorithmic issues. Many applications have been studied; Section 8 presents a review of the related literature. Finally, directions for future research are given in Section 9. Note that spatial relationships have been studied for many years, in many disciplines, including cognitive science, linguistics, geography and artificial intelligence. See, e.g., [7] [8] [13] [14] [18] [19] [39] [40]. The approach described here focuses on fuzzy models of spatial relationships (as opposed to, e.g., qualitative models) and is general only in the sense that: a variety of spatial objects can be handled (e.g., crisp or fuzzy, connected or disconnected, in raster form or in vector form); a variety of spatial information can be captured and exploited (i.e., directional, distance, topological); there is a variety of current and potential applications.

Fig. 1 How to identify the most salient relationship between two given objects A and B? Here, A and B are represented by vector data, the position of A relative to B is represented by an F-histogram (Section 3), and the 3 statements by fuzzy logic values. The answer to the question is R1.

Fig. 2 How to identify the object that best satisfies a given relationship R to a reference object B? Here, R is represented by a fuzzy binary relation, B by vector data, the relationship R to B by an F-template (Section 4), and the 3 statements by fuzzy logic values. The answer to the question is A3.

P. Matsakis, L. Wendling, and J. Ni

52

2 An Important Distinction 2.1 Relative Position vs. Relationship⎯Example Consider two points p and q in the 2D space. Possible representations of the position of p relative to q are the tuple (xp, yp, xq, yq), whose elements are the Cartesian coordinates of p and q; the pair (xqp, yqp), whose elements are the Cartesian coordinates of the vector qp; the pair (ρqp,θqp), whose elements are the polar coordinates of qp; the angle θqp; etc. The first representation is trivial. The second one, (xqp, yqp), is much more interesting. Although some information about p and q is lost, there is no loss of information about the position of p relative to q (assuming that relative position is invariant to translation). The third representation has the same characteristic. However, it is better suited for the assessment of distance relationships. These relationships cannot be assessed from the fourth representation, θqp. Too much information is lost. Nonetheless, θqp is a very compact representation, well suited for the assessment of directional relationships. For example, assuming that angular coordinates belong to (−π,π] and that the polar axis is horizontal and pointing to the right, we may consider that the degree of truth of the statement “p is to the right of q” is min{1, max{0,1−2|θqp|/π}}. In other words, the fuzzy relation R defined by

2 ⎧ ⎫⎫ ⎧ R( p, q) = min ⎨1, max ⎨ 0,1 − θqp ⎬ ⎬ π ⎭⎭ ⎩ ⎩

(1)

is a fuzzy model of the binary directional relationship “right”. If θqp= 0 then R(p, q)=1, i.e., p is definitely to the right of q. If θqp= π/2 then R(p, q)=0, i.e., p is definitely not to the right of q. In the end, once the relative position θqp has been calculated given the Cartesian coordinates of p and q (a painful task if you are using only pen and paper), the statements “p is to the right of q”, “p is above q”, “p is in direction 45° of q”, etc., can be evaluated with comparatively much less effort. The link with Fig. 1 should now be clear to the reader. Note that this example can be easily adapted to the 3D case.

2.2 Relationship vs. Relationship to a Reference Object⎯Example Here, an object is a “friendly” set of points in a rectangular region R of the 2D space, i.e., it is a nonempty, bounded, connected, regular closed set of points, included in R. Consider two objects A and B. Let |B| be the area of B. For any two points p and q, let |qp| be the distance between p and q. We call

d(A, B) = inf p∈A, q∈B qp

(2)

A General Approach to the Fuzzy Modeling of Spatial Relationships

53

the distance between A and B, and we call s(B) = 2

B π

(3)

the size of B (it is the diameter of a disk whose area is |B|). The fuzzy relation R defined by ⎧ d(A, B) ⎫ R(A, B) = max ⎨0,1− (4) ⎬, s(B) t ⎭ ⎩

where t denotes a positive real number, is a fuzzy model of the binary distance relationship “close”. If the distance between A and B is 0, then R(A, B)=1, i.e., A is definitely close to B. If the distance between A and B is t times larger than B, then R(A, B)=0, i.e., A is not close at all to B. Now, given B, assume we are asked to evaluate the statement “A is close to B” for a large number of objects A. Going through (2) and (4) every time would be inefficient. A better strategy is to compute the function dB defined on R by d B ( p) = infq∈B qp

(5)

d(A, B) = inf p∈A d B ( p) .

(6)

⎧ d ( p) ⎫ d B ( p) = max ⎨0,1− B ⎬ s(B) t ⎭ ⎩

(7)

R (A, B) = sup p ∈A d B ( p) .

(8)

and then use the fact that

Or, compute the function

and use the fact that

_ Once dB has been computed, the statements “A1 is close to B”, “A2 is close to B”, _ etc., can be readily evaluated. dB and dB are two possible representations of the unary distance relationship “close to B”. See the link with Fig. 2. Note that dB is usually known as a distance map. Again, this example can be easily adapted to the 3D case.

3 F-Histograms One of the two concepts at the core of the general approach described in this paper is the concept of the F-histogram. Its basis was laid a decade ago [23]. Since then, of course, the concept has evolved and matured. The idea and assumption behind

P. Matsakis, L. Wendling, and J. Ni

54

it are that acceptable representations of relative positions can be obtained by reducing the handling of all 3D and 2D objects to the handling of 1D entities. Notation and terminology are as follows. S is the Euclidean space. A direction θ is a unit vector. θ⊥ is the subspace orthogonal to θ that passes through the origin ω (an arbitrary point of S). The expression Sθ(p) denotes the line in direction θ that passes through the point p. Now, consider a fuzzy subset A of S. The membership α degree of p in A is μA(p). For any α∈[0,1], the α-cut of A is A ={p ∈S | μA(p)≥α}. The (fuzzy) intersection of A with Sθ(p) is denoted by Aθ(p) and called a section of A. An object is a fuzzy subset A of S such that any μA(p) belongs to the set α {α1,α2,…,αk+1}, with 1=α1>α2>…>αk+1=0, and any (Aθ(p)) i has a finite number of connected components. Consider two objects A and B. Consider a function F that accepts argument values of the form (θ, Aθ(p), Bθ(p)). The F-histogram associated with the pair (A, B) is a function FAB of θ. Its intended purpose is to represent, in some way, the position of A with respect to B. The histogram value FAB(θ) is defined as a combination of the F(θ, Aθ(p), Bθ(p)) values, for all p in θ⊥. See (9) and Fig. 3, where ope stands for the combination operator. Figure 4 is related, but will be commented in Section 4. F AB (θ) = ope p ∈θ⊥ F (θ, Aθ ( p), Bθ ( p))

(9)

Typically, F and FAB are real functions, the combination operator ope is the addition, and F AB (θ) = ∫

p∈θ⊥

F (θ, Aθ ( p), Bθ ( p)) dp .

(10)

The key point, then, is how to choose F. First, we might want to reduce the handling of fuzzy sections I and J to that of crisp sections, through some other function F: F (θ, I, J ) = ∑ i=1 ∑ j=1 (αi − α i+1 )(α j − α j+1 ) F(θ, I αi , J k

k

αj

).

(11)

Second, we might want to reduce the handling of crisp sections I and J to that of their connected components I1, I2, …, Im and J1, J2, …, Jn: F(θ, I, J ) = ∑ i=1 ∑ j=1 f (θ, Ii , J j ) . m

n

(12)

A General Approach to the Fuzzy Modeling of Spatial Relationships

Fig. 3 Principle of the calculation of the F-histogram F

Fig. 4 Principle of the calculation of the F-template F

55

AB

RB

Further reduction can be expressed as f (θ, I, J ) = ∫

∫

p∈I q∈J

ϕ (θ, p, q) dp dq ,

where I and J are (crisp) singletons, segments, lines or half-lines.

(13)

P. Matsakis, L. Wendling, and J. Ni

56

Note that for any fuzzy sections I and J, we then have F (θ, I, J ) = ∫

∫

p∈S q∈S

μ I ( p) μ J (q) ϕ (θ, p, q) dp dq .

(14)

FAB can also be referred to as the F-histogram FAB, the f-histogram f AB, or the ϕ-histogram ϕAB, depending on whether (11), (11) and (12), or (11), (12) and (13) hold. This categorization is illustrated by Fig. 5. Two properties are worth noticing at this point: F AB = ∑ f

∑ j=1 (αi − α i+1 )(α j − α j+1 ) F A k

( U i =1 Ai )( U j =1 B j ) m

and

k i=1 n

= ∑ i=1 ∑ j=1 f m

n

Ai B j

αi

B

αj

,

(15)

(16)

where A1, A2, …, Am are pairwise disjoint objects, and B1, B2, …, Bn too. Now, for any real number r, consider the function ϕr defined by: ϕr(θ, p, q)=1/|qp|r if p≠q and if θ is the direction of the vector qp; ϕr(θ, p, q)=0 otherwise. The ϕr-histogram ϕrAB is called a force histogram. The reason for the term force (and for the symbols F, F, f and ϕ, which all refer to the first letter of the words function and force) is the following. For any direction θ, the value ϕrAB (θ) can be seen as the scalar resultant of elementary physical forces. These forces (which are additive vector quantities) are exerted by the points of A on those of B, and each tends to move B in direction θ. Assume r =2. The forces then correspond to gravitational forces. This is according to Newton’s law of gravity, which states that every particle attracts every other particle with a force inversely proportional to the square of the distance between them. Under the above assumption, it is as if the objects A and B had mass and density: the area (2D case) or volumetric (3D case) mass density of A at point p is μA(p); the density of B at q is μB(q). Note that in the 2D case A and B can be seen as flat metal plates of constant and negligible thickness.

Fig. 5 Categorization of F-histograms F-histograms include F-histograms, which include f-histograms, etc.

A General Approach to the Fuzzy Modeling of Spatial Relationships

57

4 F-Templates How to identify, in a scene, the object that best satisfies a given relationship to a reference object? This question, which is one of the two fundamental questions that arise when dealing with spatial relationships (Section 1), defines an object localization task. One theory supported by cognitive experiments is that people accomplish this task by parsing space around the reference object into good regions (where the object being sought is more likely to be), acceptable, and unacceptable regions (where the object being sought cannot be) [15] [22]. These regions blend into one another and form a so-called spatial template [22], which assigns each point in space a value between 0 (unacceptable region) and 1 (good region). In other words, a spatial template is a fuzzy subset of the Euclidean space that represents a relationship to a reference object. The concept of the F-template was introduced in a series of three conference papers [29] [52] [10]. The idea and assumption behind it are that acceptable representations of relationships to reference objects can be obtained by reducing the handling of all 3D and 2D objects to the handling of 1D entities. A formal definition of the F-template is given in Section 4.1, and it is followed by an important example in Section 4.2.

4.1 Definition Here, the line in direction θ that passes through the point p is denoted by Sp(θ) (instead of Sθ(p), as in Section 3). The (fuzzy) intersection of Sp(θ) with a fuzzy subset A of S is denoted by Ap(θ) (instead of Aθ(p)). Consider a spatial relationship R and an object B. Consider a function F that accepts argument values of the form (p, R, Bp(θ)). The F-template associated with the pair (R, B) is a function F RB of p. Its intended purpose is to represent, in some way, the relationship R to the reference object B. The template value F RB (p) is defined as a combination of the

F(p, R, Bp(θ)) values, for all θ. See (17) and Fig. 4, where ope stands for the combination operator. F RB ( p) = opeθ F ( p, R, B p (θ))

(17)

There is obviously a duality between the F-template and the F-histogram, and it echoes the duality between the two fundamental questions mentioned in Section 1. Compare Fig. 4 with Fig. 3, and Fig. 2 with Fig. 1. Compare (17) with (9). In (17), θ varies and p does not. In (9), p varies and θ does not. Replace any subset of A with R, replace p with θ and θ with p, and (9) transforms into (17). Typically, F and F RB are real functions with output values in the range [0,1], the template F RB is a fuzzy subset of the Euclidean space, and F RB (p) aims to represent the degree to which p satisfies the relationship R to the reference object B.

58

P. Matsakis, L. Wendling, and J. Ni

4.2 An Important Example: Basic Directional Templates A spatial template that represents a directional relationship to a reference object may be called a directional (spatial) template. To emphasize the analogy with the wellknown distance maps (mentioned in Section 2.2), we may also call it a directional map. In [2], Bloch introduces the concept of the fuzzy landscape. A fuzzy landscape is a specific example of directional template, which does not sacrifice the geometry of the reference object (the object is not approximated through, e.g., its centroid, or its minimum bounding rectangle or box). Moreover, the defining equation (whose roots can be traced to earlier works [34] [20]) is very simple and intuitive. Because of this and the fact that the term template was coined earlier, and also to increase precision in language, we prefer to talk of basic directional templates, or basic directional maps, instead of fuzzy landscapes. The basic directional template induced by the object B in direction δ associates the value 2 ⎧ ⎧ ⎫⎫ sup q∈S −{ p} μ B (q) min ⎨1, max ⎨0,1− ∠(qp,δ) ⎬ ⎬ π ⎩ ⎭⎭ ⎩

(18)

with each point p, where ∠(qp, δ) denotes the angle between the two vectors qp and δ. Compare (18) with (1). In the case of raster data, the exact algorithm that naturally results from (18) is straightforward but computationally expensive. Reference [2] describes a much faster approximation algorithm, inspired by chamfer methods. Consider, e.g., a 2D image. The pixels are examined sequentially, from top to bottom and left to right, and then from bottom to top and right to left. Each time a pixel is examined, it is assigned a value whose calculation also involves the pixel’s neighbors. As shown in [2], a basic directional template can be seen as the morphological dilation of the reference object by a fuzzy structuring element. The idea behind the algorithm is to perform the fuzzy dilation with a limited support for the structuring element. According to Bloch, “most approaches [e.g., the F-histogram / template approach] reduce the representation of objects to points, segments or projections” [3] while hers “takes morphological information about the shapes… into account” [2], “considers the objects as a whole and therefore better accounts for their shape” [3]. The argument does not hold water, since basic directional templates can be proved to be Ftemplates [29]. As a result, they can be calculated by reducing objects to segments, using an F-template approach [29] [52]. An extensive experiment [30] has shown that this approach should be preferred to the morphological one in the case of 2D raster data, but that the opposite holds in the case of 3D raster data. 2D vector data can only be handled using the F-template approach, and there is yet no algorithm for 3D vector data. Once the basic directional template induced by B in direction δ has been computed, the degree of truth of the statement “A is in direction δ of B” (i.e., “A is in relationship R with B” where R denotes the relationship “in direction δ”) can be calculated for any object A, in comparatively no time, using a fuzzy pattern-matching approach [12] [2].

A General Approach to the Fuzzy Modeling of Spatial Relationships

59

5 F-Histograms from Spatial Correlation Most work on F-histograms has focused on force histograms. The reasons are multiple, as explained in Section 5.1. Force histograms can actually be generated from spatial correlations; this is an important concept, covered in Section 5.2.

5.1 Interest in Force Histograms Force histograms are relative position descriptors with high discriminative power [25]. Moreover, the way they change when the objects are affine-transformed is known [25] [36]. This is an important issue in computer vision and pattern recognition, especially because it is intrinsically linked to the design of widely used affine invariant descriptors. Remember that affine transformations include, e.g., translations, rotations, scalings, reflections and stretches. Let aff be any invertible affine transformation. It can be written as the composition of a translation with a linear transformation lin (an affine transformation such that lin(ω)=ω). It is a common convention to see lin as a matrix (a 2×2 matrix if S is of dimension 2; a 3×3 matrix if S is of dimension 3). Likewise, vectors can be seen as column matrices and vice versa. As shown in [36], for any real number r, any objects A and B, and any direction θ, [A ] aff [B ] ϕ aff (θ) = det(lin) r

lin −1 ⋅ θ

r −1

ϕ rAB (θ') .

(19)

In this equation, det(lin) is the determinant of the matrix lin and⎟det(lin) ⎜ its absolute value; the symbol . denotes matrix multiplication;⎟ lin−1 . θ ⎜ is the norm of

the vector lin−1 . θ ; the direction θ' is the unit vector (lin−1 . θ) /⎟ lin−1 . θ ⎜. The importance of having a property such as (19) is discussed in [25] and illustrated through experiments with synthetic and real data. Another reason for the special interest in force histograms is that they lend themselves, with great flexibility, to the modeling of directional relationships by fuzzy binary relations [26]. The main methods that can be used to achieve this are the aggregation method [20], the compatibility method [34], and the method based on force categorization [27]. The fuzzy relations then satisfy four fundamental properties, which express the following intuitive ideas: if the objects in hand are sufficiently far apart, each one can be seen as a single point in space; the directional relationships are not sensitive to scale changes; all directions have the same importance; the semantic inverse principle [16] is respected (e.g., object A is to the left of object B as much as B is to the right of A). As a corollary of these properties, it is possible to determine how the fuzzy relations react when the objects are similarity-transformed. There is, of course, a link with (19), since similarity transformations are particular affine transformations. Note that the four abovementioned properties form the axiomatic basis upon which the concept of the histogram of forces was actually developed [23].

P. Matsakis, L. Wendling, and J. Ni

60

Directional relationships are not, however, the only spatial relationships that can be assessed from force histograms. Reference [42] describes a fuzzy model of inner-adjacency. The position of A relative to B is then represented by ϕrA(B−A) instead of ϕrAB. Reference [45] describes a fuzzy model of surroundedness. The underlying assumption is that A is connected and does not intersect the convex hull of B. Reference [24] describes a fuzzy model of betweenness. Although the preposition “between” usually denotes a ternary relationship, its model in [24] is a fuzzy binary relation. A sentence such as “A is between B and C ” is read “A is between B∪C ”. The position of A relative to B and C is represented by ϕrA(B∪C). Most work on F-histograms has focused on force histograms, but not all. Consider ϕ2AB. It has interesting characteristics [33]. Usually, however, it is not defined anywhere if A and B intersect, because the integral in (10) then diverges. As shown in [23], ϕ-histograms that are not force histograms make it possible to overcome this limitation while preserving the abovementioned characteristics. Reference [23] also suggests f-(non-ϕ-)histograms for the handling of convex objects. The fuzzy model of surroundedness mentioned in the previous paragraph suits the application considered in [45], but only because the objects there satisfy certain conditions. Another model would otherwise be necessary. Its design could be based on F-(non-f-)histograms, instead of force histograms. This is a promising avenue, as pointed in [24]. Finally, [31] describes F-(non-F-)histograms for the combined extraction of directional and topological relationship information. The particularity of these histograms is that they are coupled with Allen relations [1] using fuzzy set theory. Various systems rely on them to capture the essence of the relative positions of objects with natural language descriptions [32] [55] [56].

5.2 Spatial Correlation In [27], a natural language description of the relative position between two objects A and B is generated from the force histograms ϕ 0 AB and ϕ 2 AB. The fact is that ϕ 0 AB and ϕ 2 AB have very different and interesting characteristics, which complement one another. As this example shows, it may be useful to calculate two or more force histograms associated with the same pair of objects. These histograms are obviously not totally independent from each other. Part of the calculation of one might therefore be common to all and yield an intermediate result, interesting if only for efficiency purposes. The same idea was expressed in Section 1; the intermediate result was seen as a quantitative representation of the relative position between two objects. Here, the intermediate result is a spatial correlation. Compare Fig. 6 with Fig. 1. Figure 7 is related, but will be commented in Section 6. The spatial correlation between A and B provides raw information about the position of A relative to B. It is the function ψ AB defined by ψ AB (v) = ∫

q∈S

μ A (q + v) μ B (q) dq ,

(20)

A General Approach to the Fuzzy Modeling of Spatial Relationships

61

where v denotes any vector and + denotes point-vector addition. All force AB histograms ϕ AB as follows: r associated with A and B can be derived from ψ ϕ rAB (θ) = ∫

+∞ ψ AB (uθ)

0

ur

du .

(21)

Reference [36] shows that (20) and (21) lead to different algorithms than (10) and (14) and are better adapted to the solving of some theoretical issues.

Fig. 6 F-histograms from spatial correlation

Fig. 7 F-templates from force field

6 F-Templates from Force Field Basic directional templates have been used for spatial reasoning, object localization and identification, structural and model-based pattern recognition [4] [9] [21] [48]. They have, however, important flaws. They are overly sensitive to outliers. Elongated reference objects pose problems, and concave objects as well [28]. The main reason is that basic directional templates make use of angle information but ignore distance information. According to cognitive experiments [22] [17] [15], the former is indeed of primary importance, but the latter also contributes in shaping a directional template. For a given angular deviation, the membership degrees are not constant. They fluctuate slightly, depending on the distance to the reference object. Moreover, the fluctuation varies from one angular deviation to another. Finally, when sufficiently far from the object, all the membership degrees drop. For example, if you were told that the soccer ball was

P. Matsakis, L. Wendling, and J. Ni

62

to the right of the bench, you would not look for it hundreds of feet from the bench. One may wonder whether angle information and distance information could be processed in separate steps. In [52], the authors argue that the answer is negative, and they show how directional F-templates can embed distance information to elegantly overcome the abovementioned flaws. Their work is based on the following results: (i) basic directional templates are F-templates [29]; (ii) distance _ maps like dB and dB (Section 2.2) are F-templates too [52]; a binary operation ⊗ and two F-templates p

opeθ F1(p,R, Bp(θ)) and p

opeθ F2(p,R, Bp(θ))

define a new F-template p opeθ F1(p,R, Bp(θ)) ⊗ F2(p,R, Bp(θ)). Now, assume different directional relationships to the same reference object need to be considered. Assume they are represented by directional templates. These templates are obviously not totally independent from each other. Part of the calculation of one might therefore be common to all and yield an intermediate result, interesting if only for efficiency purposes. The same idea was expressed in Section 1; the intermediate result was seen as a quantitative representation of a relationship to a reference object. Here, the idea is coupled with the desire to exploit the duality between F-templates and F-histograms; the intermediate result is a force field. Compare Fig. 7 with Fig. 2, and Fig. 7 with Fig. 6. Once the force field has been computed, F-templates that represent directional relationships to the reference object can be derived from the field in negligible time. Basic directional templates, and the templates described in [52], cannot be calculated using such a two-step procedure. The force field induced by B is the function ψ Br defined as follows: ψ rB ( p) = ∫

q ∈S

μ B (q)

qp qp

r +1

dq

(22)

The reason for the term force is the same as in Section 3. The object B is seen as an object with mass and density: the density of B at point q is μB(q). The vector ψ Br ( p) is the force exerted on B by a particle of mass 1 located at p. The force field-based template induced by B in direction δ makes use of both angle and B distance information. It is a function ϕ R r that may be defined as   rB ( p)i
  RB ( p) = max 0,

,  r B  (q)i
sup qS r  

(23)

where y denotes the dot product and R is the relationship “in direction δ”. Once B ϕR r has been computed, the degree of truth of the statement “A is in relationship

R with B” (i.e., “A is in direction δ of B”) can be calculated for any object A, the

A General Approach to the Fuzzy Modeling of Spatial Relationships

63

same way as mentioned in Section 4.2. Preliminary experiments [28] [38], where the characteristics of force field-based templates are examined and compared with those of basic directional templates, show the interest of the approach. Note that B RB the connection between the two pairs (ψ AB, ϕ AB r ) and (ψ r , ϕ r ) in Figs. 6 and 7 can be elegantly expressed by the equation below:

∫θ ϕ r −1 (θ) θ dθ = ∫ p μ A ( p) ψ r ( p) d p B

AB

(24)

7 On the Design of Efficient algorithms F-histograms and F-templates lend themselves to the design of efficient algorithms, whether the Euclidean space is of dimension two or three, the objects are crisp or fuzzy, in raster or vector form. Section 7.1 illustrates some of the typical steps in the design process. These steps are briefly described in Section 7.2, from a higher perspective. An important issue (the selection of a set of reference directions) is covered more extensively in Section 7.3.

7.1 Illustrative Example How can (10) and (14) be adapted to the case of 2D raster data? Consider two objects A and B, a direction θ and a point p. As illustrated in Fig. 8a, the line Sθ(p) might pass through some pixels i and j of A and B, with nonzero membership degrees μA(i) and μB(j). These pixels can be determined by rasterizing Sθ(p) using a line-drawing algorithm. They project on Sθ(p) as segments Ii and Jj. Let I=Aθ(p) and J=Bθ(p). The value of F (θ, I, J ) may be calculated as follows: F (θ, I, J ) = ∑ i ∑ j μ A (i) μ B ( j)

∫ p∈I ∫q∈J i

ϕ (θ, p, q) dp dq

(25)

j

(25) then replaces (14). Moreover, the integral in (10) can be approximated by a finite sum; (10) may be replaced with F AB (θ) = εθ ∑ k ∈Z F (θ, Aθ ( pk ), Bθ ( pk )) ,

(26)

where εθ and the points pk are as suggested in Fig. 9a. Symbolic computation of the double integral in (25) yields closed-form expressions that do not depend on A nor B. See Fig. 10. In the end, numerical computation of F (θ, I, J ) ⎯ and F AB (θ) ⎯ translates into multiple instantiations of these expressions.

P. Matsakis, L. Wendling, and J. Ni

64

A J1 I4 I1

I2

A

J2

J1

I5

I2 B

B

I3

I1 (a)

S (p)

(b)

S (p)

Fig. 8 The sections Aθ(p) and Bθ(p) are decomposed into segments Ii and Jj. In the case of fuzzy objects (left), all segments are of the same length. In the case of crisp objects (right), the segments Ii are mutually disjoint, and the segments Jj also.

A

S (pk+1) S (pk) S (pk
1)

A

B



B (a)

(b)

Fig. 9 In the case of raster data (left), the lines Sθ(pk) partition the objects into sets of adjacent pixels; the distance between two consecutive lines is constant. In the case of vector data (right), the lines pass through the vertices of the objects and partition the objects into trapezoids; the distance between two consecutive lines varies.

For crisp objects, (25) can be rewritten as follows:

F (θ, I, J ) = ∑ i ∑ j

∫ p∈I ∫q∈J i

ϕ (θ, p, q) dp dq ,

(27)

j

where the segments Ii and Jj are now as illustrated in Fig. 8b. In this case, symbolic computation of the double integral yields more expressions than as in Fig. 10. F AB (θ) , however, computes much faster, since each instantiation corresponds to the process of a batch of pairs of object pixels instead of the process of a single pair. Actually, (27) can be used in place of (25) whether the objects are crisp or fuzzy: the idea is to exploit (15), i.e., it is to handle the fuzzy objects through their level cuts (which are crisp objects). Equations (15) and (27) lead to shorter processing times than (25) if one object is crisp and the other fuzzy with few membership degrees. Note that all of the above holds in the case of 3D raster data: replace the word ‘pixel’ with ‘voxel’, and the sum in (26) with a

A General Approach to the Fuzzy Modeling of Spatial Relationships

 pI qJ

u2

i

Ii Jj



u=1 Ii

Jj



u=0 Ii =Jj



u
1 Jj

j

65


r (, p, q) dp dq

r=1 
[(u
1).ln(u
1)
2u.ln(u)+(u+1).ln(u+1)] r=2 2.ln(u)
ln(u
1)
ln(u+1) r1 and r2
2
r [(u
1)2
r
2u2
r+(u+1)2
r] / [(1
r)(2
r)] r=1 2
ln(2) r1 and r