Acknowledgements

This thesis is submitted for the degree of Cand Scient at the Department of Informatics

University of Oslo

The work has b een carried out at the Division of Information Technology at SINTEF SI

Department of Industrial Mathematics I would like to thank all the sta of the division for

the stimulating environment they have supplied b oth professionally and so cially Without

mentioning your names you should all know how thankful I am for your friendship and the

answers you have provided to all my questions

I would like to thank my sup ervisors Erlend Arge and Morten Dhlen for their inspiring

guidance in general and the advices given on mathematical issues in particular I am esp ecially

grateful to them for giving me sucient freedom to ensure the creativity of the work

A sp ecial thank to Bjrn Skjellaug at the Department for Co op erative Systems who has

b een an informal advisor on the data mo deling asp ects of my work He has also guided me in

the art of writing scientic rep orts and generously encouraged me when I needed it at most

To David Skogan and Olav Mork Bjrnas I have very much appreciated our vivid dis

cussions and your comments to selected parts of the thesis

At last but not least I want to express my sincere gratitude for the p ersonal supp ort

given by my wife Hilde Bro dahl and the patience she has showed during the work She has

also b een of great help in pro of reading of the thesis and in giving a helping hand in other

editorial pro cesses

Oslo November

Gunnar Misund

SI Center for Industrial Research Oslo is part of the SINTEF Group which employs ab out research scientists

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Contents

Acknowledgments i

Introduction ix

Prologue

I CARTOGRAPHY AND GIS

Outline

Cartography

Representing the Earth

The Paper Map Mo del

Generalization

Cartographic information

Generalizing top ographic information

Generalizing thematic information

Cartographic generalization

Geographic Information Systems

Managing Spatiotemp oral Information

Computer Aided Cartography

Automated Generalization

Map Mo dels in GIS

Topographic mo dels

Thematic mo dels

GIS standards

An Augmented Map Concept

Challenges

Augmenting the Paper Map Mo del

Summary

II MULTIMODELS

Outline

The Multiple Nature of Geographic Information

Examples of Multiple Geographic Information

Text

Parametric curves

Functions

Multiple Mo deling

Vector approaches

Tessellation approaches

Topological approaches

Temporal reasoning

Integrated Mo deling

The Multimo del

The Multimo del Concept

Digital Mo dels

Denitions

Mo del arithmetics

Approximation and renement

Digital Multimo dels

Asp ects of Multimo dels

Some Categories of Multimo dels

Explicit Multimo del

Multiedition mo dels

Multiscale mo dels

MULTIMOD A generic ob ject mo del

Examples of Multimo dels

Designing Multimo dels

Piecewise Linear Curves

Piecewise Linear Surfaces

Summary

III METAMAP

Outline

Integration of Geographic Information

Strategies in spatial information integration

Topology as information integration

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Metamap The Conceptual Mo del

Geographic entities

Geographic duality

The Metamap element

Topographic elements

Thematic elements

Topological structures

Map top ology

Primary top ology

Secondary top ology

Tertiary top ology

The Metamap

Ob ject mo del

Metamap as an augmented map concept

MINIMAP Towards an Implementation

Metamap mo deling

MINIMAP

Geographic elements

Topographic elements

Thematic elements

Topological structures

Ob ject mo del

Summary

IV CONCLUSIONS AND IMPLEMENTATIONS

Results

Future Research

Epilogue

A MULTIMOD A Simple Multimo del Library

A The generic library

A Customizing MULTIMOD

B MINIMAP A Simple Metamap Library

B The library

B Examples

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

List of Figures

Contour map

Crossing contours meter tolerance corresp onding to ca

Dislo cation causing oshore road

Primitive sea chart

Roman road map

TOmap

Transfer of information

Simplication

Combination

Deformation

Generalization of text

Generalization of pattern

A Geographic Information System

Textual description as thematic timevarying information

Initial map

Generalized edition

Propagation of change in initial mo del

Description of tidal variation in three resolution

Delta vectors

Adding initial vector and dierences

Change in initial vector and the comp onent

Data reduction of delta mo del

Pseudoscaleless structure

Quadtree

Simp el temp oral mo deling

Function an its approximation

Renement of approximant

Main structure of MULTIMOD

MULTIMOD a generic Multimo del library

The degrees of freedom in mo deling

viii

Geographic and nonspatial entities

The Metamap element

Metamap as metamo del

MINIMAP an ob ject mo del

Contour map after heavy data reduction of terrain

A Initial PL curve

A Approximants of initial PLC

A Detail of AD approximation

A Decomp osed delta representation of PLC

A Decomp osed delta representation of PLC detail I

A Decomp osed delta representation of PLC detail II

A Original Delauney triangulation

A Reduced triangulation with p oints

A Surface dened over p oints

A Customization of MULTIMOD

B Contour plot of the Metamap

B D visualization of the Metamap

B Simple generalization

B Group of elements

B Propagation of generalization

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Introduction

The Ptolemaic paradox

In the rst printed version of Geography by the Alexandrian multiscientist Claudius

2 3

Ptolemy was published see Bro for details of Ptolemy and Geography

The b o ok was written in the second century AD and is a compilation of the contemporary

knowledge ab out the Earth also including a treatise on cartography Ptolemy describ es how

to design maps for example how to make pro jections from the spherical surface Geography

also describ es ab out places in the then known world Geography is recognized as

the rst atlas and this sp ecial form of presenting geographic information has changed very

little since the days of Ptolemy The b o ok and the cartographic traditions it was based on

was forgotten in the Western civilization during the Middle Ages In this p erio d the Earth

was considered to b e a circular or even sometimes rectangular disc and most maps were

merely presenting legends phantasy and religious views of the world Fortunately Arab and

Byzantine scholars and copyists kept the Ptolemiac tradition alive during the p erio d from

AD to AD

Before the turn of the fteenth century not less than seven folio editions of Geogra

phy also called Cosmography was published exp ensively illustrated and in most cases

supplemented with maps After the rediscovery of these imp ortant writings cartography

exp erienced a revival after the years of standstill The demand for b etter and more

comprehensive maps mostly due to the discovery and the b eginning exploitation of new

4 5

territories was matched by ecient supply made p ossible by Gutenberg and the rise of

mass media The combination of new application areas for the map and novel technology

lead to quite a revolution in map making

Now ab out half a millenni um after the rediscovery of Ptolemy and the introduction

Regretfully only references to AngloAmerican and Norwegian literature are made throughout the thesis

This is indeed not indicating that relevant literature in other languages do not exist but reects the fact that

the author do not master other languages well enough to include such references

Claudius Ptolemy is p erhaps more known as the father of the astronomical system where the planets

circles around the xed Earth This is describ ed in his Megiste Syntaxis The Great System also called

Amalgest He also wrote volumes of music theory which represent our main knowledge of ancient western

music theory

One of the imp ortant events in this context was the discovery of America in However Cristopher

Columbus exp edition represented only the culmination of a series of remarkable discoveries made within the

last part of the fteenth century

Johann Gutenberg Germany invented at ab out AD the art of printing b o oks with

movable types

x Introduction

of the printing technology we exp erience a similar revolution As so ciety has grown more

complex and our exploitation of the Earth is reaching the limits where fatal and nonreversible

damage threatens new demands have emerged calling for wider and more intensive and

advanced use of maps The demands are as during the map revival in the Renaissance met

by the introduction of a new technology Now we encounter computer aided cartography or

CAC for short in naval navigation in pro duction of limited editions of customized maps in

land resources assessment in military missile guidance systems and in global environmental

surveillance The term Geographical Information Systems GIS for short covers many of

the applications utilizing a digital map

Still as indicated by Burrough in Bur and to some degree shown in section it is

the same map mo del that is the foundation in computer aided cartography as it was in manual

cartography in the Renaissance which again was based on the ancient mo del describ ed by

Ptolemy

One of the main problems in traditional cartography is how to pro ject the spherical surface

of Earth onto a planar medium All maps are essentially distorted representations and one

of the consequences of this is that it is dicult to compare information given in two dierent

pro jections For this reason many cartographers have promoted large glob es as the most ideal

maps Still the ma jority of CAC applications op erates in planar co ordinates Cartographers

have nally got access to a to ol making it trivial to store compute and analyze geographic

information directly related to the spherical surface For some reasons this opp ortunity has

not yet b een fully taken advantage of

This is an example on what will b e called the Ptolemaic paradox in this thesis that the

contemporary and quite sophisticated information technology is not b eing fully exploited in

computer aided cartography Even though computers in several cases are able to handle new

problems or to oer new solutions to existing problems the main ob jective for the use of

information technology in contemporary cartography has b een to make existing cartographic

metho ds more ecient and accurate

This mismatch b etween advanced available to ols and the limited and quite simple mo del

of the world they are applied on constitutes the main motivation for the thesis As shown

in Prologue the Ptolemaic paradox is not only restricting the p ossibilitie s in CAC but it

also gives rise to anomalies that may cause malfunctioning This is due to the fact that

the traditional map is designed for manual treatment and is certainly not prepared for the

semiautomatic pro cedures introduced in computer aided cartography

Scop e

The overall scop e of the thesis is to identify and to some degree solve selected problems due

to the Ptolemaic paradox Details of the scop e are outlined as follows

The thesis should provide enhancements and additions to the traditional map concept

This augmented map concept is to b e understo o d as a framework within which more

realistic mo dels of spatiotemp oral information may b e developed and implemented

The matter is even worse the surface is as known not a sphere but a complicated geometric ob ject close

to an ellipsoid

The scop e of the thesis has gradually evolved during the work

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

The augmented map concept should provide ecient supp ort for cartographic general

ization which is a central issue in cartography

In the augmented map mo del spatial and nonspatial information should b e treated in

an integrated manner and as equally imp ortant asp ects of the geographic entities

The emphasis will b e on the spatial asp ects of cartography Still selected issues con

cerning nonspatial information have to b e discussed to ensure a certain degree of com

prehensiveness

A limited ob jectoriented implementation should b e carried out to illustrate some of the

main features of the concept

The thesis should according to this scop e develop an augmented map concept that is con

ceptually comprehensive but fragmented with resp ect to detail esp ecially in the nonspatial

domain The scop e also implies the accomplishment of a complete pro cess from a complex

problem to a computer program

Outline

The thesis is organized as follows

Prologue is a brief presentation of a few problems caused by the Ptolemaic paradox

Some questions are asked that initiate the quest for an augmented map concept

Part I CARTOGRAPHY AND GIS gives a short introduction to some basic issues

in cartography b oth traditional and computer aided Cartographic generalization is

introduced as p erhaps the most central asp ect of cartography and is briey discussed

The traditional map mo del is characterized and termed the Paper Map Model The

idea of augmenting the map concept is introduced to facilitate the development of a new

generation of systems designed for management of spatiotemp oral information

In Part II MULTIMODELS we introduce and develop the Multimodel concept as a

exible mechanism designed to structure and to a certain degree solve some fundamental

asp ects of the generalization problems identied in Part I The problems are asso ciated

with the management of the multitude of dierent scales moments or intervals in time

and editions The Multimo del oers an homogeneous way to handle mo del variants if

p ossible in a consistent and compact manner

Part III METAMAP is devoted to the elab oration of Metamap our contribution to

the augmentation of the traditional map concept Metamap is an ob jectoriented high

level framework oering a exible metho d for structuring spatiotemp oral information

The Multimo del principle introduced in Part II is a key notion in Metamap Metamap

is initially presented at a conceptual level as a metamo del A limited version of this

mo del is rened into an ob ject mo del called MINIMAP

Part IV CONCLUSIONS AND IMPLEMENTATIONS summarizes the thesis We

give some conclusions and make suggestions on further research on Multimo dels and

Metamap A generic Multimo del customized for geographic information called MUL

xii Introduction

TIMOD is implemented giving examples of text records and piecewise linear curves

and surfaces as Multimo dels A mo dest implementation of the Metamap mo del called

MINIMAP is carried through to demonstrate key asp ects of an augmented map concept

We close the thesis with the Epilogue by recalling the questions given in Prologue and

suggesting some answers

Geographic information science

The increasing interest and activity around the use and development of geographic information

systems converges according to some researchers for example Rhind Go o dchild and Maguire

RGM page into a new discipline of its own The new disciplin e will in the thesis

b e referred to as Geographic Information Science Recently Canada has suggested that ISO

International Standardisation Organization should include this eld termed geomatics

in their standardization eorts GI science is now b eing studied at several universities and

colleges around the world as an indep endent sub ject and not only as part of courses in

geography geo desy land resources assessment or computer science

One of the characteristics of GI science is the interdisciplinary nature of the sub ject This

is reected in the thesis as the dierent parts are dep ending on dierent disciplines

Part I CARTOGRAPHY AND GIS is dominated by traditional cartography and ap

plications of GI systems but requires no sp ecial knowledge from the reader

However Part II MULTIMODELS which is founded on computer aided geometric

design and mathematical decomp osition theory and to a certain degree ob jectoriented

analysis and design assumes that the reader has some basic understanding of the main

principles in these elds

In Part III METAMAP knowledge from traditional cartography ob jectoriented anal

ysis and design and GIS mo deling are the main building blo cks Readers not trained in

these disciplines will hop efully still gain some insight in the area by reading the text

In Part IV CONCLUSIONS AND IMPLEMENTATIONS examples are given on the

craft of programming a computer in an ob jectoriented fashion However readers not

p ossessing this sp ecial vocational skill will hop efully still enjoy the examples given on

the use of the application esp ecially after reading the previous sections or parts of

them

According to the interdisciplinary nature of GI science the emphasis in the thesis is rather

on the framework the augmented map concept than on the sp ecic problems solved within

Since GIS may b e interpreted b oth as Geographic Information Systems and Geographic Information

Science the terms GI systems and GI science will b e used when the interpretation is not clear from the

context In the thesis GI systems is restricted to the physical computer based programs and systems of

programs used in the management of spatiotemp oral information

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

xiii

GIS activities at SINTEF SI

The work on the thesis has b een carried out during and at SINTEF SI Division of

Information Technology The thesis is founded on the knowledge and traditions represented

by this research environment Department of Industrial Mathematics has a longstanding

international reputation in geometric mo deling in general and spline technology in particular

The Department of Co op erative Systems has b een involved in many international research

programmes concerning ob jectoriented mo deling and integration of system architectures

Both departments are part of the Division of Information Technology and have b een involved

in GI activities since the early s

The two departments have since the middle of taken part in preparations of launching

a ve year national research programme devoted to geographical information technology

During the preparations some of the foundational issues treated in this thesis have b een

prop osed as basis for parts of the technological developments in the programme However the

thesis is to b e considered as an indep endent contribution to the development of more sophis

ticated real world mo dels for the use in GI systems The rst publication of the Multimo del

and Metamap concepts which are the authors terms is found as a high level conceptual

description in a preliminary technical rep ort MS published in June

xiv Introduction

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Chapter

Prologue

In the autumn of dr Erlend Arge and dr Morten Dhlen at SINTEFSI developed an

enhanced version of the well known Douglas Peucker DP algorithm designed for simpli

cation of piecewise linear curves In short an approximation or caricature of the curve is

generated b eing represented with less p oints than the original The new curve deviates from

the original within a given tolerance In this way b oth a smo othing or reduction of noise

and a data reduction are achieved Arge and Dhlen also contributed with a new algorithm

called the Intersecting Cones Algorithm AD The development of the algorithms were

motivated by a pro ject where some of the goals were to develop evaluate and implement

routines for use in an ECDIS

Figure Contour map

In nautical navigation it is typical that it is necessary to view the same area in dierent

ECDIS Electronic Chart Display Information System used for onb oard route planning and navigation

at sea

Prologue

scales ranging from small scale o ceancrossing charts to large scale harb or charts In an

ECDIS there are constraints on how long it should take to refresh a digital display of a chart

Int Much of the information in nautical charts esp ecially of coastal waters is represented

as curves either coastal contours or depth contours

In this context the imp ortance of ecient algorithms for reducing the amount of data

needed to represent these curves b ecomes obvious Even with extremely fast hardware the

ECDIS will not meet the p erformance requirements if the contours are to b e generated from

to o large data sets

I had the pleasure of working with the evaluation of the new algorithms compared to some

of the traditional algorithms ADWM The testing included reduction of height contours

in top ographic maps such as the one in gure

During the testing some side eects o ccurred due to the fact that each curve in the maps

where treated separately indep endent of the other curves in the data sets The anomalies

could b e classied as violation of the topology of the maps In this context top ology is

referring to the geometric relations b etween the curves in the maps A discussion of the

notion of top ology is found in section and

The mishaps were of two main categories and motivated the formulation of the two

questions b elow

Figure Crossing contours meter tolerance corresp onding to ca

Question Crossing contours

Figure shows the contour map in in gure after simplication with the Douglas

Peucker algorithm The tolerance is set to meters which corresponds to the maximal error

al lowed at a scale of approximately Two of the contours are chosen and slightly

enlarged In the circle we see that the contours are intersecting This same phenomenon

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Figure Crossing contours meter tolerance corresp onding to ca

but more exaggerated may be observed in gure This is the same map reduced with a

tolerance of meters corresponding to a scale of around

In many GIS applications height contours are stored as nonintersecting closed polygons

see for example Bur chapter The system wil l interpret the triangle formed by the

intersection as a closed polygon but wil l encounter serious trouble in deciding which height

to assign to the new contour Such inconsistencies may lead to system crash or other serious

malfunction

Is there any way to structure the map information that guarantees that contour topology

wil l be maintained during a simplication process

Question Dislo cation

Figure il lustrates another violation of topology due to line simplication The map

represents the shoreline of an island in addition to a road After data reduction of the map

the geometry of the island has degenerated to a linear feature and the road is reduced to a

line segment located o the island Clearly this is not a desirable result and may lead to

strange eects when the system nds an oshore road

What kind of representation of geographic objects such as roads and shorelines could help

preventing dislocation during simplication

A third problem was addressed in the ECDIS pro ject mentioned earlier The simplication

pro cedures resulted in several variants of logically the same chart each characterized by a

GIS Geographic Information System see chapter

Indeed not only structure is imp ortant when solving problems like this the design and usage of algorithms

is equally imp ortant Still this thesis is based on the assumption that the structuring of a problem is the rst

step to take in problem solving pro cesses

This may b e regarded as a pathological example using extremely large tolerances However similar but

less pronounced anomalies are frequently encountered in simplicati on of cartographic curves

Prologue

ROAD

ISLAND

Figure Dislo cation causing oshore road

sp ecic tolerance corresp onding to a given scale of the map In an ECDIS one of the functions

is the ability to zo om in and out of the current map This is achieved by jumping from one

scale to another according to a set of threshold tolerances and gives rise to the following

problem

Question Multi scale structures

Given a set of cartographic contours performing line simplication according to a set of

given tolerances yields a collection of variants of logically the same map diering only with

respect to scale We may call it a multi scale map

Is there any ecient way to represent this and related multi scale structures

The questions suggest that the representation of the digital maps is to o primitive and

inadequate for this sp ecial purp ose that is to pro duce variants of dierent scales from one

original map by the means of traditional line simplication algorithms

With question and in mind some questions of more fundamental character arise

which are the main sources for the inspiration b ehind the results obtained in the thesis

Question Augmentation of the map concept

Is the traditional map concept ful ly capable to be a foundation for digital systems ded

icated to the management of geographic information

If not in what ways are the traditional maps inadequate

Is it possible to augment the map concept such that it could withstand the impact of

the wave of information technology

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

The questions and span a vast area of knowledge exp erience and research They

have b een asked b efore and answers have b een given P A Burrough states it this way

Bur

Now it is time that GISusers should ask if the data structures that current

commercial GIS oer are really what is needed and if not then please would

someone pick up this interesting and complex challenge to provide something

b etter

It is far b eyond the scop e of the thesis to meet such a challenge in its full extent Still

this is an attempt to take a few rst steps on a path through the interdisciplinary wilderness

of GI science Hop efully the path will lead to an augmented map concept

The rst thing to do is to take a closer lo ok at the traditional map

Prologue

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Part I

CARTOGRAPHY AND GIS

Outline

The overall purp ose of the thesis is to identify and to a certain degree solve problems due to

the Ptolemaic paradox ie that computer based geographic systems widely utilize a literally

medieval and in fact ancient map concept see the Introduction Part I is therefore devoted

to the science of maps cartography and to computer aided systems which rely heavily on the

map mo del

In chapter we take a closer lo ok at traditional cartography A general denition of maps

is given The traditional map concept is analyzed and termed the Paper Map Mo del It is

shown to b e in many ways limited compared to the general denition of the map

We then investigate the notion of cartographic generalization which is claimed to b e a

fundamental asp ect in the management of geographic information Generalization is basically

referring to the pro cess of abstracting and representing real world phenomena in a cartographic

setting

Some details are given on generalization of b oth spatial and nonspatial information and

three main classes of generalization are prop osed scale generalization time generalization and

edition generalization A formal denition of generalization is given based on considerations

of information theoretical nature

Chapter gives a brief survey of the use of computers in handling geographic information

A few denitions of Geographic Information Systems are given and some of the ma jor research

trends in computer aided cartography over the past thirty years are mentioned A distinction

is made on generalization in traditional cartography termed visual generalization and in

CAC called analytic generalization

We then discuss the map mo dels utilized by GI systems b oth for representing top ographic

and thematic information and comment the extensive vectorraster debate We claim that

the ancient Paper Map Mo del is the core in most GI systems To supp ort this assertion

we give some examples from a widely accepted and used GIS standard the Vector Pro duct

Format VPF

We close the part by listing some of the many challenges in GI science and prop ose an

augmentation of the Paper Map Mo del as a step towards a b etter foundation for computer

aided management of geographic information

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Chapter

Cartography

In The Multilingual Dictionary of Technical Terms in Cartography McoCI we nd the

following denition of cartography

Denition Cartography The art science and technology of making maps together

with their study as scientic documents and works of art In this context maps may be

regarded as including al l types of maps plans charts and sections threedimensional models

and globes representing the Earth or any celestial body at any scale

This is a very exible and broad denition and extend the eld of cartography b eyond the

common interpretation of the sub ject In the development of the Metamap Part III the

exibili ty will b e of great advantage

The statement is also an implicit denition of map which in many senses is an invitation

to augment the traditional map concept By adding the asp ect of time we get the following

denition of a map which this thesis can b e said to b e based up on

Denition Map A map is a model of the Earth or any celestial body or part of it

It is any representation in dimensions or any planar projection of a such at any scale

represented over a span of time

In the thesis only topics related to the scientic and technologically asp ects will b e treated

The artistic p ersp ective of map making will also hop efully b enet from the improvements

suggested in the sections to come Before analyzing the traditional map concept we will give

some examples in order to broaden the view of the map

Representing the Earth

The need to make representations of our physical surroundings according to denition one

can exp ect to b e as old as mankind itself

There exist a wide variety of suchs representations but they all share the following char

acteristics

They are supp orted by a physical medium such as pap er and magnetic tap e

Cartography

The representation is realized by the means of some sort of co ding

The users must b e able to deco de the representation

The gures and illustrate the range of dierent types of maps They show that

maps are dep endent of what kind of world view the designers and users represent Cultural

background physical needs religious ideas and the purp ose of the maps are all imp ortant

factors contributing to the world view

Figure Primitive sea chart

The chart in gure was made by seagoing natives of the Marshall Islands some time

in the th century The chart consists of a framework of palm leaf b ers tied together

with leather rop es Tiny shells are scattered over the framework representing islands and

coral reefs The branches have a function b eyond supp orting the shells they indicate ma jor

wavefronts and phenomena signicant to nautical navigation

This representation is far from our common understanding of a sea chart First of all the

medium is quite dierent from pap er Secondly it emphasizes top ology the relations b etween

elements of the map rather than top ography the geometric description

The Roman road map in g the so called Peutinger Table originates from the th

or th century It is a copy of a map made in the rst century AD It is another example in

which top ology is the main ob jective The Roman Empire is squeezed into a by inches

pap er roll totally ignoring the top ographic distortions All roads lead to Rome and this

map is undoubtly well suited for a division of Roman soldiers returning from a mission in the

outskirts of the Empire See Rai Part One for more information on the Peutinger Table

and the Marshall Island sea chart

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Representing the Earth

Figure Roman road map

PART I CARTOGRAPHY AND GIS

Cartography

Figure TOmap

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

The Paper Map Mo del

The only familiar feature of the TOmap in gure in is the circular disc reecting the

spherical nature of Earth The name TO stems from the Tlike shap e inscrib ed in an O The

vertical bar in the T represents the Mediterranian the left part of the top bar is the water

systems originating from the river of Danube and the right part is the Nile The landmasses

are divided into three segments Asia on top Africa to the right and Europ e to the left The

double outer ring is the river Ocean and in the inner ring there is information on the direction

of prevailing winds and observations of celestial b o dies In RSM chapter The History of

Mapmaking p a thorough description is given of the TOmaps

The TOmap is an extreme abstraction of the then known world and is a striking example

of the pro cess called cartographic generalization which roughly sp eaking is how to simplify

and reduce the scale of a representation of a part of or the whole Earth The topic is discussed

in some detail in section

In spite of its infantile and almost ridiculous simplicity the TOmap provided the Mediter

ranian sailor with signicant and in some cases sucient information on how to navigate in

these waters The TOmaps and related representations were commonly used as illustrations

in manuscripts from a few hundred years BC up to the Middle Ages In fact the rst known

printed map was such a map It app ears in a b o ok dated The text is a copy of an

explanation of the world by St Isido or of Seville written in the th century TB

The Paper Map Mo del

The general map concept covers a wide range of representations of the Earth as illustrated

by the examples given in section and stated in denition Still traditional cartography

is mainly concerned with what will b e called the Paper Map Model in the thesis This is

essentially the real world or part of it represented on a planar medium usually pap er

with the help of graphic attributes such as lines dots text color pattern etc

Burrough in his Principles of Geographic Information Systems for Land Resource As

sessment Bur chapter takes on this limiting approach when he denes a map

A map is a set of p oints lines and areas that are dened b oth by their lo cation in

space with reference to a co ordinate system and by their nonspatial attributes

Burroughs view of the map is clearly fo cused on a planar pro jection even if he also adds

A map is usually represented in two dimensions but there is no reason to exclude

higher dimensions except through the diculty of p ortraying them on a at piece

of pap er

To obtain a denition of the Paper Map Mo del it is useful to start with some basic

ideas from traditional cartography It is common to classify maps into two main categories

topographic maps and thematic maps see RSM p AS p Ass p

Topographic maps represent the terrain and a limited number of visible top ographic fea

tures Height contours and curves drawing for example shorelines are the most common

mo deling to ols

The Greek word for map or chart is originally meaning leaf of papyrus

PART I CARTOGRAPHY AND GIS

Cartography

Thematic maps fo cus on one or a few selected themes The top ography is classied

according to the selected themes typically resulting in a partitioning of the top ography

into curves or areas Each theme is co ded according to a legend for instance that

areas with the highest soil fertility is colored dark green Maps of sewage and drainage

systems soil fertility overviews aerial and nautical navigation charts and visualization

of demographic variables in an urban region are examples of thematic maps

There is however no sharp distinction b etween the two classes The classication moti

vates a corresp onding structuring of the information in a map claiming that the information

may b e classied either as top ographic or thematic This dichotomy is characteristic when

handling geographic phenomena The logically same thing or entity may b e describ ed ge

ometrically according to its shap e and app earance or interpreted or classied according to

some predened criteria This geographic duality will b e further stressed during the develop

ment of Metamap in Part III

The top ographic information describ es essentially the shap e or geometry of the world and

thereby mo dels the spherelike surface of the Earth It is common to achieve this in one of

three ways or in a combination

Horizontal cross sections with constant height related to the sea level height contours

Each contour is an op en or closed curve and has to b e asso ciated with the corresp onding

height value in some way

Proles or vertical sections as generated by multibeam echo sounders in hydrographic

surveys

Points asso ciated with height value Single soundings in sea charts are examples of

p oints representing surfaces

The top ography in traditional cartography is thereby mo deled by p oints and op en or

closed curves in the plane Height contours are often not closed due to intersection with the

b orders of the map or missing data commonly encountered in sea charts All the metho ds

mentioned ab ove are kinds of sampling of a continuous surface and the user generates the

surface by mentally interpolation and extrap olation There may b e additional information

such as dierent color co ding of the various height intervals but they are essentially derived

from height contours and p oint samples Such additional information may also b e classied

as thematic information

The thematic information is asso ciated to parts or the whole of the planar pro jection of

the top ographic surface and is lo cated by

p oints

The legend is in traditional cartography an explanation of what the various graphical attributes such as

patterns symbols and color are representing in the thematic domain The term arouse during the Middle

Ages where p erhaps the most imp ortant part of a map was the elab orated and colorful illustrati ons of stories

myths and legends asso ciated with the places on the map

Strictly sp eaking the height contours are generated by sectioning the terrain with spherical osets accord

ing to the geoid the complex geometric description of the earth at constant zero height

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

The Paper Map Mo del

op en or closed curves or

regions planar surfaces represented as the interior of closed curves The regions may

b e complex eg with holes

Burrough Bur emphasizes the common and limited attitude towards geographic by

stating

All geographic data can b e reduced to three basic top ological concepts the p oint

the line and the area

We observe that Ptolemy in his Geography see the Introduction in fact demonstrated a

broader approach by at least including textual descriptions as an imp ortant part of geographic

data Geography which has b een the prototype of the atlas for years contains for

example a description of ab out places in the then known habitable world

The information itself is with few exceptions represented with one of or a combination of

graphic attributes such as text color and pattern according to some given legend or standard

How to choose the appropriate visual variables to represent the information graphically is

an imp ortant issue in traditional cartography Interested readers should consult the rich

literature in this eld such as Ans RSM and Cur

The themes may b e regarded as classications or interpretations of distinct physical parts

of the world An area b ounded by a closed p olygon may b e classied as a national park It is

obvious that the same part of the world may have several dierent thematic interpretations

A certain part of the national park may also b e describ ed as a primeval forest This fact is the

main motivation for the overlay concept It is common practice among mapping authorities to

supply maps separated into a number of overlays or foils The overlays are then combined

in the most convenient way for the dierent users and customers

Usually there is one overlay p erhaps several representing the top ography most often

supplying coastlines and height contours The thematic foils represent waters systems roads

county b orders etc

The map is an attempt to mo del the reality Still it is indeed not a complete mo del but

an abstraction The most obvious abstraction is that the map is a scale reduced version of the

reality In addition it is a selection of the huge number of p ossible top ographic elements and

their still larger number of thematic interpretations At last the selected and scale reduced

features are presented in a customized version dep endent of the purp ose of the map The

same geographic area may lo ok quit dierent in a road map compared to a map designed for

land resources assessment Even if they are based up on the same selection of top ographic and

thematic features the use of colors texture and graphical symbols can make the app earance

of the maps quite dierent

All these three pro cesses scale reduction selection of top ographic and thematic elements

and customization are encompassed by the concept of cartographic generalization In section

some generalization metho ds will b e briey discussed

Based on the discussion ab ove and the fact that pap er maps are abstractions representing

limited parts of reality b ounded by rectangles not in reality but a rectangle in the given

planar pro jection we prop ose the following characterization of the traditional pap er map

mo del

PART I CARTOGRAPHY AND GIS

Cartography

Denition Paper Map Mo del PMM The Paper Map Model is a planar representa

tion of the real world or part of it with the fol lowing characteristics

Decomposition of the reality into cartographic elements which represent topographic

or thematic information

Utilization of points curves and areas combined with graphic attributes to represent

the reality

Each thematic element is located to a distinct part of the topography A part of the

topography may have several thematic interpretations

Representation of the reality at a given scale or resolution and at a given moment

Representation of the reality in a specic edition or generalized version

Tessellation of the reality into rectangles according to the projection used

Coding of the graphic attributes according to a legend either provided by the map or

given as some sort of common understanding or agreement

The Paper Map Mo del will in this thesis some times b e referred to as PMM Note that the

mo del complies with denition of a map limited to a certain moment or interval in time

and b eing a D pro jection of the real world or part of it

In chapter it is claimed that the traditional Paper Map Mo del is the core of most GI

systems to day It will also b e noted that the PMM imp oses severe limitations on the systems

esp ecially with resp ect to the new generation GI systems Still we will show that the mo del

is well suited for additions and enhancements that may lead to an augmented map concept

In the next sections some details will b e given on a central topic in cartography the

generalization pro cess

Generalization

In section a map was dened essentially to b e a representation of the Earth Such a

representation has inevitably to b e pro ceeded by some sort of abstraction It is this abstraction

pro cess that is commonly referred to as cartographic generalization This is a sophisticated

discipli ne relying b oth on theoretical insight vocational skills and sound understanding of

the various uses of maps

A review of ve dierent generalization mo dels is given in McM indicating the multi

tude of approaches developed to describ e the pro cess formally The following sections fo cus

on the underlying structures of the generalization problem and not on the pro cess itself

Some selected issues in cartographic generalization is discussed helping to sort out tractable

problems and p ossible metho ds for structuring geographic information in a way suitable for

generalization

Before giving details on various generalization pro cedures we present some formalism

related to generalization as a to ol in controlling and manipulating of cartographic information

Some maps mo del variation over time for instance a historical map showing the rise and fall of the Roman

Empire as a phenomena represented at several distinct moments in time in the same map

The term generalization in this context must not b e mixed up with the same term used in data mo deling

metho ds and programming languages

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Generalization

Cartographic information

Cartography is essentially concerned with compilation organization storage and distribution

of any types of lo cational information ie information p ossible to asso ciate to a distinct

spatial description of the reality The following decomp osition of cartographic information is

slightly mo died taken from AS

Denote information as I and using the subscripts total explicit implicit topographic

and thematic we have that

I I I

tot exp imp

The explicit information I is further decomp osed as

exp

I I I

exp topo thema

I is essentially derived from top ographic descriptions such as the shap e of a coastal

topo

contour or the area of a lake

I is supplied by the co ding of the graphic symbols used in the map such as color

thema

pattern and text fonts I can b e considered as the part of the information that would

thema

b ecome meaningless without a legend or some prerequisite knowledge of the graphic language

I is a result of a synergy pro cess b etween the various elements of the explicit informa

imp

tion If separate cartographic ob jects all carrying distinct explicit information together by

synthesis generate new information not initially present this kind of information is classied

as implicit information

Needless to say the concept of implicit information is closely related to the skills and

exp erience of the map reader It is not a trivial task to analyze such information eg by the

means of computers

As stated ab ove one of the main purp oses of maps is to transfer information The receiver

is traditionally the human user but now see chapter the use of computers in transferring

information is rapidly growing

In many contexts it is imp ortant to p erform the transfer as ecient as p ossible i e to

transfer maximum information during a minimal span of time

A naive solution to the problem could b e to represent as much information technically

p ossible limited by such factors as the resolution of the display medium The amount of

transferred information will however in most cases not b e prop ortional to the density of

the displayed data Figure illustrates a p ossible scenario of the correlation b etween the

information density of the map and the amount of absorb ed information by the receiver

or user The essence of the illustration is that if we increase the information density the

transferred amount of information that is the fraction of the total information displayed that

the user will absorb will reach a maximum after a Sshap ed development In rare cases the

graph will converge to the maximum where absorb ed data equals the displayed data Usually

the amount of absorb ed information will start to decrease when increasing density b eyond

the critical p oint Too much information confuses the reader and obscures information at

more basic levels Ultimately at the p oint where the display or pap er is completely lled

with elements there is not any transfer at all

PART I CARTOGRAPHY AND GIS

Cartography

All information

Relevant information

All information absorbed ABSORBED INFORMATION

INFORMATION DENSITY Optimal density − all information

Optimal density −

relevant information

Figure Transfer of information

Another asp ect to consider is that not all the information may have the same relevance to

the user Figure illustrates that the amount of relevant absorb ed information may b ehave

dierent from the transfer of all information

In cartography one of the fundamental goals is to optimize the information density so that

a maximum amount of relevant information will b e absorb ed by the user This optimization

pro cedure is hard to formalize since the measurements of b oth information density and

absorb ed information will always b e of a heuristic nature As a rule it will b e dicult if not

imp ossible to nd the optimal solution Still with the pro cess of cartographic generalization

cartographers are constantly trying to solve this optimization problem

Generalizing top ographic information

In the thesis a distinction b etween top ographic and thematic generalization will b e made

Topographic generalization deals with the geometric descriptions of physically recognizable

elements of the map Thematic generalization on the other hand is concerned with how the

thematic information asso ciated with the top ography can b e represented at dierent scales

in various editions and at several moments or intervals in time

We will now give three examples of top ographic generalization

Simplication is a reduction of the complexity of linear features also referred to as

smo othing Figure shows a coastline in a given scale X The two smaller maps

are of scale X The map at left is generated by simply reducing the size of the

original X The coastline is unnecessary detailed The map at right is pro duced

by simplication of the curve representing the coastline yielding a presentation that

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Generalization

1 : X

1 : 4X

Figure Simplication

is b etter than the one to the left Better in this context means aesthetically more

pleasing and more eciently p erformed information transfer As noted earlier this

kind of measurements are of typical heuristic nature

1 : X

1 : 4X

Figure Combination

Combination is the merging of two or more ob jects into a single one

Figure presents three islands and a coastline and two editions of a scaled down

version An alternative could b e to merge the three islands into a single new one as it

is done at the map to the right

Deformation is a arbitrary change of the original geometry of an ob ject Figure shows

PART I CARTOGRAPHY AND GIS

Cartography

1 : 1

1 : 4X

1 : 1

1 : X

1 : 4X

Figure Deformation

yet another variation over the sea chart theme Here to emphasize sailing channels on

b oth sides of the island it is squeezed from two sides yielding a relatively thinner

presentation of the island

In these examples the three generalization operators are asso ciated with scaling down

existing maps Still the op erators esp ecially combination and deformation are also used

when dierent editions are pro duced from the same map The scale is then unchanged but

the maps are manipulated or edited to t the purp ose of the map A navigation chart and

a map designed according to recreational activities may emphasize quite dierent asp ects of

the same area In addition we might think that dierent versions were pro duced in order to

mo del variations over time for example how the contours of the islands changed according

to the tide

Other top ographic generalization op erators do indeed exist such as selection which se

lects a subset of the cartographic ob jects in question and displacement which translates or

and rotates an ob ject See AS Ans RSM and BM for further details on

generalization op erators

Generalizing thematic information

As with the top ographic information thematic information is also sub ject to cartographic

generalization Throughout the thesis the emphasize is on the top ographic information rather

than on thematic issues Still it is necessary for the development of b oth the Multimo del

Part II and the Metamap Part III concepts to briey touch relevant asp ects of thematic

information

Thematic information may b e of near say any kind The only condition is that the in

formation has to b e asso ciated to a top ographic element in some way In traditional maps

there is a limited number of p ossible representations for this kind of information The most

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Generalization

1 : X BIGTOWN 1 : X 1 : X Majorriver

1 : 4X Majorriver BIGTOWN LITTLETOWN

Minorriver

Tinyriver

1 : X BIGTOWN Majorriver SMALLTOWN

LITTLETOWN Minorriver

Tinyriver

1 : 4X SMALLTOWN

Figure Generalization of text

common is plain textual information such as names on cities rivers and lakes Other p ossi

bilities is the use of graphic attributes such as color and pattern that are to seen as co ding

according to a legend or a given standard In the traditional atlas additional information is

given in tables illustrations and textual descriptions

Figure shows an example of generalization of textual information In the generalized

X edition at right names of minor features are simply omitted to make a more readable

map This pro cedure corresp onds to the selection op erator in the previous section

In gure a geographic area is classied in an original map according to levels of soil

fertility Plain scale reduction yields a confusing picture at left obscuring main trends In

the right map the information is aggregated to two levels and this results in a map easier to

comprehend This is essentially the same pro cess that the combination op erator p erforms on

top ographic information

It is not hard to realize that the generalization op erators for thematic information may

dier substantially from those used in top ographic generalization Still they share the com

mon purp ose to optimize transfer of information Thus all later references to generalization

in the thesis include b oth top ographic and thematic information unless something else is

explicitly stated

Cartographic generalization

As stated earlier generalization is needed b oth as the scale of the map is changed when the

map is customized into a sp ecic edition and in mo deling temporal changes Thus we may

introduce the following classication of the various generalization pro cesses Given a map

In fact there is a top ographic asp ect in this kind of thematic generalization since it includes merging of

areas into larger ones

PART I CARTOGRAPHY AND GIS

Cartography

1 : X

1 : 4X

Figure Generalization of pattern

there are essentially three main categories of generalization we may want to p erform and of

course combinations of them

Change the scale This may also b e considered as a change of resolution or accuracy

of the map information

Customize it within the same scale The customized versions is to b e considered as

dierent editions or variants of essentially the same piece of information

Adjust it to represent a certain moment or interval in time

In fact we will se later in chapter that the edition and time asp ects of generalization

involve basically identical op erations even if the motivation and the pro cess as such are quite

dierent

International Cartographic Asso ciation ICA has made the following explanation of gen

eralization

the selection and simplied representation of detail appropriate to scale andor

purp ose of the map

We observe that temp oral changes is not considered part of the generalization pro cess Nev

ertheless we take the lib erty of claiming that temp oral changes should b e encompassed by

the generalization concept

With this description and the observations made during the last sections in mind we

make a more precise statement on the nature of cartographic generalization

Denition Generalization Cartographic generalization is the process of optimizing the

information density of maps according to denition under the constraints provided by

scale or resolution

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Generalization

edition dened by map purpose and skil lexperience of the user and aesthetic guidelines

and

moment or interval in time

Generalization may be performed on both topographic and thematic information

According to the three main aspects of generalization the terms scale generalization edi

tion generalization and time generalization wil l occasionally be used in the thesis when refer

ring specically to one of the three aspects of the generalization process

The denition motivates key asp ects of the development of the Multimo del structure in

chapter This structure aims to supp ort certain stages in the generalization pro cess as it

will oer a compact and consistent representation of a set of generalized maps

Occasionally in this thesis in certain contexts the term generalization will b e used in a limited setting

referring only to the edition asp ect and not scale or time

PART I CARTOGRAPHY AND GIS

Cartography

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Chapter

Geographic Information Systems

Computerized handling of spatial data by the use of Geographic Information Systems has

b ecome an imp ortant decision supp ort to ol in a wide range of areas such as environmental

surveillance route planning land resources assessment and aerial and nautical navigation

As hardware and software technology during the last two decades has grown more mature

the demand for additional functionality higher capacity and advanced user interfaces in GIS

has grown accordingly

During the past few years much attention has b een paid b oth from advanced users and

leading vendors to design and develop the new generation geographic information systems

In many ways this thesis may b e regarded as a contribution to this ongoing eort

In this chapter we give some details on GIS in general and computer aided cartography

in particular

Managing Spatiotemp oral Information

Geographic Information Systems is the common term covering software capable of various

degrees of managing spatiotemp oral information There are many explanations of the concept

and Maguire in Mag page lists some of them

A system for capturing storing checking manipulating analyzing and displaying data

which are spatially referenced to the Earth

Any manual or computer based set of pro cedures used to store and manipulate geo

graphicly referenced data

An information technology which stores analyzes and displays b oth spatial and non

spatial data

A p owerful set of to ols for collecting storing retrieving at will transforming and dis

playing spatial data from the real world

A decision supp ort system involving the integration of spatially referenced data in a

problemsolving environment

A system with advanced geomo deling capabilities

The explanations illustrates the great variety of uses of GI systems and the dierent exp ec

tations to how the systems should p erform Even though none of the denitions explicitly

Geographic Information Systems

mentions the asp ect of time it is obvious that much of the spatial referenced information will

vary over time Thus no mistake will b e made by suggesting a more general denition

Denition GIS A Geographical Information System hand les spatiotemporal informa

tion in a structured manner utilizing a model of the real world or of a part of it

Note that a GI system not necessarily is a digital system a traditional map is clearly a

GIS according to this denition In particular one might characterize the traditional atlas as

probably the most common GI system to day However in this thesis the use of the term

GIS will refer to a computer implemented system unless something else is explicitly stated

GI systems dier from other information systems by the fact that the information to b e

handled is of spatiotemp oral character ie that it is related to b oth space and time

The information system part of a GIS is concerned with actually storing in a database

retrieving and analyzing the information This is a research eld of its own and discussions

on such asp ects eg whether ob jectoriented databases are more suited for implementations

of GI systems than traditional relational databases is b eyond the scop e of the thesis The

fo cus is rather on the structure of the real world mo del than on how to store and retrieve

the data in such a mo del These two asp ects of a GI system are of course not disjoint but

are closely interrelated and are indeed mutually dep endent Still one may view a GIS as a

core consisting of the real world mo del surrounded by an information system The latter

acts as an interface b etween storage sources users applications and the real world mo del as

illustrated in gure

USERS

INFO

REAL WORLD MODEL

SYSTEM

APPLICATIONS

Figure A Geographic Information System

Recalling denition of the map we may very well characterize the core in a GI system

as a map It then b ecomes clear that cartography plays a fundamental role in GI science In

computer based GI systems computer aided cartography is thereby of vital imp ortance The

topic is briey discussed in the next section

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Computer Aided Cartography

One of the characteristics of cartography is the huge amount of data involved The advantages

of computerizing some of the pro cesses were early recognized From the late s the use

of computers in cartography has increased steadily Still the complexity of b oth data and the

use of them until now have limited computerization to certain areas like storage and simple

analysis

Advances in computer technology in the late s like large volume storage devices fast

pro cessors high resolution graphic displays sophisticated printing devices and ob jectoriented

mo deling and programming to ols have now made it p ossible to address unsolved problems

It is convenient to distinguish b etween certain tasks in the area of CAC

Data capture the pro cess of gathering primary cartographic information from sources

such as

geo detic surveys using digital instruments and recording equipment

digital soundings from multi b eam echo sounders

digital analysis of aerial and satellite images and

scanning of existing printed maps

Compiling and ordering of spatial data according to some kind of mo del such as a

top ographic map with height contours

Storage using digital storage technology like magnetic tap e or optical discs

Production of traditional printed maps Advanced drawing programs and standard

CADCAM applications are frequently used to compile and edit maps digitally prior to

printing them by traditional techniques

Analysis such as computation of areas and distances and nding the shortest route in

a network of roads are a typical tasks well suited for computerization

The op erations ab ove are not necessarily exclusively asso ciated with the map domain of

a GIS but may partly b e asso ciated to the information system which facilitate the retrieval

pro cedures and other management op erations

Automated generalization has b een one of the goals in CAC In BM three ep o chs of

research in this eld are identied

Period I

Focus on algorithm development with emphasis on line simplication

Early in the p erio d exp eriments using raster images

Later fo cus on vector representation and top ological data structures

Period II late s

Algorithmic eciency

Investigation of metho ds to deal with the scale dep endent nature of geographic

phenomena

Period III rescent

Formalization of cartographic knowledge

Comprehensive mo dels

Knowledge based systems

The Metamap development in Part III is based on trends emerging in Period III advanced

computer aided design CAGD ob jectoriented mo deling and last but not least traditional

PART I CARTOGRAPHY AND GIS

Geographic Information Systems

cartographic knowledge

Automated Generalization

Generalization is a slow and lab our consuming pro cess and it has b een put much eort in

enabling computer systems to automate cartographic generalization In limited and fairly sim

ple applications mainly concerned with top ographic information see AS for an example

this task has b een accomplished to a certain degree A collection of articles on contemporary

research in the eld is found in BM Still as Freeman simply puts it in the preface in this

collection

This automated map generalization has b een dicult to achieve

In other areas such as computer aided geometric design CAGD there is a common census

on the need for human interaction in complex computerized pro cesses This should undoubtly

also apply to computer aided mapping

In CAC an additional dimension is encountered in the generalization concept In deni

tion section generalization is formulated as an optimization problem In traditional

cartography it is a human user that is the target for the information transfer from the map

This may b e the case in CAC but it may b e as well a computer that is retrieving informa

tion from the map representation in the GI system From this motivation it is natural to

make a distinction b etween visual and analytic generalization Visual generalization b ecomes

equivalent to traditional generalization and analytic generalization may b e dened as follows

slightly dierent from denition

Denition Analytic generalization Analytic generalization is the process of optimiz

ing the information density in digital represented maps according to denition under the

constraints provided by

scale or resolution according system specic variables such as processor speed nu

merical accuracy and algorithmic constraints

edition dened by map purpose and type of application accessing the map and

moment or interval in time

Analytic generalization may be performed on both topographic and thematic information

In CAC the term scale b ecomes sort of meaningless Scale is dened as the ratio b etween the the size of

a geographic ob ject as it is represented in the display medium traditionall y pap er and the the size in reality

In CAC there is a characteristic indep endence of the digital representation and the displayed representation

Thus the scale of the same digital map would vary according to if it was printed by a plotter or edited on a

computer screen

Instead the term resolution may b e used as corresp onding to scale The digital map is always a discretization

of some real phenomena and the resolution is prop ortional to how dense or detailed the real world is sampled

in the map The scale concept may b e generalized to cover this asp ect such as the scale express the level of

accuracy in the map There would then b e p ossible to dene some bidirectional mapping b etween scale and

resolution

For this reason in this thesis the term scale will also b e applied to digital maps and GIS even if it would

b e more correct to use resolution This is done to b e consistent to the basic idea in the thesis to augment

the traditional map concept

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Map Mo dels in GIS

In a GIS generalization will often b e a combination of b oth the visual and the analytic

considerations since accessing the real world mo del in most cases involve b oth computer

pro cessing and visual insp ection of the result

Map Mo dels in GIS

There exist a variety of map mo dels used in GI systems and a multitude of ways to de

scrib e them Frank prop oses to dierentiate b etween three levels of description of real world

phenomena Fra

Concepts Ideas notions and relations b etween them that are used by humans to orga

nize and structure their p erception of reality

Data models A comprehensive set of conceptual to ols to b e used to structure data

Data structures Detailed and low level descriptions of storage structures

This approach is quite common Papers discussing related issues in GIS are often making

these distinctions but usually at an implicit level

In the early days of GIS mo deling the emphasis was rather on the more low level data

structures than conceptual frameworks This implied that notions asso ciated to implemen

tation issues b ecame valid as characterizations of GI systems at a conceptual level Burrogh

states that this has b een a ma jor constraint in many dierent application areas using GIS

esp ecially natural resources study Bur page Recently however there has b een a

signicant shift towards the conceptual asp ects in GI science

As indicated in section and a map handles b oth top ographic and thematic

information Still many pap ers concerning GIS mo deling fo cus on the top ographic or more

frequently called geometric or spatial information The treatment of thematic issues is most

often secondary treated at lower levels typically reduced to discussing how to implement

attributes in databases

In the next sections some asp ects of the conceptual mo dels characterizing the GIS scene

will b e discussed At this p oint there is no need to give details on the implementation level

but selected issues relating to data mo dels are touched briey

Topographic mo dels

There seems to emerge a fundamental main classication of how to describ e the top ography

2 3

of the real world Frank et Mark use the terms Kantian and Descartian to describ e the

two world views FM page

A Kantian also called feature based p oint of view implies an emphasis on the objects

that ll the geographic space In this way space or lo cation b ecomes an attribute of

an ob ject

From a Descartian viewp oint also called lo cation based each p oint in space is describ ed

by which ob jects that are encountered The ob jects are in this manner prop erties of

Immanuel Kant German philosopher known as the father of Criticism

Rene Descarte French philosopher and mathematician Emphasized the duality in the

existence distinguis hi ng b etween spiritual and physical asp ects of the world Father of the mechanistic world

view

PART I CARTOGRAPHY AND GIS

Geographic Information Systems

the lo cation

Frank et Mark p oint out that b oth views are used interchangeably in real life dep ending on

what is more suitable for the task at hand

Burrough Bur is probably having the KantianDescartian distinction in mind when

describing the top layer in a six level schematic overview of stages on the way from the real

world to a graphical implementation mo del He claims that reality may consist of fully dened

and fully denable ob jectsentities or incompletely dened or incompletely denable spatial

entities

Go o dchild Go o denes the fundamental element of geographic information as the

tuple T fx y h t z z g giving the values of n spatial variables where z at the

1 n i

geographic lo cation x y at height h at the given moment t This denes a eld over the

entire spatiotemp oral D space and he terms it the eld model

He juxtap ositions this mo del to what he call the object model Here a set of discrete

ob jects is represented by a set of tuples fi a a g where i is an ob ject and a through

1 m 1

a are m attributes of the ob ject Lo cation is describ ed by a set of tuples fx y o o g

m 1 i

where o is a binary variable indication the presense or absence of ob ject i at lo cation x y

Go o dchild notes that this ob jecteld dichotomy is a long standing issue in cartography

The dichotomy has also motivated a classication of GIS that is p erhaps the most widely

used to day into vector based and raster based GI systems

Vector representation is the term for mo deling top ographic ob jects by the means of p oints

piecewise linear curves and p olygons all structures which may b e expressed as vectors of

geographic co ordinates This corresp onds to the ob ject view of the world

Raster representation originates conceptually from a eld view of the world A raster

represents a limited version of a eld in the sense that in a raster each cell is commonly

asso ciated with only one value The raster is in its nature a regular tessellation of the

top ographic surface most common rectangular or quadratic but also triangular tessellations

so called trixels are used together with more exotic versions such as hexagonal tessellation

See HB for a discussion of mo dels raster enco ding and Mag for an example of a GI

system based on raster representation

As data mo dels the raster and the vector concepts represent truly dierent approaches

to the world following the Descartian resp ectively Kantian approach However as data

structures one might argue as Burrough in Bur page that to a certain degree the

two representations are equivalent Raster representations may b e transformed into a vector

representation and vice versa allowing for a loss of information esp ecially in conversions from

vector to raster

Following this approach we claim that the ob ject view in geographic mo deling encom

passes the eld mo del and supp ort the assertion with the following argument

Assume we have a tessellated mo del or in other words a piecewise constant eld mo del

eg a raster representation This mo del may b e transformed to an ob ject mo del without loss

of information in one of two ways

A more prosaic reason to employ the raster structure is that one of the main input sources in GIS is

digitall y scanned pap er maps Scanning yields directly a raster image from where contour lines an other

ob jects may b e extracted and classied

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Map Mo dels in GIS

We may dene the ob ject spatially as the b oundary dened by the tessellated area and

interior trivially without loss of information Further let the tessellation b ecome the

thematic description of this ob ject thus we have encapsulated the information carried

in the tessellation The spatial description in addition to this thematic information will

together represent the information in the original eld mo del

One and each of the tessellated cells may dene spatially a tiny ob ject and the value

of that cell may trivially b e assigned to the ob ject as thematic information The set of

all cellob jects together with their thematic values will together carry exactly the same

information as the eld

Thematic mo dels

Thematic information is secondary treated in contemporary works on GI science The em

phasis is clearly towards spatial issues This is reected in many GI systems that provide

thematic information as attributes to spatial features Such attributes are usually of quite

primitive character eg alphanumeric co des and xedlength text strings The opp osite

approach is rarely seen in GIS that lo cation is an attribute in the thematic ob jects

The two characterization of a GI system could b e termed spatial orientation resp ectively

thematic orientation An example of such a thematic oriented information system could b e

a prop erty management system where the main purp ose was to handle information such as

technical data on buildings the size of the estates and tenants and their rent paying status

In such a system the spatial information could b e reduced to a single attribute in fact it

could very well only b e an implicit reference such as the street address

The degree of spatial resp ectively thematic orientation makes a continuous range of in

formation systems from clean cut spatial systems for example top ographic maps through

main stream GI systems to pure general information systems

The Paper Map Mo del PMM is also clearly spatial oriented Thematic information is

typically represented by text and numbers color and pattern co des and sp ecial graphical

symbols constrained by eg the size of the map and printing techniques

The concept of information integration is often used in discussion of how to integrate to

p ographic and thematic information Shepherd reviews dierent ways of connecting thematic

and top ographic information b oth traditional and alternative She Issues concerning

information integration will b e further discussed under the development of Metamap Part

III

GIS standards

Most standards relating to GI are aimed at low levels of GIS mo deling Some are limited to

data structure denitions others includes issues concerning the data mo del level Very few

if any at all discuss conceptual relations

VPF Vector Product Format is an example of a widely used GIS standard VPF The

standard is developed by an ad ho c organization in NATO Digital Geographic Information

Working Group DGIWG and is also known as DIGESTC Digital Geographic Exchange

Standard Annex C The standard is freely distributed and has b een accepted in many

PART I CARTOGRAPHY AND GIS

Geographic Information Systems

application areas not only within the military communities and also outside NATO The

well known DCW Digital Chart of the World distributed as public domain data included

necessary software is an example of a pro duct based on VPF

VPF is a low level standard in the sense that it is p ossible to implement the standard

more or less directly in an arbitrary relational data base This is p erhaps one of the reasons

b ehind its p opularity

A brief review of VPF reveals the connection b etween the standard and the Paper Map

Mo del

VPF is a representation of a planar pro jection of the reality as indicated by the use of

pro jection co des VPF App endix G table page

On page VPF the statement Realworld ob jects are referred to as entities or

features shows that VPF follows the ob ject approach to GI mo deling just as the

Paper Map Mo del

The geometric primitives of VPF are no des edges and faces VPF page and this

corresp onds to the p oints curves and areas of the PMM

In Kot page it is fo cused on the levels of reality that are the foundation of the

VPF view thematic coverages and primitive geometry comp onents within the themes

This corresp onds nicely with PMM where each thematic element is lo cated to the

top ography and where a single top ographic element can have several thematic inter

pretations

Regarding scale or resolution VPF do es not asso ciate any such information at all to a

certain map other than optionally and implicit in data quality tables VPF page

In cases where data quality information is not added VPF is inferior to PMM VPF

do not handle phenomena varying over time except from mo deling them explicitly as

dierent data sets

Dierent editions due to generalization has also to b e represented in separate data sets

as in the PMM

As PMM VPF is based on tiling the geographic space into a set of smaller rectangular

areas VPF page Some mechanisms are provided to make the connections

b etween consecutive tiles as seemless as p ossible

As a conclusion VPF complies closely to the Paper Map Mo del dened in section except

for minor insignicant deviations

In section the attention was drawn towards the concept of generalization one of the

most basic characterizations of cartography In a nutshell generalization is the pro cess of

generating dierent variants of logically the same geographic reality diering in scale edition

and time

Still VPF and most of the other commonly accepted standards do not consider this

imp ortant issue explicitly It oers only one solution to the problem to pro duce the dierent

generalizations and to describ e and store them separately even if they only dier in some

minor details

One exception might b e the SDM Schlumberger Data Mo del used by the p etroleum industry in activities

asso ciated with EP exploration and pro duction This standard handles dierent versions of certain sets

of data like deviating interpretations of the same collection of seismic data The mechanisms are simple and

rudimentary but still they try to encounter the problem explicitl y see sch chapter page

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Map Mo dels in GIS

The characterization of one single standard will of course not apply to all other existing

standards but a thorough analysis of the main standards would probably yield the same

result that they more or less describ e dierent implementations of the Paper Map Mo del

Kottmans compact survey on GI standards supp ort this assumption Kot

As a conclusion one might say that GIS to day is dominated by the Paper Map Mo del

In the next chapter it is claimed that this imp oses severe limitations on GI systems and

that such systems should b enet from adopting a map mo del closer related to the real world

instead of relying on an ecient implementation of the traditional map concept

In the next chapter we will make the rst general description of an augmented map

concept

There are trends within the standardization b o dies to develop the basis of GI systems into more compre

hensive mo dels An example of this is the work done by CEN Comite Europeen de Normalisation TC

Working Group The goal is to supply the Europ ean countries with a high level GI standard by the

end of the century Their revised philosophy pap er of June constitutes a promising foundation for a

standard closer to reality Com

PART I CARTOGRAPHY AND GIS

Geographic Information Systems

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Chapter

An Augmented Map Concept

In this chapter we consider some of the challenges the core map mo del in a GIS is facing

We then make a sketch of an augmentation of the map concept

Challenges

GIS is emerging as an imp ortant to ol in a wide variety of application areas Articles on the

use of GIS in the following areas are found in MGR

So cioeconomic applications

Land information systems

Car navigation systems

Market analysis

Population counting

Environmental applications

Soil information

Integration of geoscientic data

Multinational environmental GIS

Global GIS databases

Management applications

Land resources information systems

GIS in urban planning

Integrated planning information systems

These and other applications represent indeed new challenges for the core of GI systems the

map mo del Traditional cartography has b een used for similar tasks b efore but not at the

scale and complexity as the ab ove application areas represent

To meet the new challenges a GIS has to oer a wide sp ectrum of functionality Rap er

and Maguire make the following classication of GIS functionality RM

Data capture

An Augmented Map Concept

Transfer

Validation and editing

Structuring

Restructuring

Generalization

Transformation

Query

Integration

Analysis

Presentation

Some of these functions are already found in traditional cartography but together they rep

resent an integrated basis of functionality that would b e dicult if not imp ossible to fully

realize within the Paper Map Mo del

Due to the new challenges Rhind Go o dchild and Maguire RGM foresee that the

recent research will supply

data mo dels to handle D and time dep endence and complex interactions b etween

ob jects

supp ort for complex analytical applications including tracking of data lineage to ols for

visual interactions with the stages in the analysis pro cess propagation of uncertainty

supp ort for quality assurance and quality controlQAQC esp ecially in GIS applications

where litigation is a constant problem

supp ort for multiple media unstructured images b oth digital and NTSC text and

sound

integration of GIS with the capabilities of GPS for data collection and compilers

to ols for visualizing D and timedep endent data

to ols for data compilation particularly in D

improved techniques for conducting functional requirements studies evaluating costs

and b enets b enchmarking and other asp ects of the GIS acquisition and pro ject man

agement pro cess

On the background of section one might add

supp ort for generalization by the means of structures for consistent and compact man

agement of dierent scales and editions

Indeed such results imply the utilization of a reality representation far more sophisticated

than that provided by the Paper Map Mo del

The Ptolemiac paradox presented in the Introduction together with the recognition of

generalization as on of the fundamental mechanisms in cartography and the challenges out

lined ab ove constitutes the main motivation for the introduction of an augmented map con

cept

Augmenting the Paper Map Mo del

Considering the discussion in the last sections it b ecomes clear that GI systems need some

thing more sophisticated than the Paper Map Mo del as a core map mo del Today the

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Augmenting the Paper Map Mo del

ma jority of GI systems are based on the PMM and this map mo del is stretched to its lim

its and often b eyond in many applications As briey touched in the Prologue this may

cause inconsistencies and anomalies that leads to errors malfunctions and other unpredictly

misb ehavior of the system

In this thesis the problem with the inadequate core mo del is encountered by augmenting

the traditional map concept the PMM rather than starting totally from scratch This

approach takes advantage of the rich literature exp erience and technology asso ciated with

traditional cartography and merges it together with advanced data mo deling techniques and

state of the art technology in CAGD into an enhanced map mo del One of the goals will

b e to maintain where it is p ossible the terminology and concepts used by the cartographic

community However the concepts and terminology will b e enhanced and supplied with the

additions needed to meet the challenges and requirements provided by GI science

The various addition and enhancements are merged with the PMM as dened in section

into the Augmented Map Model AMM which is dened at conceptual level b elow

Denition Augmented Map Mo del AMM An Augmented Map Model is a realistic

representation of the real world or part of it with the fol lowing characteristics

Decomposition of the reality into cartographic objects which represent both topographic

and thematic information

Utilization of advanced and exible structures both spatial and thematic which in

combination represent the reality

Each thematic element is located to a distinct part of the topography A part of the

topography may have several thematic interpretations

Representation of the reality integrating a range of dierent scales or resolutions at

several moments or intervals in time and in a multitude of editions or generalized

versions

Seamless representation independent of tessel lations

Facilitates the usage of a wide range of presentations independent of the internal

representation of the reality model

In Part III we will design the augmented map concept Metamap founded on denition

An implementation of a limited Metamap case will b e carried through in app endix B

PART I CARTOGRAPHY AND GIS

An Augmented Map Concept

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Summary

We op ened this part by a general and indeed broad denition of the map By dening and

studying the Paper Map Mo del we found that this was a narrow and limited map concept

mainly due to technological constraints

Contemporary information technology has removed some of the barriers encountered in

cartography since the Middle Ages A geographic information system may b e considered as

a computer based atlas adding some new functionality and capacity Still the Paper Map

Mo del was shown to dominate the GI scene

The GI research during the past thirty years has b een dominated by problems concerning

databases and how to store and index large amounts of information ecient implementation

of low level data structures advanced visualization techniques and a neverending discussion

of the low level data structures raster vs vector b oth representations of primitive geometric

ob jects We nd that this orientation somewhat diverts the attention from more fundamental

issues concerning the core map mo del for example how to handle the various asp ects of

cartographic generalization in a computer based environment

The exp ectations to the p erformance of future GI systems regarding capacity application

areas and functionality implies that such systems have to manage huge amounts of inhomo

geneous information b oth top ographic and thematic in a compact and consistent manner

We claim that the Paper Map Mo del is not capable of meeting these challenges it was

designed for use in a completely dierent technological setting Still we see the advantages

of making additions and enhancements to the traditional map concept rather than discard a

rich and sophisticated tradition evolved during a couple of millenniums

This motivates the notion of augmenting the traditional map concept in order to take

full advantage of the information technology and thereby meet the somewhat overwhelming

new challenges in geographic information management within the framework of traditional

cartography

The rest of the thesis is devoted to investigate a p ossible augmented map concept accord

ing to denition The investigation is divided in two parts

In Part II we study the consequences of cartographic generalization We develop a metho d

for homogeneous management of sets of variants which we call the Multimo del concept We

pay sp ecial attention to structural issues asso ciated to consistent and compact representation

of variants

The problem of integrating spatial and nonspatial information is treated in Part III

where we introduce the Metamap as an augmented map concept Metamap is designed to

fully exploit the exibility provided by the Multimo del concept

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Part II

MULTIMODELS

Outline

As a rst step towards an augmented map concept according to denition this part in

vestigates metho ds and concepts for the integration of multiple representations of geographic

information

In order to increase the sparse supply of techniques dedicated to managing a set of vari

ants of an initial mo del we will introduce and develop in some detail the concept of the

Multimodel as a general mechanism for structuring a set of multiple representations of an

initial mo del

To motivate the elab oration of the Multimo del we will give some examples of mo del

variants encountered in a GIS setting The variations in the mo dels are due to the inherently

multiple nature of geographic information and to the pro cesses of cartographic generalization

as describ ed in Part I

Before presenting our own approach to the problem of multiple mo deling we briey outline

a few existing metho ds developed in this research area It will b e fo cused on what categories

of geographic information the dierent approaches encompass an what kind of variations

they cover The notions of compactness and consistency are introduced in order to obtain an

additional classication of metho ds in multiple mo deling

The Multimo del development is initiated by formulating the concept at a general level

Then a denition of the digital mo del is made in order to facilitate the exploration of the

Multimo del in a computer aided setting Certain prop erties dierentiating various digital

mo dels are highlighted On this basis some fundamentally dierent categories of Multimo dels

are outlined We concentrate on describing basic op erations on the Multimo del and discuss

what degree of compactness and consistency the dierent Multimo dels oer

The Multimo del concept is then presented as an ob ject mo del of a generic class library

which will b e the basis for a limited Multimo del implementation in app endix A

We close the part with the description of an informal metho dology for Multimo deling

The metho d is applied to piecewise linear curves and surfaces b oth examples of geometric

ob jects highly relevant to geographic information systems

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Chapter

The Multiple Nature of

Geographic Information

Geographic information may b e characterized by its multiple nature In section it was

illustrated that maps over the same part of the reality may dier substantially The map is

dep ending on a wide range of factors such as map purp ose cultural background of the map

compiler and available technology

As indicated by the Ptolemiac Paradox stated in the Introduction map makers have until

recently b een restricted to various metho ds of planar representations of the geographic reality

Denition of the Paper Map Mo del characterizes this traditional map concept which has

b een developed into highly sophisticated to ols and pro ducts which indeed have served their

purp oses well

However the Paper Map Mo del is a single representation in the sense that it is mo deling

the given geographic information in a single scale as a single generalized edition and at a single

moment or interval in time Thus to achieve a more complete picture of the reality one has

to relate to a set of dierent maps These maps will together yield a multiple representation

The Ptolemiac Paradox implies that in spite of the information technology revolution

exp erienced in the last part of our century it may b e claimed that no signicant improve

ments have b een introduced in computer aided cartography to handle the multiple nature

of geographic information in a more structured and integrated manner This is not entirely

true some achievements in this direction have indeed b een made and a few of these will b e

outlined in section

Examples of Multiple Geographic Information

To motivate the further treatment of mo deling multiple geographic entities three examples

on geographic ob jects will b e investigated in order to illustrate variations over time according

to scale and due to edition generalization

However some maps do indeed mo del time eg a certain historic map describing the rise and fall of the

Roman Empire

The Multiple Nature of Geographic Information

Text

Assume we have a setting where a GIS is used to illuminate certain asp ects of the historical

development in Europ e One of the natural entities in such a context would b e a nation

In addition to spatial descriptions of for example b orders and coastlines we may want to

supply textual descriptions of the nations Let us assume that the texts are essays written

by dierent historians The texts would certainly not have any inherent common structure

b eyond that they are collections of groups of characters

Since our purp ose is to mo del the development of the nations of Europ e through a certain

time span it is natural to op erate with a number of variants representing signicant moments

or p erio ds Each of the variants may dier more or less in the spatial description but the

dierences in the textual description would probably b e substantial b oth by length and

contents

This set of dierent pieces of text constitutes together a multiple thematic description of

the nation

...... 1814 − 1905 1905 − 1930 1930 − 1940 1940 − 1945 ......

jgdfigjlg hjglsjgjsk gagjpagkakgkapgargpawtj ggdasgkadsgsdakgldasg gkjgpdsgkdgkdghsre daslgkdagdagdas[sdg gopgodasgiodasgddpgpoa gkdfghdkskhsrpghsre sdjgisdagdasgdsgdsg dfshds[hdfhdfshd jsgjsogpos kogkdgkdsgkkd;lsg; dgohdsgkkhkdf[hdf dgfjgjdagkakgasgj dfghkodhds[ jglkgflsgfldslgdsgjls jogjasdgpasgdspgdsgo gjdsgpadsogodsgpsgpo gjoigjsjgsjgjdasigh gdasgsjgiasigijg rekodkhdsh[d gjdfogjdagdagspgpsgposg gjopigpdsgdspgsdo ghdsgoisigdsgdsoigdagh gjsigjoigisajgsd dhdshskhdfh gdasgodsjgpdasgdasgp dsjiofgadsogdasjgads ejoigasgpodsgdagogpoas \gdsifgjodasgsdg sgosdpgdsjgds jgjgjdsagksa \gdsjdsjgadsjgpoasdgdspjodsi jdsogjdsjgadsgjsadjp sogpsgpsogpas jigjdsjgdasgjidas dsfghsoghdsghdhasgds]hgdahgd' godsigdsgshgds nsdoifghsgoidsghdsoh

ihdsfghsgoashgasogadsohgs

Figure Textual description as thematic timevarying information

There are no obvious smart way to structure this inhomogeneous set of mo dels other

than arranging them in an array where the indexes corresp ond to the time span they represent

as illustrated in gure

Parametric curves

Supp ose we were mo deling a railway network with the help of a GIS The railway in our

case is extremely simple consisting of only one branch from Beginville to Stoptown In

b etween the railroad passes just outside Halfway Village A map is pro duced as shown in

gure

However in this particular scale the map gives the impression that the railroad passes

through Halfway Village and not just outside To highlight the distinction a generalized

variant is pro duced by altering a little part of the parametric curve representing the railroad

A parametric curve in the plane is dened such that for every value of a parameter t in a given denition

interval a b there exists a p oint xt y t in the plane where x and y are functions x y a b IR

In contrast to a explicit curve dened by a real function a parametric curve may describ e lo ops circles

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Examples of Multiple Geographic Information

Figure Initial map

as shown in gure

Figure Generalized edition

If we wanted b oth editions of the map we might store each map separately as commonly

practiced in most GI systems to day This is however not ecient since the maps are

identically except for the minor changes introduced by the generalization Assume that the

curves represent the variants of the railroad as vectors of equal length where the elements are

p oints in the plane Indeed it is straightforward to subtract the original from the generalized

vector thus obtaining a dierence or delta vector The interesting part of the original vector

is as follows

Original x

vector y

The corresp onding generalized vector is as follows

Generalized x

vector y

selfintersections and other complex geometries

PART I I MULTIMODELS

The Multiple Nature of Geographic Information

The delta vector is characterized by the large amount of zero elements

Delta x

vector y

It is straightforward to realize that the generalized vector may b e generated by adding the

delta vector to the original vector

Two advantages is achieved by this representation

Compactness If we nd a way to store a sequence of zerop oints that takes less space

than a corresp onding sequence of arbitrary p oints we obtain representation that is more

compact than the explicit representation See for example Nel for an overview of

such data compression techniques

We observe that the nonzero numbers in the delta curve ab ove are smaller than the

corresp onding explicit values In fact it would b e p ossible to represent the delta values

with fewer bits than the explicit representation The integer would require bits

including the sign in contrast to bits needed to represent Similar eects may

o ccur in delta representation of other ob jects than curves and may b e taken advantage

of in order to store the dierences in a compact manner

Consistency If there is need for an up dating of all editions of a map let say the railroad

station in Beginville was moved to another part of the town we may say that the mo dels

are dep endent In our case the changes need only to b e applied to the initial mo del

Due to the construction of the other editions by summation of the initial mo del and

the corresp onding dierences the changes will automatically propagate to the various

editions Such representations will b e referred to as consistent if they automatically

maintain the dep endencies b etween the various mo dels This is illustrated in gure

The delta mo dels are visualized as dierences added to the originals where the nonzero

sequences in the delta vectors are highlighted

Section will supply more details on compact and consistent representation

Functions

In many applications it would b e interesting to have information on the tidal variations of

the sea As known these variations vary across the glob e and are dep endent of many factors

ranging from the constellation of the mo on and the sun to meteorological conditions For

simplicity let us assume that the tide is a diurnal variation ie that it varies p erio dically

within a cycle of exactly hour with only one ebb and one high water o curring during

the p erio d If the hour of the day is mapp ed to a real number in the range the tidal

variations may b e describ ed by the function T IR giving the sea level in meters

relative to some reference level

There are several ways to mo del this variation in a GIS The tide could b e embedded

in the spatial description of the sea as dierent time variants see section for a brief

discussion on temp oral mo deling Another p ossibility which we will choose is to let the

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Examples of Multiple Geographic Information

INITIAL MODEL GENERALIZATION GENERALIZED EDITION AS DIFFERENCE

UPDATE AS GENERALIZATION GENERALIZED EDITION DIFFERENCE AS DIFFERENCE WITH UPDATE OF

INITIAL MODEL

Figure Propagation of change in initial mo del

tidal function T b e part of the thematic description of the o cean

Assume in addition that our application should b e used to pro duce digital sea charts in

three dierent scales for the use in an ECDIS one small scale version for deep sea navigation

let say one scale for coastal navigation and a large scale harb or chart

in Let us further assume that the system by using the tidal function T is able to

pro duce versions of the three scales according to time of the day To accomplish this task in

an ecient manner we may assume that the ECDIS system needs to access the tidal function

in three degrees of accuracy corresp onding to the dierent scales

Figure shows three variants of a tidal function T describ ed as piecewise linear functions

dened by a vector of nine samples of T t t f g The vector v is

the coarsest description corresp onding to the smallest scale describing a constant sea level of

meters ab ove a given zero sea level

The vector v mo dels T in medium resolution showing that the tidal variation yields a

high water at t corresp onding to a level of meters and an ebb of meters

o ccurring at t

The most detailed variant of T corresp onding to the largest scale is given by v This

vector mo dels a sinelike function giving an even more detailed picture of the tide

The obvious way to represent the set of the three variants of T is to store three arrays of

function values explicitly In this manner no relations b etween the vectors are obtained

This is an example of the consequences of the duality of geographic information see section In many

cases a certain asp ect of a geographic phenomenon may b e considered b oth from the thematic and spatial

p oint of view resulting in equally go o d descriptions

ECDIS Electronic Chart Display Information System

PART I I MULTIMODELS

The Multiple Nature of Geographic Information

4 'f0' 'f1' 'f2' 3

Sea level in meters -1

-2

-3

-4 0 5 10 15 20 25

Time in hours

Figure Description of tidal variation in three resolution

Delta representation

An alternative approach to the explicit representation illustrated in gure is to decompose

the vectorrepresented functions Since the vectors are all of the same length it is p ossible

to dene the deltavectors or dierencevectors v v and v v as shown in

1 1 0 2 2 1

gure

If the deltavectors and are stored together with the initial vector v we have an

1 2 0

alternative implicit representation of the three vectors as

v v

1 0 1

v v

2 0 1 2

The construction of v as v is illustrated in gure

2 0 1 2

Consistency

Apparently at rst glance the delta representation seems a little unmotivated and useless

However supp ose that reference zero level was changed with a constant value thus af

fecting all the representations of T If we had an explicit representation b oth v v and v

0 1 2

had to b e up dated to yield the new vectors v v v v and v v

0 0 0 1 1 0 2 2 0

In contrast with an implicit representation only v has to b e up dated b ecause b oth

v and v is dened relative to the initial vector In other words v and v are dep endent

1 2 1 2

of v The following shows that the structure is consistent since the change v and v

0 0 1 2

automatically propagates to

v v v v

0 1 0 0 1 1 0 1

v v v v

1 1 2 0 0 1 2 2 0 2

This is not unrealistic the various hydrographic oces do in fact use a variety of such zero levels

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Examples of Multiple Geographic Information

4 'g1=f1-f0' 'g2=f2-f1' 3

Sea level in meters -1

-2

-3

-4 0 5 10 15 20 25

Time in hours

Figure Delta vectors

4 'f0' 'g1=f1-f0' 'g2=f2-f1' 3 'f2=f0+g1+g2'

Sea level in meters -1

-2

-3

-4 0 5 10 15 20 25

Time in hours

Figure Adding initial vector and dierences

PART I I MULTIMODELS

The Multiple Nature of Geographic Information

Further assume that the decomp osition was motivated from the fact that the tidal func

tion T varied slightly across the glob e and that this variation was due only to variations

in the comp onent such that we for example had two variants of this comp onent and

1 1

In an explicit representation two main variants of the tidal function T each represented

in three scales had to b e stored In a delta representation however only the had to b e

stored in addition to v and

0 1 2

4 'f0^' 'g1^' 'g2' 3 'f2^=f0^+g1^+g2'

Sea level in meters -1

-2

-3

-4 0 5 10 15 20 25

Time in hours

Figure Change in initial vector and the comp onent

Figure illustrates how changes of b oth the initial vector v and of the comp onent

0 1

aect the most detailed vector v which is obtained by the addition v This

2 0 0 1 2

example shows the consistency in the deltarepresentation ie that changes and up dates

automatically propagate to dep ending mo dels

Compactness

There is another prop erty asso ciated with the deltarepresentation concerning the amount of

storage needed for the representations

An explicit storage scheme uses in our example a storage equivalent to function

values Without any mo dication this is exactly the same that is needed with the delta

representation However due to the correlation of the dierent curves we observe in gure

a fairly large amount of zeros in the deltacurves and In the dierence every

1 2 1

second value is a zero and displays zero values In fact the total number of nonzeros

in the complete deltarepresentation of the three curves is in contrast to the values

required by the explicit representation

Indeed the new initial vector v and the new comp onent may also b e expressed in delta representations

but in order to avoid confusion we do not prop ose more complicated storage schemes at this p oint of the thesis

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Examples of Multiple Geographic Information

If we were able to store the zeros and sequences of zeros using less space than required for

the storage of corresp onding arbitrary values we would achieve a more compact representa

tion As mentioned earlier in the section such metho ds exist but it is b eyond the scop e of

the thesis to investigate them more closely

Data reduction asp ects

There are still more redundant information in the deltarepresentation The vector v is essen

tially representing a straight line segment needing just two values to b e uniquely determined

leaving seven values redundant Accordingly needs only four values and eight Figure

1 2

illustrates these three data reduced representations of the initial vector and the dierences

The p oints used in the reduced variants are marked

4 'f0,reduced' 'g1,reduced' 'g2,reduced' 3

Sea level in meters -1

-2

-3

-4 0 5 10 15 20 25

Time in hours

Figure

The data reduction may seem a little out of place since it is no longer p ossible to subtract

or add the new vectors as they are of dierent length Thus we must introduce some extra

machinery to overcome this minor obstacle

Our goal is to bring the reduced variants over to the space spanned by all vectors of length

representing function values for all t f g In other words we want

to rene the coarse mo dels into more detailed ones such that the construction by summation

may b e applied We want to design a renement operator R such that

v Rv R

1 0 1

v Rv R R

2 0 1 2

The op erator has to lift v and into the space to which the original vectors b elong

0 1 2

In this case the op erator may b e trivially dened by inserting missing values computed

by linear interpolation see section for more information on such renement This

decomposition approach included the notation is adapted from mathematical decomp osition

PART I I MULTIMODELS

The Multiple Nature of Geographic Information

theory describ ed in eg Dhlen and Lyche DL and applications of such techniques in

computer aided cartography as outlined by Arge and Dhlen AD In section the

decomp osition concept is examined more closely

This tidal function and its dierent representations provides an example of thematic in

formation varying according to scale In addition the information was decomp osed into a

representation consisting of the initial representation and a set of deltamo dels The comp o

nents were reduced such that they were represented using fewer function values than initially

and a metho d for p erforming arithmetic op erations on these reduced mo dels was introduced

In the next section more general approaches to mo deling multiple geographic information

will b e discussed

Multiple Mo deling

In the article Generalization of Spatial Databases Muller Mul discusses scaleless and

scaledependent databases as dierent ways of structuring a set of variants with dierent

scales representing the same geographic area

The scaleless or scale independent database is a single representation of the most detailed

variant of the map From this representation it should ideally b e p ossible to generate

arbitrary views at any desired scale or variant according to a given resolution in real time

The real time scaling functionality is often referred to as zo oming The complexity asso ciated

to the pro cess of geographic generalization as discussed in section is most likely the

main motivation for the following statement from Muller

This futuristic notion of scaleindep ende nt spatial databases has yet to b e realized

Questions concerning such real time generalization will not b e addressed in the thesis

A more realistic approach according to Muller is the pseudoscaleless database This is a

pyramidal structure of dierent levels corresp onding to certain scales He stresses that each

level should b e accessible without duplication of data In many ways this can b e viewed as a

discretization of an ideally continuous range of scales see gure

The third category of multiple representation is the scaledependent database where the

dierent scales are stored explicitly resulting in an overhead of redundant information

Muller highlights the following advantages of scaleless databases

Avoids duplication in storage

Enables pro duction of exible scaledep endent outputs any scale is available say

Ensures consistency and integrity b etween the various scale outputs

The principle of avoidance of duplication will in this thesis b e referred to as compactness

and is considered as one of the main goals in mo deling multiple structures

Another ma jor ob jective in our treatment of representation of multiple information is the

the notion of consistency Basically in a consistent multiple representation an up date in any

part of the mo del will automatically propagate to those parts of the mo del that is dep ending

Still some b elieve that this concept is not to o futuristic Aasgaard Aas as an example has studied

some selected real time generalization pro cedures of spatial ob jects

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Multiple Mo deling

1 : 1 1 : 25.000 1 : 100.000 1 : 500.000 1 : MAX

Figure Pseudoscaleless structure

on the up dated part The scaledep endent structure where each level is represented explicitly

and indep endently is certainly not to b e considered as consistent

Muller restricts his characterizations to concern spatial ob jects varying according to scale

However in this thesis Mullers classications will b e generalized to apply to nonspatial infor

mation not only multiply represented regarding to scale but also varying due to dierences

in time and editions In other words we are interested in representations of any kind of ge

ographic information that may give rise to a multiple structure when sub jected to any form

of cartographic generalization according to denition in Part I

Thus in this thesis the classication of the various multiple structures as scaleless pseudo

scaleless and scale dep endent and the concept of consistency and compactness will b e applied

to b oth spatial and nonspatial information varying according to scale time and edition

In the next section we will briey outline a few existing approaches to multiple mo deling

of geographic information Except for Kuhn and Bruegger BK all the metho ds and

structures are limited to handle spatial information only and most often restricted to the

most primitive ob jects such as p olygonal curves

Vector approaches

A vector based data mo del of geographic information is essentially using p oints arcs and

areas as the fundamental geometric entities of which more complex ob jects are constructed

see section

Several metho ds for representing a vector based ob ject in a range of scales corresp onding

to a set of given tolerances have b een prop osed Two such metho ds are outlined b elow one

concerning piecewise linear plane curves the other fo cusing on TINs Triangulated Irregular

Networks Both metho ds handle scale generalizations only temp oral or edition based variants

are out of the scop e

PART I I MULTIMODELS

The Multiple Nature of Geographic Information

Multiscale line trees

Several approaches have b een made to structure multiple cartographic curves or piecewise

linear curves as pro duced by successively p erforming scale generalization of an initial curve

Jones and Abraham JA suggest a multiscale line tree for this purp ose The well

known metho d of Douglas and Peucker DP is used to generate a set of curves according to

a growing tolerance such that each curve is represented by a subset of the p oints representing

the successor

These curves are organize by building a straightforward tree structure where the no des

contains p ointers to no des in the successing level The b ottom level representing the original

curve or the nest resolution contains p ointers to the data representing the curve

The metho d clearly yields a more compact structure storing only the original curve and a

set of p ointers to the data and consistent in the sense that a change in the original p oints will

aects approximated versions but the tolerance requirements may no longer b e met after such

a change In addition the structure provides fast access to the dierent layers of the three

and thereby ecient reconstruction of a certain approximant since only no des corresp onding

to those in the nal curve will b e visited however they may b e visited more than once see

for example PS Introduction which includes a discussion of the tree and related data

structures

Hierarchical TINs

In computer aided cartography the triangulated irregular network TIN has b een a p opular

structure for representing digital elevation mo dels A TIN is a complete tessellation of a part

of the plane into a number of mutually disjoint triangles of dierent size and shap e Each

vertex in the tessellation is asso ciated with a value corresp onding to the elevation of the

terrain Since a plane in space is uniquely determined by three p oints the TIN easily yields

a piecewise linear surface that may represent the top ographic surface of the Earth Since the

triangles may dier in size and shap e great exibility is oered in mo deling the multitude of

dierent classes of terrain ranging from undulating deserts to ragged mountain areas

Since the late s much eort has b een given to the design of structures capable of

ecient handling of multiscale TINs Floriani gives an overview over hierarchical surface

mo dels and presents a pyramidal TIN structure which to a certain degree may b e considered

b oth compact and consistent Flo

More details on TINs in general and their multiple representations in particular will b e

given in section

Tessellation approaches

In a raster based mo del the world is tessellated into mutually disjoint cells of equal size

exhausting the space to b e describ ed The most simple and most common approach is to

divide the plane into a set of squares Each cell is then assigned a set of attributes which are

to b e understo o d as constant within the cell see section

By successively grouping four cells into a new cell a so called quadtree is constructed Un

der the assumption that the most detailed tessellation the basis mo del contains quadratic

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Multiple Mo deling

cells the quadtree will have n levels If the coarsest level consisting of one cell is dened

to b e level level i corresp onds to a tessellation consisting of cells see gure The

quadtree structures are also widely used in digital image pro cessing

Level 0

Level 1

Level 2

Figure Quadtree

Each level of the quadtree may b e dened to corresp ond to a certain resolution which

again may b e asso ciated to a sp ecic scale The quadtree then represents a discretization of

a continuous range of resolutions thus yielding a multiresolution structure

By generalizing the twodimensional quadtree to three dimensions the o ctree is obtained

A volume partioned by cub es yields a n level o ctree

Other types of tessellations have also b een prop osed and implemented Considering the

spherical nature of the Earth hierarchical tessellations by regular triangles have b een claimed

as more appropriate than quad and o ctrees in global GI systems For a closer lo ok at the

various tessellation approaches see GS and references therein

The tessellation approaches only concerns spatial information at multiple scales The

temp oral and generalization asp ects are not covered by these metho ds Tessellated structures

are neither compact nor consistent since each level is represented explicitly and indep endent

of the other levels

Topological approaches

The multiscale line tree and the hierarchical TIN fo cused on single geometric primitives and

do not take into account the relations b etween the various ob jects

Topological mo dels see section may b e considered as enhanced vector mo dels The

various geometric ob jects are interconnected by top ological relations thereby yielding a more

coherent and consistent structure

Kuhn and Bruegger BK describ es the multiple topological representation MTR as

a hierarchical structure of a multiscale top ological mo del The concept is introduced to

PART I I MULTIMODELS

The Multiple Nature of Geographic Information

overcome the problems of accessing large ob jects aggregated by many smaller entities which

in a single scale system may take unacceptable long time

In the MTR approach several layers of dierent top ological structures of the same ge

ographic area are constructed by the means of hierarchical relations Kuhn and Bruegger

emphasize that the extraction of data can b e ecient at any level of abstraction but that

insertion of new ob jects is time consuming due to the redundancy in the structure forcing

new data to b e inserted in several representations instead of just one The MTR approach is

then neither compact nor consistent

Temporal reasoning

During the past years there has b een a growing activity in the eld of mo deling time in

general often called temporal reasoning and in particular how spatial ob jects are varying

over time frequently referred to as spatiotemporal modeling For a general bibliography on

this large research area see ATSS Spatiotemp oral problem in GIS has b een addressed

in various pap ers see eg AlTaha and Barrera ATB and references therein

The discussions in the eld of temp oral reasoning reveal a vast area of research were only

some few initial steps have b een made towards a thorough understanding of the sub ject

In this thesis a very simple approach is made towards time mo deling Even if it may b e

p ossible to mo del time as a continuous pro cess this thesis is restricted to mo del time either

as a discrete p oint or an interval in the onedimensional time space The various geographic

ob jects are considered to exist in dierent variants asso ciated to a p oint or interval in time see

gure Thus investigating more sophisticated asp ects of temp oral reasoning is b eyond

the scop e

t_0 t_jt_i t_k t_max

ON THIS DAY, BEGINVILLE GOT A NEW RAILWAY STATION......

......

Figure Simp el temp oral mo deling

This time slice approach has the advantage that each time variant may b e thought of as

sp ecial kind of generalization according to denition of the initial mo del representing the

rst moment in time The development of the Multimo del concept in the sections to come

will take advantage of this approach

In fact we have already made an exception in mo deling the tidal variation section as a continuous

p erio dic function

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Integrated Mo deling

In this context integrated mo deling refers to metho ds and structures able to represent mul

tiple mo dels varying not only according to one parameter say a tolerance corresp onding to a

scale but rather a set of variables In geographic mo deling there is need for metho ds capable

of structuring spatial and nonspatial information that vary according to b oth time space

and edition

In the previous sections examples have b een given on contemporary approaches to mul

tiple representations of spatial ob jects varying according to scale or time In spite of the

imp ortance of edition generalization as p ointed out in section in b oth manual and

computer aided cartography there have b een few attempts to design multiple representa

tions integrating encompassing multiple editions Still fewer works present integrated views of

the three main asp ects of generalization ie scale edition and time In addition little has

b een done to investigate multiple representations of the thematic or nonspatial information

within GI systems

However an approach to integrated mo deling is given by Guptill Gup Both spatial

and thematic information is considered varying over time and represented in several scales

The approach is somewhat limited in the sense that it is investigated in detail how a sp ecic

geographical data mo del the Digital Enhanced Line Graph DLGE can b e implemented

by two dierent relational database management systems The mo del provides an integrated

representation of a set of time and scale variations of b oth spatial and thematic entities The

metho d is neither consistent nor compact since the variants are represented quite explicitly

without any dep endencies

Arge and Dhlen AD prop oses another approach to integrated mo deling outlining

a metho d for structuring several variants of spatial information diering b oth according to

scale and limited edition generalization The high level structures and algorithms presented

are motivated from mathematical decomp osition and oers indeed b oth a more compact and

consistent representation The Multimo del concept to b e developed in chapter may b e

viewed as a generalization and extension of the concepts introduced by Arge and Dhlen

AD for example by incorp orating time generalization and including nonspatial ob jects

Developed and used by US Geological Survey

PART I I MULTIMODELS

The Multiple Nature of Geographic Information

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Chapter

The Multimo del

In chapter several examples of the multiple nature of geographic information were given

Various approaches towards the integration of versions based on the same mo del referred to

as multiple mo deling were briey outlined In this chapter the concept of the Multimodel

is introduced and developed as a mechanism for structuring multiple representations of ge

ographic information varying b oth according to time edition and scale With the help of

the Multimo del technique b oth spatial and nonspatial information can b e handled as far as

p ossible in a compact and consistent manner

The use formalism and notation in this chapter is adapted from mathematical decomp osi

tion theory as describ ed in eg Dhlen and Lyche DL and application of such techniques

in computer aided cartography as outlined by Arge and Dhlen AD

The Multimo del is rstly introduced and dened at a general conceptual level Then in

order to investigate the use of Multimo dels in computer based systems a denition of the

digital model is given A discussion is made on the p ossibility of applying the binary op erations

addition and subtraction and the unary op erations approximation and renement on a set

of digital mo dels

Based on this discussion a few categories of Multimo dels are prop osed as follows

Explicit Multimo del

Implicit Multimo del

Multiedition mo dels

Pseudodelta mo del

Delta mo del

Multiscale mo dels

Selection mo del

Decomp osed mo del

Multiresolution mo del

The chapter closes with a description of the composite Multimodels exemplied by an

ob ject mo del of a comp osite Multimo del customized for use in geographic mo deling called

GeoModel

The general Multimo del is not restricted to the digital domain

The Multimo del

The Multimo del Concept

As a rst step towards the Multimo del let us consider the scale problem in GIS If we have

a geographic ob ject V we can dene the scale as the exact representation of the

phenomenon Further we have p otentially an innite set of variants of the ob ject which

corresp onds to any scale x where x i

Consider a subset of n of all these p ossible scale variants of V as the indexed set

W fV V V g we may dene a mapping I X W that for any value of the

0 1 n

parameter k and thus for any scale x asso ciates one an only one mo del from W We

may typically dene I such that every V corresp onds to an interval in i In the example

of the tidal function in section the variants of the tidal function could b e dened to

corresp ond to the intervals i i and i

In this manner we can structure a set of variants of the initial mo del V such that for

every scale we may access a certain version that corresp onds to the scale This approach is

consistent with the notion of the pseudoscaleless structure describ ed by Muller Mul see

We will call this structure a Multimodel and dene it as a set of parameters a set of

n mo dels and the mapping that for every parameter picks a unique variant

W hfV V V g X I i

0 1 n

The Multimo del is essentially a function that according to a given parameter or set of

parameters see section generates one and only one of many p ossible variants of a

certain mo del It is imp ortant to note that a Multimo del is selfcontained in the sense that

it contains all information needed to generate the various variants The Multimo del concept

is formally dened as follows

Denition Multimo del Assume we have a set of model variants and a set of pa

rameters X

The Multimodel is formal ly dened as

W h X I i

where I is the mapping I X such that

I x V for al l x X

A Multimodel is said to be dep endent if a change in one of the models in W is assumed

to aect any of the other models according to some dened relation among the models A

dependent Multimodel is said to be consistent if such a change is automatically fol lowed up

by the needed changes in the depending models

Note that denition is not restricted to the digital domain In order to enable us to

study more closely the use of the Multimo del concept in computer systems the next section

gives a description and a denition of the digital model

In the scale problem outlined ab ove the set X was the interval i but in other cases the parameter

set may b e quite dierent

See the railway case in section for an example of dep endency

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Digital Mo dels

The text the parametric curves and the functions investigated in are all examples on

digital models This section formalizes the notion of the digital mo del The term digital is

used to emphasize that we are concerned with mo dels that are p ossible to implement in a

computer based system We will investigate some p ossible binary and unary op erations on

sets of digital mo dels

Denitions

A digital mo del is essentially a nite ordered set of attributes or data and a transformation

that maps the attributes to a set of real world mo dels The transformation may typically

consist of a computer program or subroutine that deco des and transforms the attributes into

a comprehensible presentation

Let a denote an ordered vector of n attributes a a a Further dene A to

1 2 n i

b e the set of all p ossible instances or values of the sp ecic attribute a and let b e the

set of all p ossible attribute vectors with the same number and types of attributes

fA A A g The set will o ccasionally b e referred to as the attribute space Note

1 2 n

that some or all of the attribute sets A s may b e identical In the example of the parametric

curve in section all the attributes were elements in IR and thus the A s were equal

Further let V b e the real world mo del eg the text section as printed on

w or ld

the map or the function section as it is plotted on the screen Let b e the set

w or ld

of all p ossible real world mo del mo dels

A transformation T is then dened as a mapping

w or ld

that takes the attribute vector as an argument and transforms the data to a real world

mo del T a V

w or ld

A digital mo del V is formally dened as follows

Denition Digital Mo del An attribute vector a a a a of digital ly rep

1 2 n

resented features a s together with a transformation T is called a Digital Model

i w or ld

Formally it is dened as the pair

V ha T i

The denition is recursive in the sense that the attributes a s may be ful ly dened digital

models

a f g where

i 1 2

The attribute vector and the transformation is dened correspondingly to T and a

A digital mo del by this denition is a discretization of an analog or real world mo del

We will some times use the terms V and a as they were equivalent and let the context decide

which to use The transformation T is in many cases quite trivial as in the example of textual

mo dels in section

Before we start investigating dierent op erations on digital mo dels some denitions con

cerning the likeness of digital mo dels will b e stated

PART I I MULTIMODELS

The Multimo del

Denition Likeness of digital mo dels Assume we have two models

0 0 0

V ha T i and V ha T i

The attributes are elements in the attribute spaces and We characterize the models as

outlined below

Noncompatible

0 0 5

The two models V and V are said to be noncompatible if T T

Two digital models one representing an image the other a piece of text are clearly non

compatible since their transformations indeed are dierent The two transformation are

dened over dierent sets of attribute spaces and However note that models

with attribute vectors from the same space and even with identical vectors may stil l be

noncompatible An example of this is an array of real values representing a function

sampled according to a given set of argument values We may have that T interprets

the attributes as a piecewise linear function while T constructs a cubic interpolant to

the data These two digital models are considered noncompatible

Compatible

Accordingly the models are compatible if their transformations are identical T T

Note that the attribute vectors may be of dierent attribute spaces assuming

that the transformation accept both the attribute vectors in question as arguments Two

vectors of points in the plane but of dierent length supplied with a transformation

that interprets the vectors as piecewise linear curves are considered to be compatible

Equivalent

If the models share a common transformation and have attribute vectors from the same

0 0

attribute space T T and then they are dened to be equivalent

Two vectors of points in the plane with the same number of elements together with

a transformation that generates piecewise linear curves are equivalent Note that the

values of the dierent attributes may vary from one model to the other

We have now established the framework needed for the investigation of op erations on

digital mo dels

Mo del arithmetics

The parametric curve section and the function section may b oth b e considered

as digital mo dels In order to construct dierence mo dels and generate variants as sums of

an initial mo del and corresp onding dierences we applied the op erations subtraction and

addition

We were able to p erform these arithmetic op erations of two reasons

The term mo del will b e used instead of the more precise digital mo del if the context do not require a

distinction

0 0

Two transformations T and T are said to b e equal if T a T a for all a

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Digital Mo dels

For each of the attributes ie p oints in plane and reals we could dene addition and

subtraction in a meaningful manner trivially in the case of the real values dening the

functions and a bit more advanced as p ointwise op erations in the parametric curve

example

The number and type of the attributes of the mo dels in question were identical thus the

mo dels were equivalent according to denition It was then straightforward to dene

arithmetic op erations on the attributevectors as elementwise addition and subtraction

In the case of the parametric curve the addition of two attributevectors were dened

0 0 0 0

g a a a a a a fa a

n 2 1

where again the addition of the elements a x IR y IR was dened as

i i i

0 0 0

y y x x a a

i i i

Based on these examples we dene addition and subtraction of digital mo dels as follows

Denition Mo del arithmetics Assume that we have two equivalent models accord

ing to denition

0 0

V ha T i and V ha T i

Assume further that the set in which the attribute vectors a and a are members is a

group under the addition operator

We trivial ly dene the addition operator as

0 0

V V ha a T i

The subtraction operator is dened as common as addition of the inverse element

0 0

V V V V

0 0

where the inverse model V has an attribute vector of inverse attributes a fa a a g

1 2 n

0 0 0

The new model V V or V V is then equivalent to both V and V

The notion of the group is a fundamental mathematical structure simple but far from uninteresting The

formal denition of a group is given here and readers interested in details of group theory is referred to eg

Herstein Her

Denition The Group The nonempty set is forming a group if there is a binary operator such

that

0 0

V V V V Closure

0 00 0 00 0 00

V V V V V V V V V Associativity

j V V V V Existence of identity

V j V V V V V Existence of inverses

PART I I MULTIMODELS

The Multimo del

0 0

The new mo del V V is equivalent to V and V since it inherits the transformation T

by denition and since the new attribute vector a a is a member of the attribute space

due to the closure of the under

As we have seen the ordinary addition and subtraction op erations are easily applied to

equivalent mo dels where the attribute vectors are members of a group A set of mo dels with

prop erly dened arithmetic op erators will b e called dierence mo dels

Many of our familiar mathematical structures are indeed groups like the integers the reals

the rationals all under the common addition vectors and matrices under their resp ectively

dened addition and functions of various kinds

In the next section we will investigate op erations of a more complex nature the approx

imation and the renement

Approximation and renement

In section we studied a set of three variants of a function simulating tidal variations

in a primitive manner They were regarded as mo dels of dierent resolution or scale Let us

investigate the tidal function in a slightly dierent setting

Consider the function f and the data reduced or approximated variant f see gure

Both functions may b e considered as digital mo dels represented as two curves with p oint

vectors a and a with resp ectively p oints in IR Both mo dels share the transformation

T which interprets the vectors as piecewise linear curves Note that this transformation

plc

is able to handle an arbitrary vector of p oints fq q g The two mo dels are clearly

1 n

3 'f' 'f,approximated' 2.5

1.5

0.5 f(x)

-0.5

-1

-1.5

-2 0 5 10 15 20 25

Figure Function an its approximation

compatible according to denition However the functions are not equivalent since the

attribute vectors are elements in dierent attribute spaces The initial mo del f is represented

Throughout the thesis the notation will b e used to emphasize that the mo del in question is approxi

mated

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Digital Mo dels

by nine p oints representing samples of the tidal function

a f g

and the approximated and datareduced variant has ve attributes

a f g

We may dene the attribute space from which a is an element as follows

f g g g g g

where g is any univariate function dened on the interval and the attribute space

as spanned by

f g g g g g g

where again g is any arbitrary function on the interval in question

The pro cess of approximating and reducing the data of the mo del f may b e formalized

as a mapping

that picks every second p oint from f starting with the rst thus reducing or approximating

the attribute vector of f to a truncated vector in another attribute space of lower dimen

sionality The notation P indicates that the approximation pro cess is to b e regarded as a

projection from one attribute set down to another set consisting of vectors of less length

The mo dels f and and the approximant f are not compatible and we are unable to eg

take the dierence a a In order to make f equivalent to f we introduce the renement

op erator R as the mapping

This renement op erator may b e dened by linear interpolation such that

a +a a +a

Ra a fa a a g

1 2 5

2 2

Figure illustrates the insertion of p oints resulting in a renement of the approximant f

The attribute set is spanned by all p ossible function sampled on the given argument values

Since R consists of the same functions but with some of the function values restricted to

the linear interpolation scheme outlined ab ove we clearly have that R

We now generalize the examples ab ove to a denition of decomp osable mo dels

Denition Decomp osable mo del A model is said to be decomposable if we are able

to approximate and rene it as fol lows

We will use the notation to emphasize that the mo del in question is rened

PART I I MULTIMODELS

The Multimo del

3 'f' 'Approximant,refined' 2.5 'Points,interpolated'

1.5

0.5 f(x)

-0.5

-1

-1.5

-2 0 5 10 15 20 25

Figure Renement of approximant

Approximation of a digital model V ha T i is a mapping of the attribute vector a as

an element of the attribute space to the attribute space under the assumption

i j

that the transformation T are valid for elements in

i j

such that the approximant V is generated as

V hP a T i

P is the identity mapping The approximant is compatible according to denition

to the original model but not equivalent

Renement is a lifting or mapping of a digital model V ha T i of the attribute set

to a larger attribute space

j i

under the assumption that T is valid in The rened model V is produced as

a T g V fR

P V is equivalent to model V Note that the approximated and then rened model R

but not necessarily equal to this since R P

i i

The design of approximation and renement op erators is indeed not a trivial task In

the example of the tidal function the op erators were quite simple More sophisticated and

complex op erators will b e briey investigated in section and

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Digital Multimo dels

In many applications it is interesting and frequently necessary to estimate how go o d a

certain approximation is That is we need a to ol for measuring the distance b etween two

mo dels V and V which is to b e considered as the error of the approximation For instance

in the case of the tidal function the error might for instance b e dened to b e the largest

absolute value of the dierences of all corresp onding function values let a a x y

i i i

distance f Rf max jy y j

i=1

As a to ol for measuring the error of an approximation we use the notion of the metric to

dene a set of metric models

Denition Metric mo dels A set of models is said to be metric if there exists a

0 00

function V V IR such that for al l V V and V in the fol lowing holds

V V

0 0

V V V V

0 0

V V V V

0 00 00 0

V V V V V V

The distance function distance f Rf dened ab ove is easily veried to b e a metric In a

set of metric decomp osable mo dels we will b e able to generate approximations and estimate

the asso ciated error as the distance b etween the original and the approximated mo del In

many application areas and indeed GI science this ability is of vital imp ortance

The discussion and various denitions concerning the digital mo dels will play an imp ortant

role in the elab oration of the Multimo del concept during the next few sections

Digital Multimo dels

In section denition gave a description of Multimo del that was not restricted to the

digital domain Since we in section have studied the digital mo del we can now give a

more detailed description of the digital Multimo del

A digital Multimo del is essentially a nite discrete and indexed set of digital mo dels

fV V V g asso ciated with a set of index parameters X f ng The function I

0 1 n

is the trivial mapping

I x V

In other words the digital Multimo del is an indexed set of mo dels which may b e directly

accessed according to the index In many applications as in the example with the scale

intervals in section we have to introduce an additional mapping from an arbitrary set of

parameters to the index set X In the further treatment of the Multimo del we will assume

that such transformations are provided

Since the mapping I is trivial the digital Multimo del is reduced to

W hV V V i

0 1 n

PART I I MULTIMODELS

The Multimo del

Note that the mo del set W may b e represented in various ways assuming that the semantics

of the ordered set are maintained ie that it is p ossible to p erform op erations as insert

delete access and so on

Asp ects of Multimo dels

In this section we take a closer lo ok at some asp ects of Multimo dels that will come at hand

when developing the concept at a more detailed level

Multimo del op erations

In an application a MultiMo delhas to oer a broad range of functionality However at this

level we will only b e interested in three kinds of pro cedures which all are highly dep endent

on the internal representation of the set of mo dels

Assume that we have a Multimo del W we may want to design pro cedures for p erform

ing basic op erations such as initialize insert delete app end merge and so on However

to illuminate certain fundamental asp ects of Multimo dels we will restrict the investigation

to concern pro cedures for inserting new mo dels for accessing a particular mo del and for

up dating or changing an existing mo del

insertV For simplicity we restrict the insertion to app end the mo del V to the

existing set W such that we get the new set W fV V V V V g

0 1 n n+1

reconstructk The reconstruction or accessing pro cedure takes a parameter k

f ng and based on this it shall pro duce and return the mo del V from W This

is to b e considered as the most fundamental op eration on a Multimo del

updatek V This pro cedure p erforms an up dating of the mo del corresp onding to the

parameter value k such that mo del V replaces the old mo del Since k corresp onds to

mo del V we get the new mo del set W fV V V V V g

k 0 1 k n

Based on the characterization in section of mo dels as noncompatible compatible or

equivalent we will prop ose three main classes of Multimo dels

Consistency and compactness

In this section we introduce the notions of consistency and compactness In denition a

Multimo del is said to b e dep endent if a change in one mo del is supp osed to imply changes

in other mo dels In the digital domain we make a restricted denition of dep endency The

digital Multimo del W fV g is said to b e dep endent if a change in mo del V is supp osed to

i i

aect the consecutive mo dels fV V V g according to some predened rules

i+1 i+2 n

Further W is said to b e consistent if the representation is such that a change in V implies

the necessary up dates of the consecutive mo dels without taking any explicit actions against

Note that this kind of change is irreversible If it is imp ortant to maintain the history of a set of changes

it would b e b etter to p erform the up date as the introduction of a new edition of the mo del

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Some Categories of Multimo dels

each of these mo dels This is in many applications a desirable ability recall eg the example

of the railway in section See section and for further details on dep endency

and consistency

Another imp ortant issue certainly in GI systems but also in other areas is to represent

the information handled by the system as compact as p ossible in the sense that the data

takes as little storage space as p ossible

The most straightforward way to store the data in a Multimo del is to represent each

mo del explicitly If a certain representation of the set of mo dels takes less storage space than

the explicit representation we say that the representation is more compact than the explicit

mo del More formally let

kV k

bit

b e the number of bits required to store the representation of the mo del V digitally and let

kW k kV k

bit

i=1

b e a measure of the space required for the storage of W The representation W is

alter nativ e

more compact if

kW k kW k

alter nativ e explicit

where W is the explicit representation of W

explicit

As exp erienced in section a certain representation the delta mo del scheme yielded a

structure where the mo dels of the curves contained a large amount of zeros There exist many

techniques for storing such ob jects using less space than storing an arbitrary corresp onding

mo del see eg Nel Thus if a certain Multimo del structure contains representations with

a larger content of zero or void attributes than the originals we consider such a structure to

b e inherently compact

Another p otentially compact Multimo del is a representation where the attributes on the

average are of smaller magnitude than in the original In the parametric curve example in

section we saw that the delta representation of the p oint vectors were of small magnitude

compared to the explicitly represented vectors

Based on the characterization of mo dels as noncompatible compatible or equivalent we

prop ose three main classes of Multimo dels the explicit Multimodel the multiedition model

and the multiscale model The last two structures may b e considered as variants of the

implicit Multimodel These Multimo dels will dier in the way the mo del set is structured and

represented and in the algorithms asso ciated to the three basic op erations insert reconstruct

and up date We will also make comments on the degree of consistency and compactness they

oer

Some Categories of Multimo dels

The collection of the mo dels in W may b e represented and implemented in various ways as

long as the representation ob eys the semantics of the ordered set ie that it supplies the

With this denition we dont take in account the additional overhead introduced by storing a number of

mo dels as a collection

PART I I MULTIMODELS

The Multimo del

needed op erators to maintain the indexed set In our case we have limited the op erations to

insertion reconstruction and up dating

It is p ossible to identify three main classes of representation of a Multimo del W

Explicit representation is a direct representation of each and one of the mo dels without

any relations b etween the representations of the variants This will also b e referred to

as the trivial representation of a Multimo del

Implicit representation is the characterization of any metho d that do not rely exclusively

on explicit techniques but represent the mo dels using some sort of relations b etween

the variants We have two main categories of implicit Multimo dels

A multiedition model is a collection of equivalent mo dels ie sharing the same transfor

mation and having attribute vectors from the same space The multiedition structure

is useful when it is interesting to maintain a set of variants of an initial mo del all

describ ed with the same degree of accuracy or with the same resolution Within geo

graphic mo deling the multiedition mo dels should b e useful for representing the time

and edition variants pro duced in a cartographic generalization pro cedure recall deni

tion in section

Multiscale models are essentially sets of compatible and nonequivalent mo dels The

dierent mo dels are to b e considered as variants of an initial mo del with decreasing

accuracy represented with less data than the original mo del Such structures are par

ticularly interesting when maintaining the essentially same geographic information in

dierent scales or resolutions

The trivial Multimo del and variants of multiedition and multiscale mo dels are outlined

in the next sections The list of the techniques describ ed is by no means exhaustive many

other approaches may b e equally interesting or imp ortant The main goal is to suggest a few

quite dierent directions which may lead to practical and ecient Multimo del structures

We pay sp ecial attention to the design of the op erations insertion reconstruction and

up dating and of the degree of compactness and consistency they oer For all the examples

we assume the Multimo del is indexed as

W hV V V i

0 1 n

We start with a closer lo ok at the trivial Multimo del structure

Explicit Multimo del

The explicit Multimo del will b e useful when the collection of mo dels in question is a set of

chalk and cheese ie that in spite any apparently likeness of the mo dels they may neither

b e regarded as deltamo dels nor decomp osable mo dels

We do not have much freedom in the design of a Multimo del of such inhomogeneous

mo dels The text case in section was an example of this class of mo dels even if the

dierent pieces of text in fact were compatible

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Some Categories of Multimo dels

Assume we have a set of n explicitly represented mo dels

i W hfV g

i=0

The insertion reconstruction and up dating pro cedures b ecome quite trivial and may b e

expressed algorithmically as follows

Algorithm Insertion in explicit Multimodel

insertV

if W

V V n

else

V V

n+1

n n

Algorithm simply app ends the new mo del explicit to the Multimo del Generating a

particular mo del is equally simply p erformed as a direct access of the mo del

Algorithm Reconstruction in explicit Multimodel

reconstructk

return V

If the Multimo del is dep endent according to denition the up dating pro cedure has to

check all successors to maintain consistency The pro cedure

checkthis model changed model p erforms the check and the p ossible changes

Algorithm Update in explicit Multimodel

updatek V

V V

if W is dependent

for i k n

checkV V

Since all the mo dels V are explicitly represented this is indeed not a compact represen

tation Any up date of one of the mo dels implies a corresp onding check of all consecutive

mo dels V i k and the mo del is clearly not consistent

The explicit Multimo del is a primitive construction but may b e useful in cases where

noncompatible mo dels are to b e handled in a homogeneous way The existence of the trivial

Multimo del assures that any set of digital mo dels may b e handled as a Multimo del

In the algorithms presented in the thesis we assume that the data structures used in the algorithms

to follow allows dynamic expansion ie that elements may b e added to existing structures Checking of

invariants initiali zin g of data structures or other details may b e omitted to stress the main structural issues

PART I I MULTIMODELS

The Multimo del

Multiedition mo dels

Assume we have a set of n equivalent mo dels fa g ie that all the mo dels share the

i=0

transformation T and are elements of the same attribute set where all attribute vectors

have m elements a a a a

i i i i

1 2 m

In addition assume that it is p ossible to represent the dierence b etween two mo dels

Then we may express the representation of the Multimo del W as V and V expressed as

j i

T i hf g

i=0

where and V In other words W is represented by the initial mo del V and a

i 0 0 0

sequence of dierences of consecutive mo dels This sp ecial kind of Multimo del will b e referred

to as a multiedition model Temporal variations of geographic information and generalized

editions of a geographic entity are candidates for this particular Multimo del structure The

railway case in section supplied examples of b oth a temp oral change and a change due

to the construction of a new edition which were shown to b e well suited for multiedition

representation by the means of a delta op erator

it is p ossible to make a more precise interpre With the help of the dierence op erator

tation of dep endency and consistency W is said to b e dependent if a change of mo del V to

V is supp osed to yield the new Multimo del representation

hV V V V V i

1 2 k k +1 n

^ ^

k i i

we mean the dierence b etween V and V In other for all i k With such that

i i

in mo del k should b e added to all the following mo dels words the change

A dep endent multiedition mo del is consistent if the representation and the asso ciated

up date pro cedure is designed in such a manner that no explicit actions has to b e taken to

the mo dels following the up dated mo del ie that the change will automatically propagate

through the structure

We will now investigate two classes of multiedition mo dels the pseudo delta model and

is realized the delta model diering in how the op erator

Pseudo delta mo del

is dened as In the pseudodelta mo del the delta or dierenceop erator

fd d d g

i i i i

1 2 m

where d is dened as

if a a

i i1

a if a a

i i i1

j j

The element is to b e interpreted as an empty or void attribute In app endix A we nd

an example of a pseudodelta mo del in the implementation of the record

This will not necessarily imply that we have a fully dened arithmetic

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Some Categories of Multimo dels

Insertion of a mo del V with an attribute vector a of m elements fa a a g is p er

1 2 m

formed as outlined b elow Note that since all mo dels in a pseudodelta mo del are equivalent

the n attribute vectors corresp onding to f g including the one to b e inserted

0 1 n

a are all elements in

Algorithm Insertion in pseudodelta model

insertV

if W

V V n

else

for j m

i n

while d and i

i i

if d a

i j

d a

n+1 j

else

n+1

n n

Algorithm runs through the attributes of the mo del to b e inserted and checks each

attribute against the rst nonzero corresp onding attribute iterating backwards from the

last to the rst mo del in the existing structure If the attributes are identical a is inserted

as the n th attribute of this kind else the attribute of the new mo del is inserted

The reconstruction algorithm works in a similar fashion

Algorithm Reconstruction in pseudodelta model

reconstructk

for j m

i k

while d and i

i i

a d

j i

return a fa a a g

1 2 m

Algorithm will for every V attributes work its way backwards from the k th level in the

structure until encountering a nonzero attribute which will b e inserted as the corresp onding

attribute in the vector to b e returned

Finally we present the up dating algorithm

PART I I MULTIMODELS

The Multimo del

Algorithm Update in pseudodelta model

updatek V

for j m

if d a

k j

d a

k j

Check in worst case all prior attributes

in the model such that no successing

attributes are identical

The attributes in the vector on the k th level are compared to the corresp onding attributes

in the up dating vector and replaced if they dier In addition all the attributes of this kind

included the up dated have to b e checked and p ossibly up dated

If not all the mo dels in the pseudomo del structure are completely dierent from the

successor the dierences will contain a certain amount of zero elements thus clearly yielding

a compact representation The step in algorithm shows that the structure is not

consistent

Some mo dels p ermit a more sophisticated implementation of the deltafunction and this

will b e investigated in the following section

Delta mo del

Assume we have a set of equivalent mo dels with a well dened arithmetic ie a set of

dierence mo dels see section In addition we restrict the mo dels not only to b e

elements of a group see denition under but the group shall also b e ab elian or

commutative The identity element will b e noted as and the inverse elements will b e

noted V

is in this case trivially dened as V V The implicit The dierence op erator

i i i1

representation will b e the initial mo del and the sequence of dierences

T i hf g

i=0

where V

0 0

The insertion algorithm app ends a mo del V by generating the mo del V and app end the

new dierence V V

A set of completely dierent mo dels would indeed represent a pathological example of a multiedition

mo del

0 0 0

A group under is ab elian if for any V V we have that V V V V The term ab elian

honors the Norwegian mathematician Nils Henrik Ab el most famous for proving the imp ossibi li ty

of solving quintic or higher degree equations by means of ordinary arithmetic op erations including ro ot

extraction

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Some Categories of Multimo dels

Algorithm Insertion in delta model

insertV

if W

V V n

else

for i n

n+1

n n

To verify that the insertion is correct we have to check if the inserted dierence

n+1

V is equivalent to V V ie that V In the algorithm is computed as

n n

We will now p erform the computation using the denition of the ab elian V V

i n 0

i=1

group and some trivial implications

i=1

V V V V V V V V V

0 1 0 2 1 n1 n2 n n1

V V V V V V V V V

0 0 1 1 2 2 n1 n1 n

V V V

0 0 n

The reconstruction algorithm uses a similar multiple addition scheme as in algorithm

starting with the initial mo del and adding dierences up to the level corresp onding to the

given parameter k

Algorithm Reconstruction in delta model

reconstructk

for i k

return

The up dating algorithm generates the mo del V and replace the old dierence

k 1 k

with the dierence b etween the new mo del and V

k 1

The scheme will work with nonab elian groups to o but with a little awkward denition of the deltas as

V V

i i i

PART I I MULTIMODELS

The Multimo del

Algorithm Update in delta model

updatek V

for i k

It is straightforward to verify algorithm and using the same pro cedure as for verifying

As with the pseudodelta mo del this structure is compact since the dierences will consist

of a number of zero elements under the assumption that no mo del completely diers from

its successor

The delta mo del is consistent with resp ect to the dep endency Note that the delta mo del

scheme will not work correctly if such a dep endency is not wanted Any up date of a certain

mo del will automatically propagate to the successing mo dels as demonstrated in algorithm

To verify this we observe that after an up date of V to V such that V V we

j j j j j

have that a mo del number k k j is reconstructed as follows

i=1

P P

j 1

i i j

i=j +1

i=1

P P

j 1

i i

V V V

j j 1 0

i=j +1

i=1

P P

j 1

i i

V V V

j j j 1 0

i=j +1

i=1

0 j

i=1

k j

Thus the change applied to the mo del V propagates to all following mo dels

j j

We have now outlined two metho ds for structuring collections of equivalent mo dels The

next sections investigates corresp onding structures for handling mo dels which are compatible

diering in having attribute vectors from attribute spaces of dierent size

Multiscale mo dels

ie that all the mo dels share Assume we have a set of n compatible mo dels fa g

i=0

the transformation T but may have attributevectors of dierent types ie for

i j

i j where is the set in which a is an element The mo dels are ordered after increasing

i i

complexity such that the number of attributes in V is larger than in V when j i

j i

In addition assume that we have an approximation op erator P see section dened

such that

P V V V

n n i

ie that the mo dels V V V are approximations of the mo del V represented with

0 1 n1 n

increasing complexity ie represented with an increasing amount of data

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Some Categories of Multimo dels

The representation of a multiscale mo del is characterized by the collection

n i

hfa g P T i

i=1 n

A set of compatible mo dels with attribute vectors from dierent attribute sets may b e

regarded as mo dels of essentially the same phenomenon represented in dierent accuracies or

resolutions In cartography and GIS it is common to asso ciate the notion of scale to accuracy

or resolution see eg the Introduction Of this reason this particular kind of a Multimo del

will b e termed a multiscale model

The notion of consistency should b e dierent interpreted when it comes to multiscale

mo dels It is not natural to dene dep endencies among the various mo dels other than the

inherently dep endency dened by the fact that each mo del is an approximation of an initial

mo del The approximation op erator is part of the Multimo del and we will therefore consider

all multiscale mo dels as consistent This is leaving compactness as the main asp ect when

investigating multiscale structures

Before investigating multiscale structures we have to make some comments on the insert

and update op erations

insert In the case of the multiedition mo del the insertion was restricted to

app end the mo del V given as the parameter to the pro cedure In a multiscale structure

insertion will b e reformulated to app end a new approximant The approximant may b e

generated either by sp ecifying the attribute space which we want the approximant

to b e a member in or if we deal with metric mo dels by giving the tolerance

updatek V The multiscale structure is a sequence of successively datareduced

mo dels constructed by applying a well dened approximation op erator on the original

mo del Any up date of the mo dels apart from the original mo del is out of the question

since such an up dating would p otentially violate the assumption that the mo del should

b e an approximant Of this reason the up date op eration b ecomes uninteresting in the

context of multiscale mo dels

The reconstruction pro cedure will b e identically dened and interpreted as for the multi

edition structure

We will now study three dierent variants of multiscale mo dels mainly diering in prop

erties of the approximation and renement op erators

Selection mo del

We may sometimes encounter a sp ecial kind of the multi scale mo del where the approximation

op erator is restricted to select a collection of the attributes of the initial mo del to yield a mo del

of lower precision This is a simple but nevertheless imp ortant scheme widely used in eg

line simplication algorithms see section

The approximation op erator may b e dened as follows

i i

P V P a a a a a a V

1 2 m i i i

1 2

n n

PART I I MULTIMODELS

The Multimo del

where m and every a is an element in a a a and where the ordering of the

i 1 2 m

original vector is maintained P is dened as the identity op erator

This sp ecial Multimo del may b e mo deled extremely compact by using an integer array

of the same length as the original attribute vector a a and storing only the original

mo del V

The representation of selection mo del is expressed as

ha P T i

where a a a a

1 2 m

The insertion algorithm runs as follows under the assumption that W and that is

initialize d to zeros if only one single mo del is represented by W ie that n The notation

P indicates that the approximation is dep endent of the input of the pro cedure as discussed

in the previous section

Algorithm Insertion in selection model

insert

a P a

for i m

if a a

i i

n n

We assume that the approximated mo del is element of a space of lower dimension

n+1

ality than

The reconstruction use the information embedded in the vector and picks the elements

in a accordingly In app endix A we implement piecewise linear curves as selection mo dels

Algorithm Reconstruction in selection model

reconstructk

for j m

if n k

return f g

1 2

The selection mo del is extremely compact In addition to the original mo del only the

vector consisting of m integers is needed Further the storage demand is not dep endent on

the number of mo dels represented

There exists indeed more timeecient representations of the selection mo del In our case

the running time of the generation algorithm is O m where m is the number of attributes in

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Some Categories of Multimo dels

the original mo del A treelike organization of p ointers to the data as the linetree describ ed

in JA may reduce the average running time to O log m but as p ointed out earlier our

emphasis is towards compactness rather than runtime eciency

Note that variations can made over the selection mo del scheme The dierent mo dels may

i i

b e pro duced as stepwise approximations such that V P in contrast to V P as we

i i

i+1

suggested However such variations requires only minor changes in the dierent pro cedures

asso ciated with the structure

Note that the selection mo del did not require a renement op erator such as the decom

p osed mo del outlined in the next section

Decomp osed mo del

This section is an adaptation of the metho ds and algorithms in Dhlen and Lyche DL

The article investigates asp ects of decomp osition of well dened mathematical structures

thus we have to make some mo dications to t a more general Multimo del concept

Assume that we have a set of decomp osable mo dels fV g ie that we have b oth an

i=0

approximation op erator and a renement op erator The corresp onding attribute vectors are

elements in f g where we may have that

0 1 n i j

Let each mo del b e the result of an approximation that pro jects the initial mo del V to

a space of lower dimensionality ie attribute vectors of less length

V P V

i n

such that the decomp osition mo del essentially is an ordered set of mo dels of increasing pre

cision

The renement op erator see section is dened such that

V V j i R

i j

where V is element of a subset of Note that V is equivalent to V but most often

j j j j

not identical

Since we assumed that the mo dels were decomp osable and thus having addition and

subtraction at hand we may represent the mo dels as an explicit mo del V V and a set of

dierences

V V R

i1 i i

where the rened mo del R V is element in

i1 i i

The representation of the decomp osition mo del may formally b e dened as

j n

T i P R hf g

i=0 n

where V thus consisting of the initial mo del in the coarsest resolution a set of approxi

0 0

mated dierence mo dels arithmetic op erators approximation and renement op erators and

nally a common transformation

The app ending of a new approximant will essentially involve steps from the reconstruct

routine in addition to b o okkeeping actions and will not illuminate signicant asp ects of the

PART I I MULTIMODELS

The Multimo del

decomp osed mo del Thus we will not outline any details of the insert op eration for this

multiscale structure but rather concentrate on the reconstruction of a certain approximant

The reconstruction in a decomp osed mo del is to b e considered as a variant of algorithm

algoReconstructDelta the corresp onding op eration in a deltamo del

Algorithm Reconstruction in decomposed Multimodel a

reconstructk

for i k

return

Piecewise linear curves with a sp ecial kind of approximation op erator are implemented as

decomp osition mo dels in app endix A

The algorithm follows since V can b e constructed from V in the following way

i+1 i

i+1

V R

i i+1

i+1 i+1

V V V R R

i i i+1

i i

i+1

If the renement op erator satisfy

k k

i j k R R R

i j

and

j j j

0 0

V V R V V R R

i i i

where V V we may design a variant of the reconstruction algorithm Here the additions

are all p erformed in the largest attributespace and not on levels of increasing complexity

as in algorithm

Algorithm Reconstruction in decomposed Multimodel b

reconstructk

V R

for i k

return

The correctness of is veried by p erforming the addition up to a level k

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Some Categories of Multimo dels

k k

V R R

i 0

i 0 i=1

i k k

V V R V R R

i1 i 0

i1 i 0 i=1

i k k k

V R V R V R R

i1 i 0

i1 i i 0 i=1

k k k

V V R R V R

i1 i 0

i=1

i1 i 0

k k k k

V V R V R V R R

1 1 0 0

k k k

V V R V R R

k k 1 k 1

The decomp osed structure is more compact than an explicit representation under the

assumption that the magnitude of the dierences is less than the magnitude of the corre

sp onding approximants see section

As with the selection mo del structure variations can made over the decomp osed scheme

i i

for example by pro ducing the approximations as V P V instead of V P V See

i i+1 i i+1

i+1

section in Dhlen and Lyche DL for further details on variations over a similar scheme

In the selection mo del and the decomp osition mo del the approximation op erator pro jects

the mo dels from one attribute space down another known attribute space of lower dimension

2 +1

ality Such op erators could for example b e implemented as P ie an approximation

2 +1

n n1

from a space of attributes to a space of attributes by just picking every second

attribute starting with the rst In this way we do not have any control of how go o d the

approximation is ie the distance b etween the original and the approximant or in other

words the error of the data reducing pro cess In the next section we investigate a multiscale

structure for metric mo dels with approximation op erators that dep ends on given tolerances

Multiresolution mo del

A multi resolution mo del is essentially a multiscale mo del asso ciated with a set of tolerances

a metric see denition and an approximation op erator capable of p erforming constrained

data reduction according to the corresp onding tolerances

n n

Assume we have a set of metric mo dels fV g a set of tolerances f g and a con

i i

i=0 i=0

strained approximation op erator A multiresolution mo del is then the set of mo dels such

that V is the result of the stepwise approximations

V P V

i i i+1

i+1

V V V is the original mo del with highest precision such that R

i i+1 i n

We may have the alternative denition where we have a set of approximations all generated

from the original mo del

V P V

i i n

n n

V V R where the error in the approximation is measured as R

i+1 i

The notation P indicates that the approximation maps the mo del from the attribute

set to the unknown set which dep ends on the tolerance

j i i

Both the selection mo del and the decomp osed mo del may easily b e formulated as multi

scale mo dels with minor adjustments of the structures and the op erations

PART I I MULTIMODELS

The Multimo del

Comp osite mo dels

Until now we have only b een concerned with Multimo dels varying according to one single

parameter typically in a GIS setting representing scale edition or time

However in Part I we stressed that geographic information is characterized by varying

according to all these parameters simultaneously This fact motivates the introduction of

the composite Multimodel ie a Multimo del that vary according to more than one single

parameter

We will not give any further details on comp osite Multimo dels in the thesis but we observe

that such Multimo dels represent additional challenges particularly regarding consistency and

compactness

We have in the last sections outlined various asp ects of Multimo dels and investigated to

some detail a few main categories of structures for multiple mo deling

In the next section we will summarize the chapter with the description of an ob ject mo del

of a generic Multimo del class library

MULTIMOD A generic ob ject mo del

We will now summarize the elab oration of the Multimo del concept in an ob ject mo del of

a generic library structure The library will b e called MULTIMOD The main structure of

MULTIMOD is displayed in gure

ApplicationFunctionality

(Operations are to be supplied by the user) DigitalModel

setAttributes(...) {abstract} getAttributes(...) {abstract} 1+ MultiModel ...... insert(...) {abstract} reconstruct(...) {abstract}

update(...) {abstract}

Figure Main structure of MULTIMOD

The gure simply states that we have a class of ob jects named Multimodel which consists

of a number one or more of ob jects of the class DigitalModel The DigitalModels are

sp ecializations of the class ApplicationFunctionality

Rumbaugh et al gives the following description of an ob ject mo del RBP

An object model captures the static structure of a system by showing the ob jects in the system

relationship s b etween the ob jects and the attributes and the op erations that characterize each

class of ob jects

In the presentation of the Ob ject Mo deling Technique OMT Rumbaugh et al further stress that an ob ject

mo del is the most imp ortant description of a system We will use a limited subset of the OMTnotation in our

ob ject diagrams and we assume that the reader are familiar with such diagrams

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

MULTIMOD A generic ob ject mo del

We see that Multimodel has a set of op erations needed to maintain an ordered set of

variants represented by the op eration reconstruct see section for details on such op er

ations The class Multimodel is abstract and derived sp ecializations need to implement the

details of op erations

The DigitalModel is also an abstract class It is a sub class of ApplicationFunctionality

which is to b e considered as an interface to the application in which the Multimo dels are to

b e used Initially this class has no op erations the intention is that the op erations should b e

provided by the user In our example we have furnished the ApplicationFunctionality

with the op eration printMap With this structure we are ensured that regardless of how the

sub classes of the DigitalModel and the Multimodel are implemented we are always able to

make the call

AnyMultimodelreconstructindexprintMap

ie that is p ossible from any Multimo del to reconstruct a certain variant according to the pa

rameter index and print that particular mo del in a certain fashion sp ecied by printMap

If the MULTIMOD was to b e used in a GIS we should indeed have designed a comp osite

Multimo del that were customized to handle variants according to time scale and edition

as describ ed briey in section Such an extension would essentially involve mo deling

of relations b etween three single Multimo dels and is omitted in order to highlight the main

structural issues of the implementation of a Multimo del

TrivialMM DigitalModel

PseudoDiffMM PseudoDiffModel

MultiModel DifferenceMM DifferenceModel ApplicationFunctionality

SelectionMM ApproximationModel

DecomposedMM RefinementModel

Figure MULTIMOD a generic Multimo del library

In gure the main framework of MULTIMOD is extended to incorp orate the dierent

categories of digital mo dels and Multimo dels outlined in section The initial DigitalModel

is successively sp ecialized into four types

The PseudoDiffModel is characterized by a simple op erator

The DifferenceModel which is supplied with a fully developed arithmetic This implies

that the mo dels to b e derived of this class must form groups recall denition under

the particular kind of addition implemented

PART I I MULTIMODELS

The Multimo del

The ApproximationModel has an approximation op erator enabling the mo del to gen

erate approximations of itself either to an attribute space of known size or according

to a given tolerance In the last case we assume that there exists a well dened metric

or distancemeasure

The RefinementModel provides an renement op erator making it p ossible to lift the

mo del into a larger attribute space

Note that the various mo dels inherit the op erations from their resp ective sup erclasses such

that the renement mo del has b oth well dened arithmetic op erators and an approximation

op erator in addition to the renement op erator of its own Also note that the mo del categories

are abstract classes and can not b e instantiated without b eing sp ecialized into classes which

implement the various op erations

Based on the mo del categories we have designed ve sub classes of the Multimodel

The TrivialMM is simply a collection of any type of DigitalModel and is to b e con

sidered as a chalk and cheese Multimo del The basic op erations are implemented

according to algorithms and

The PseudoMM integrates equivalent mo dels with op erators as dened in the class

PseudoDiffModel such that it is p ossible to express dierences b etween mo dels Note

that this not imply a full set of arithmetic op erators The algorithms and

are implemented in the class

The DifferenceMM is handling a set of sp ecializations of the DifferenceModel under

the assumption that they are equivalent Basic op erations are implemented as outlined

in and

The SelectionMM integrates compatible mo dels or sp ecializations of ApproximationModel

They are restricted to b e the result of an approximation that selects a subset of the

original attributevector The insert and reconstruct pro cedures are implemented

according to algorithms and

The DecomposedMM is the most completely equipp ed Multimo del b eing a set of de

comp osable mo dels or sub classes of RefinementMod We have chosen to use a stepwise

reconstruction pro cedure as given by algorithm

Note that the dierent Multimo dels are not abstract classes and may b e instantiated as they

are

In chapter we will prop ose an informal metho dology for designing sp ecic Multimo dels

based on the framework outlined in this section We will also give details on the design of

some selected digital mo dels

A limited implementation of the MULTIMOD incorp orating the examples given on digital

mo dels in chapter is carried through in app endix A

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Chapter

Examples of Multimo del s

In this chapter we will use the discussion and results from chapter and to design Multi

mo dels of piecewise linear curves PLCs for short and piecewise linear surfaces dened over

triangular irregular networks called PLSs

Both these geometric structures are examples of representations of spatial information

The PLC is frequently used in GIS to mo del railway networks b orders shorelines and other

curvelike features

The PLS is frequently used in digital elevation models DEMs for short to mo del the

terrain as an explicit linear bivariate function In addition the use of the PLS to represent

thematic information such as the variation of a certain parameter over a given area is

increasing

Before investigating the PLC and the PLS we will outline a general strategy for Multi

mo deling

Designing Multimo dels

The design of a particular Multimo del should start with a thorough analysis of the ob jects in

question and their prop erties as digital mo dels Following the characterizations and denition

in chapter the analysis may go as follows

Characterize the likeness mo dels according to denition as

noncompatible

compatible or

equivalent

If the mo dels are noncompatible we have no other options than structure them as an

explicit Multimo del which may b e trivially implemented

If our mo dels are compatible we have two main sub classes to investigate the mo dels

in a multiedition setting and in a multiscale setting

We will restrict the investigation to single Multimo dels and will not give any details of design metho dology

for comp osite Multimo dels

Examples of Multimo dels

The multiedition structure requires at least a denition of a dierence op erator

to b e able to represent the dierence b etween two mo dels We are p erhaps able to

dene a pseudodelta mo del or we may nd that our mo dels in fact have well de

ned arithmetic op erators and p ossibly members of the same group Section

outlined two implementations of multiedition mo dels dep ending on the arithmetic

prop erties of the mo dels

If the mo dels are to b e accessed in variants of dierent accuracy we have to ex

amine p ossible approximation and renement op erators Section describ ed

alternative designs of multiscale structures based on certain prop erties of the

approximation op erators and p ossibly the renement pro cedures

We will now apply the describ ed metho dology on the PLC and the PLS

Piecewise Linear Curves

Denition

A piecewise linear curve may b e considered as a digital mo del The attribute vector is an

ordered set of an arbitrary number of p oints in the plane

p p p p p x y IR

1 2 m i i i

The transformation of the mo del may b e considered essentially as the following denition

of the curve C as

s C

i=1

where s is the segment dened as the convex combination of two consecutive p oints

s fx IR j x p p g

i i i+1

The formulation of the PLC as a digital mo del is then

P LC hp C i

Arithmetics

The PLC is fully supp orted with arithmetic op erators The set of all such curves C of length

m is indeed an ab elian group recall denition under the addition dened as

m 0 0

C C p p

i=1 i

0 0 0 0

where C C C and p p x x y y The zerocurve is and the

m i i i

i i i

inverse element of C is C x y

i i

i=1

The verication of hC i as a commutative group is quite trivial and thus omitted

We may also trivially dene a metric on this group as

0 0

2 2

C C max y y x x

i i

i=1

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Piecewise Linear Curves

ie the largest Euclidian distance b etween corresp onding p oints of the curves This is easily

veried as a metric according to denition

Since piecewise linear curves of same length are elements of an ab elian group it will b e

convenient to structure a Multimo del of such PLCs as a delta mo del see section and

the algorithms and We will then b enet from the compactness oered by this

structure and also from the consistency of the scheme An implementation of the PLC as a

delta mo del is carried through in app endix A

Decomp osition

There exist a multitude of metho ds designed for line simplication in computer aided cartog

raphy see eg McM for an overview and discussion of some of these algorithms We will

give some details on two such metho ds fo cusing on features related to the application of the

pro cedures as approximation op erators in a multiscale setting

The DouglasPeucker metho d DP also known as the anchorandbuoy algorithm is one

of the traditional line simplication pro cedures widely used in CAC Details on the algorithm

will not b e given here we will only lo ok into asp ects concerning the use of the op erator in a

Multimo del

If we denote all the set of all p ossible PLCs represented with m p oints as C the Douglas

Peucker may b e formulated as the selection approximation see section P C

DP m

C where C is the set of all curves represented with m p oints The notation C

() () ()

indicates that the size of the space is dep endent on the tolerance The approximant to

is an ordered the curve C is generated as C P C such that the the p oint vector p

i=1

subset of the p oints in the original curve C

Further the distance b etween the two curves or the error of the approximation should

b e less or equal to the tolerance C RC

The renement op erator R is dened as follows The missing p oints in C are generated

as the p erp endicular pro jections of the corresp onding p oints in C down to the approximant

Arge and Dhlen ADWM have prop osed a variant over the DouglasPeucker scheme

where basically the p oints in the approximant are allowed to b e slightly p erturb ed in order

to yield higher data reduction In this case the approximated vector is not a subset of the

original We will term this approximation op erator P The asso ciated renement op erator

is the same as for P

Since we have the choice of two approximation op erators we get some freedom in choosing

a multiscale structure We have a well dened metric thus we may extend our multiscale

structure to a multiresolution structure and obtain a range of mo dels corresp onding to a set

of tolerances which in turn refer to certain ranges of scales

If want to use the P op erator which picks a subset of p oints from the original curve the

selectionmo del in section b ecomes a natural choice This scheme oers a particularly

compact and simple structure see algorithms and

The P op erator may b e used in a decomp osed mo del where the given level is generated

essentially as a sum of the coarsest representations and the corresp onding dierences see the

We assume that we want the set of PLCs to b e dep endent ie that changes in a certain variant should

propagate to the following curves

PART I I MULTIMODELS

Examples of Multimo dels

two variants of reconstruction algorithms and The latter scheme assumes that the

k k

renement op erator satisfy P P P which is not needed in the former

i j i

In app endix A we nd an implementation of a selection mo del using P and a decom

p osed mo del with P

Piecewise Linear Surfaces

Denition

We will now study a surface dened as a piecewise linear function dened over a triangulated

irregular network TIN The surface will b e termed PLS Before we give a denition of

the PLS we need the denition of a triangulation There exist many formulations of the

triangulation problem and the one given here is adapted from DLR

Denition Triangulation Assume we have a set of distinct points in IR

W fv g fx y z g i m

i i i i

We denote the orthogonal projection of W to IR as V fx y g i m

i i

3 t

Let IR be a region with a polygonal boundary so that V The set T fT g

i=1

of nondegenerate disjoint triangles is a triangulation of if each x y V is on a vertex

i i

T of some triangle T and if

i j

i=1

The linear surface S is dened as the set of triangular patches

s S

i=1

where s is the planar segment dened as the barycentric combination of three p oints

s fp IR j p uT v T w T u v w u v w g

i i i i

1 2 3

where T T and T are the p oints in W which pro jections in V dene the triangle T of

i i i i

1 2 3

the triangulation T

The triangulation of V is indeed not unique but vary according to underlying triangulation

metho d and in some cases the ordering of the p oints in V See Sch for a discussion of

triangulation metho ds Let us assume we use the well known Delauney metho d see eg LS

for an outline of certain prop erties of this triangulation and decriptions of two algorithms for

constructing such a TIN

One formulation of the PLS as a digital mo del could b e

P LS hW D S i

where W is the p oint set D is a the Delauney transformation that order W into a triangula

tion and nally S which is the denition of the surface We observe that the transformation

of the mo del is actually comp osed of two separate op erations

Note that this b oundary need not b e convex

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Piecewise Linear Surfaces

Another p ossibility is to consider the PLS as a completed triangulation and the surface

denition

S i P LS hfT g

i=1

is the set of triangles and S the transformation into a surface where T fT g

i=1

Arithmetics

We observe that the rst formulation of the PLS as

P LS hW D S i

allows a trivially denition of arithmetic op erators as the attribute vector is a set of D

p oints The addition and subtraction op erators can b e dened as we did with the PLC in

assumed that the p oint sets are of equal size and the structure yields an ab elian group

However the denition

S i P LS hfT g

i=1

is not so straight forward regarding the design of arithmetic op erators

We observe that the triangle patches T s are highly correlated as a p oint in a triangle

most often is shared with several other triangles and that changes in the p oint set V may

violate the triangulation requirements However further investigation on arithmetic op erators

on such PLSs is b eyond the scop e of the thesis a go o d p oint to start would b e Flo

An implementation of a multiedition PLS as a delta mo del based on the formulation

P LS hW D S i

is carried through in app endix A

Decomp osition

During the last years several schemes for data reduction of PLSs has b een prop osed

Floriani Flo outlines two fundamentally dierent approaches concerning the relations

b etween the approximant and the original

In the subdivision approach the triangulation is constructed by recursively inserting p oints

in triangles The insertion of a new p oint splits the old triangle into three new triangles

The approximation of such a triangulation is basically p erformed by successively removing

vertices until some tolerance requirement is met thus yielding a coarser triangle network

The other approach which in Florianis work is based on the Delauney triangulation is

the construction of a pyramidal structure of surfaces of successively ner resolutions In each

step a set of p oints are inserted such that the Delauney requirements are maintained

Both approaches are to b e considered as constrained approximation op erators ie that

they pro duce data reduced mo dels of an original where the error of the approximation is less

than a given tolerance

The two metho ds would certainly require dierent multiscale schemes However the dis

cussion will not b e carried through in this thesis and we restrict ourselves to mo del multiscale

PLSs as trivial Multimo dels as implemented in app endix A The approximation op erator

PART I I MULTIMODELS

Examples of Multimo dels

to b e used is based on the socalled data dependent triangulation In this metho d the tri

angulation is not restricted to a planar pro cess but takes into account that a surface is to

b e generated over the triangulation Thus the data or the D comp onent of each p oint

b ecomes imp ortant The principles of datadep endent triangulation is outlined in DLR

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Summary

In this part the Multimo del was introduced and developed to provide a general integrated

metho d for multiple mo deling ie the structuring and management of several variants of an

initial mo del

Diering in the prop erties of the variants regarded as digital mo dels two main classes

of Multimo dels were prop osed in addition to the trivial explicit Multimo del The multi

edition mo del handles a variety of mo dels which are considered to b e of the same resolution

or accuracy The multiscale structure encompasses variants of the same mo del but diering

according to resolution or scale

The multiedition mo del was shown to b e an inherently compact structure Two sp e

cializations of the multiedition mo del were prop osed the pseudodelta mo del and the delta

mo del The former was to b e considered as an inconsistent structure while the latter indeed

was consistent

The multiscale structure was considered inherently consistent Two variants were intro

duced the extremely compact selection mo del and the decomp osed mo del which compact

ness may vary according to the mo dels involved

According to the prop osed informal metho dology of Multimo deling details were given

on mo deling piecewise linear curves PLCs and piecewise linear surfaces PLSs It was

straightforward to design b oth multiedition and multiscale mo dels of the PLC but the

PLS was shown to represent more complex problems which we did not attempt to solve

To overcome the problems one might resort to the exibility of the recursive prop erty of the

digital mo del that allows a mo del to b e partitioned into simpler structures which in turn may

b e more easily managed in the Multimo del setting However a semi multiedition mo del

was trivially designed

One might design other variants of Multimo dels than those prop osed in this part and

indeed the compactness and consistency prop erties should b e more detailed studied Empiric

research involving a variety of types of ob jects b oth spatial and nonspatial should b e carried

through to gain more rm insight in Multimo deling and its p otential

However we have managed to outline a mechanism which is capable of handling sets of

arbitrary digital mo dels in a homogeneous way Within the same framework we are able to

mo del multiple representations of b oth chalk and cheese ob jects such as texts of varying

length and contents and gradually more wellstructured mo dels such as curves and surfaces

We have prop osed a set of alternative representations diering in what types of mo dels they

are able to handle and in the degree of compactness and consistency they oer

The Multimo del is indeed motivated from but not restricted to geographic information

The Multimo del is a general approach hop efully useful in other areas characterized by the

need to manage multiple mo dels ie a base mo del along with its variants In computer aided

geometric design CAGD hierarchical surface editing is a technique where a basesurface is

edited by adding dierencesurfaces in order to create new versions of the initial surface It

is a growing and promising research area see eg KW and references therein and is an

example on multiple mo deling outside the GI domain

In Part III we will take advantage of the Multimo del concept and incorp orate it in a

suggested augmented map concept termed Metamap

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Part III

METAMAP

Outline

In this part we prop ose an augmented map concept and call it Metamap The development

is based on the discussions and results in Part I and Part II

We op en the part with a brief discussion of information integration in GIS ie basically

the pro cess of linking spatial and nonspatial information

We then claim that the notion of top ology ie the set of various relations interconnecting

ob jects of dierent kinds is covering central asp ects of information integration

The elab oration of Metamap is started with a general description at a conceptual level

We introduce the notion of the geographic element an abstract representation of the real

world phenomenon to b e mo deled indep endent of any spatial or nonspatial descriptions

We stress the imp ortance and implications of the geographic duality ie that a geographic

element have b oth a top ographic description and a thematic interpretation

Then we design the Metamap element as the main building blo ck in a Metamap our

contribution to the augmentation of the map concept We supply the Metamap with a set of

topologies as a exible structure supp orting information integration

The top ographic and thematic ob jects managed in a Metamap are supp osed to b e Multi

mo dels customized for use in a cartographic setting

Metamap is summarized as an ob ject mo del and we then discuss the compliance of the

Metamap concept according to the denition of the augmented map concept

Based on a prop osed informal metho dology for Metamap mo deling we design a simple and

limited Metamap called MINIMAP In app endix B a mo dest implementation of MINIMAP

will b e carried through to demonstrate key asp ects of b oth the Multimo del mechanisms and

the Metamap principles

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Chapter

Integration of Geographic

Information

One way of characterizing a GI system is from the information integration p oint of view This

brings our attention to how dierent pieces of information p erhaps extremely heterogeneous

and varying over time are b eing linked to spatial ob jects In a broader setting information

integration covers interconnections of the spatial ob jects and of the various thematic ob jects

Recalling that the denition section of the augmented map mo del encompasses b oth

spatial and nonspatial features information integration also b ecomes an imp ortant issue in

computer aided cartography based on such a map mo del With the supp ort of the Multi

mo del structures developed in Part II the information to b e integrated will span a range of

scale or resolutions may vary over time and b e accessible in dierent editions as a result of

generalization pro cesses One of the main goals in the Metamap development is to imp ose

some coherence and consistency to this somewhat chaotic set of fragments of information

Before describing some details on the integrating mechanisms a brief review of trends in

spatial information integration is given

Strategies in spatial information integration

Shepherd characterizes information integration in GIS like this She

Current thinking and research in GIS tends to ignore the information richness of

the real world

In the same article Shepherd gives an overview of the eld of geographic information

integration He classies the classic approaches to information integration in GIS as the two

main directions outlined b elow

The composite map model This is equivalent to the raster concept see section

where a map is a stack of regularly gridded sheets overlays sup erimp osed to the same

geographic area Each cell is given a value corresp onding to an attribute thematic

information Spatial and thematic information is in this way inseparably bundled

Integration of Geographic Information

within the same overlay The dierent overlays are integrated indirectly in the sense

that the values of the corresp onding cells together comp oses the information asso ciated

with that particular lo cation

The georelational model In this mo del top ographic and thematic information is sep

arated and stored in dierent tables linked together by a common key which is the

unique identier of the ob ject In this manner spatial features may b e accessed through

the nonspatial features and vice versa The mo del has b een adopted by many vector

based systems

It is natural to draw the parallel to the ob jecteld dichotomy describ ed in The

comp osite map mo del is clearly based on a eld mo del where each p oint in space is assigned

thematic information attributes The ob ject approach viewing the world as distinct spatial

ob jects assigned various information corresp onds to basic idea b ehind the georelational

mo del

Shepherd nds these current approaches somewhat limiting and suggests three directions

to follow in search of more consistent and coherent integration strategies

Multimedia databases

Interactive hypermedia

Virtual reality systems

Details of these approaches will not b e given here

The Metamap development will b e founded on an approach which to a certain extent is

related to the georelational mo del We nd that the separation of spatial and nonspatial

issues reects the notion of geographic duality and in section we expressed preferences

towards the ob ject approach which the georelational mo del is based on However we nd

the dep endency to relational databases severely limiting as Shephard also do es and we will

in the next section prop ose an approach to information integration based on the notion of

topology

Top ology as information integration

In a wide p ersp ective one may view top ology as a set of relations b etween ob jects In several

application areas more sp ecialized interpretations do indeed exist

In mathematics top ology is a discipli ne of its own studying eg how certain geometric

prop erties are conserved during continuous transformations see eg Des for an

introduction to the sub ject

In data communication theory top ology refers to how various no des in a network are

connected eg star top ology ring top ology and tree top ology Sta The relations

b etween the no des are mainly of one kind the connected to relation

The top ology concept as represented in the GIS standard VPF VPF section

VPF Topology may stand as an example of a common interpretation of top ology in

As suggested by the name the georelational mo del is founded on the principles of the relational database

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Topology as information integration

CAC VPF recognizes four levels of top ology which describ es relations b etween the

geometric entities no des edges and faces The lowest level level or the spaghetti

structure is in fact characterized by the lack of structure as only no des and edges and

their co ordinates are represented Level is essentially a nonplanar graph like Level

is a planar graph The richest description is given on Level where the surface is

partioned by a set of mutually exclusive and exhaustive faces where edges only meet

at no des The relations used in building these top ologies are typically start node

node right face left face right edge left edge and so forth end

Applying the top ology concept in information integration yields a rich variety of asso ciations

and it is p ossible to distinguish b etween dierent classes of relations and dierent collections

of relations constituting dierent top ologies

Consider the following example of this sp ecial use of top ology The two city ob jects Oslo

and Trondheim has each a relation is part of to the ob ject Norway Oslo is related to

south of asso ciation The cities have each two is represented as Trondheim by the is

relations giving each a spatial description and a nonspatial description Note that the

relation is south of is not only supp orting spatial inquiries but also acts as a constraint

barring Oslo to b e relo cated north of Trondheim during a generalization pro cedure if the

scale is small enough this is more likely to happ en than one should b elieve at rst glance

Such constraining relations may b ecome imp ortant to ols in various generalization pro cedures

The main motivation b ehind the design and utilization of top ological structures is to make

op erations on related ob jects or sets of them as ecient as p ossible A sparse top ology may

cause the op erations to take to o much time in contrast to a rich top ology where computations

are rapidly executed This is an example of the classical timespace tradeo in computer

science The top ological structures requires extra storage space and time to build The

tradeo implies that the design of the top ological utility system should start with a thorough

analysis of the wanted p erformance of the system eg recognization of the most frequent

used op erations and how long time they are allowed to take

The example indicates that there is indeed p ossible to build large and complex top ological

structures dealing with spatiotemp oral information and in section an attempt is made

to classify dierent top ologies useful in an augmented map concept

The next chapter initiates the construction of the Metamap framework by giving a de

scription of the mo del at a conceptual level

PART I I I METAMAP

Integration of Geographic Information

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Chapter

Metamap The Conceptual Mo del

In this chapter we present an augmented map concept according to denition The mo del

will b e called Metamap and is based on discussions and results achieved so far in the thesis

Metamap will b e treated on two levels in this chapter First a general description is given

of the main structures At this level Metamap is to b e considered as an abstract mo del and

could also b e classied as a metamodel From this metamo del it will b e p ossible to derive a

multitude of related sp ecic mo dels

At the next level the metamo del is further developed as additions and sp ecications are

added The mo del will no longer b e abstract and will b ecome suitable as a basis for the

implementation to come in app endix B To emphasize that this is one of many p ossible

realizations of the Metamap concept this particular version will b e called MINIMAP

All inn all Metamap will b e treated at three dierent levels since it will b e realized as an

implementation in app endix B There are many degrees of freedom in this pro cess The Paper

Map Mo del may b e augmented in several ways the description of a sp ecic augmentation

may b e formulated as several dierent ob ject mo dels and nally one ob ject mo del certainly

implies a variety of implementations Figure illustrates the multitude of p ossibilitie s

The rst step in the Metamap development is to identify the main building blo ck the

geographic entity

Geographic entities

According to the discussion on the KantianDescartian dichotomy in section and the

preferences expressed towards the ob ject view of the world a fundamental characterization

of Metamap will b e that it is p ossible to uniquely identify ob jects existing in the real world

Such ob jects will b e called geographic elements in the thesis Note that on this level the only

assumption made is that it is p ossible to give the ob ject a spatial characterization included

a lo cation Further details on how to p erform the spatial description of the entity is not given

The concept of space is not so straightforward as it may seem at rst glance Gatrell Gat reviews and

discusses dierent approaches to the space problem distingui shi ng b etween metric spaces such as the familiar

Euclidian space a space based on the so called Manhattan metric and nonmetric spaces eg topological

spaces and conceptual spaces

Metamap The Conceptual Mo del

PAPER MAP MODEL

? ?

METAMAP: CONCEPTUAL MODEL !!!!!!!!!!! !!!!!!!!!!! ? !!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!! ? !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ? !!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!! METAMAP: !!!!!!!!!!! OBJECT MODEL !!!!!!!!!!! !!!!!!!!!!! ? !!!!!!!!!!!

METAMAP: IMPLEMENTATION CCCCCCCCCBBBBBBBB@@@ CCCCCCCCCBBBBBBBB@@@ CCCCCCCCCBBBBBBBB@@@ CCCCCCCCCBBBBBBBB@@@ CCCCCCCCCBBBBBBBB@@@ ...... CCCCCCCCCBBBBBBBB@@@@@@@ ...... CCCCCCCCCBBBBBBBB@@@@@@@ ...... @@@@ ?

...... @@@@ ?

Figure The degrees of freedom in mo deling

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Geographic duality

GEOGRAPHIC ENTITY NONSPATIAL ENTITY

Notre Dame Liturgy

The rise and fall of Communism in USSR Communism as ideology

Bank of Switzerland Your bank account

Norway Utopia

University of Oslo Geographic Information Science

Figure Geographic and nonspatial entities

neither is it stated anything concerning how to classify interpret or describ e its nonspatial

asp ects eg which thematic information that is assigned to it

The geographic entity is a neutral entity representing the most abstract description of

a phenomenon in the real world p ossible to lo cate in space and time In its most primitive

implementation the geographic entity is equivalent to a unique identier typically a name

or some sort of co de

Stretching the idea to its limits one may reach to the conclusion that practically all

phenomena in the real world could b e formulated as geographic elements This is certainly

not true even though a large number of real world entities indeed is of geographic character

To illuminate the distinction b etween geographic entities and other real world entities some

examples are given in gure

An imp ortant feature of the geographic entity is that it may b e decomp osed into a set

of other geographic entities Norway may typically b e decomp osed into a set of counties

Going the opp osite direction Norway Sweden and Denmark may b e aggregated into a new

entity Scandinavia Note that the decomp ositionaggregation takes place on an abstract

level and is indep endent of any top ographic or thematic description

Geographic duality

The traditional map concept denition section is strongly spatially oriented and in

section it is claimed that this also applies to many GI systems

As embedded in the georelational mo del section geographic information is char

acterized by the fact that it has b oth a top ographic spatial and a thematic nonspatial

comp onent This is what the term geographic duality is referring to in the thesis

Shepherd She underlines that access to the information from the spatial domain or the

thematic domain or a combination implies increased eciency in navigation and retrieval

of spatiotemp oral information A philosophy pap er from the Europ ean standardisation or

ganization CEN Com states that the new Europ ean GI standard in progress shall b e

based up on a conceptual scheme capable of handling spatial and nonspatial identications

In Metamap spatial and nonspatial information will b e treated as truly equal asp ects of a

geographic entity

Both the spatial and nonspatial information is mandatory even though the description

may take a minimalistic form such as a p oint in space or a oneletter thematic co de How

ever the top ographic description has to b e unique while there may exist several thematic

PART I I I METAMAP

Metamap The Conceptual Mo del

classications

As an example consider the geographic entity Svenner lighthouse Indeed it is p ossi

ble to give a unique description of the building the ground plan the height of the tower the

conical shap e the exact lo cation in geographic co ordinates and so on The thematic interpre

tation may very well b e of two categories b oth as a navigational aid and a building lo dging

the lighthouse keeper These two thematic descriptions would most probably b e disjoint in

most asp ects

Integration of the geographic entity the notion of geographic duality and the Multimo del

concept in Part II yields what will b e called the Metamap element This will b ecome the

main ob ject in the Metamap concept

The Metamap element

The Metamap element is the fundamental building blo ck in the Metamap construction as

it is the representation or mo del of a real world geographic entity

The Metamap element is dened at a conceptual level as follows

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBTHEMATIC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCBBBBBBBBBBBBBBBTOPOGRAPHIC BBBBBBBBBBBBBBB CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBINFORMATION CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCBBBBBBBBBBBBBBBDESCRIPTION BBBBBBBBBBBBBBB CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

Figure The Metamap element

Denition Metamap element A Metamap element is representation of a geographic

entity It is supplied with an unique identier and a mandatory dual description of both its

unique topography or set of spatial features and the

possibly multiple thematic or nonspatial classication

A Metamap element will if allowed by the context o ccasionally b e referred to as just an element In the

MINIMAP implementation in app endix B the Metamap element will b e referred to as a geographic element

of implementational reasons

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

The Metamap element

These descriptions are Multimodels integrating a set of variants according to

dierent scales or resolutions

dierent editions generalizations and represented at

dierent moments or intervals in time

A Metamap element may be an aggregation of several elements

Figure illustrates the denition

The Metamap element may not seem particularly useful standing all alone Some struc

ture has to b e imp osed to make dierent elements play together The structure that will glue

together a collection of Metamap elements is provided by top ologies as further addressed in

section

We will give some details on the top ographic and thematic descriptions of a Metamap

element in the next two sections

Topographic elements

The top ographic description of a Metamap element is essentially an entity of its own a

top ographic or spatial element It is a unique geometric denition of the shap e and size of

the geographic entity indep endent of p ossible relations to other spatial or nonspatial ob jects

Initially there are indenitely many types of spatial elements so there is a need for a

classication of main categories of such ob jects The classication will b e inuented by the

map purp ose and the available technology for representation and manipulation of spatial

entities Nowadays the eld of computer aided geometric design CAGD is likely to b e the

main contributor of relevant metho ds exp eriences a vigorously development For an overview

of the eld see eg Far

The degree of reality characterizing an augmented map concept will heavily dep end on

how close to reality the spatial ob jects are mo deled This is due to the fact that p eoples

p erception of reality is oriented towards to the physical domain and in particular towards

spatial features

The planar pro jection of rivers coastlines and county b orders in the Paper Map Mo del

is clearly not close to our p erception of reality but rather a highly abstract derivation Still

we are so used to this map mo del that most p eople feel quite comfortable with the abstract

representation

The rst steps towards a more realistic mo del are to lib erate ourselves from the planar

pro jections and to introduce D mo deling There are many solutions and many ways to

accomplish this task and we will suggest and implement only mo dest renements see section

and app endix B The top ographic elements will b e restricted to a few and quite simple

geometric structures

Thematic elements

Ideally the thematic elements encompass information on any format that may b e handled

digitally The various multimedia data types such as plain text images audio and video

This is indeed not a trivial question how to p erceive the reality In fact the question has through a couple

of millenni ums b een a favorite theme among philosoph ers

PART I I I METAMAP

Metamap The Conceptual Mo del

t naturally as thematic elements Today most GI systems only supp ort strongly formatted

text such as records and o ccasionally digital images as eg satellite recordings

However according to the scop e of the thesis there will b e paid little attention to the rich

domain of nonspatial information To ensure the conceptual comprehensiveness of Metamap

we merely assume that it is p ossible to design and implement realistic thematic mo dels

Top ological structures

As mentioned in section the term top ology will in the thesis refer to the structures of

relations b etween various entities in a map included augmented maps

In the Metamap context four main classes of top ology will b e recognized

Map topology This is simply the relations linking Metamap elements together into

a Metamap These may b e primitive relations typically constituting an unordered

collection

Primary topology The denition of the Metamap element implies that it may b e ag

gregated of other Metamap elements This gives rise to a set of is aggregated of

relations that will b e referred to as primary top ology

Secondary topology The Metamap element has a mandatory and unique spatial de

scription and a mandatory p ossible multiple nonspatial description The structure

of the relations b etween the top ographic and the thematic domains are characterized

as secondary top ology

Tertiary topology This is describing the optional relations b etween spatial entities and

b etween nonspatial entities thus yielding two sp ecializations

topographic topology which is how the spatial ob jects are related to each other

and

thematic topology describing the various p ossible nonspatial interconnections

In the next sections some details will b e given on the web connecting the elements in a

Metamap Certain asp ects of relations as top ology is treated in MS and BS

We will now give some details on the dierent top ologies

Map top ology

The map top ology is simply the collection of Metamap elements constituting a Metamap

The most simple realization is the unordered set but more ecient structures as an ordered

set dened as a linked list may b e preferred in real applications

We stress that on this level anything but the unique identiers of the elements are known

The map top ology is therefore to b e considered to b e the rst do or to op en when dealing

with a Metamap and a mechanism to traverse the elements at the most abstract level

In order to make denition of Metamap sensible the map top ology is mandatory

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Topological structures

Primary top ology

The primary top ology is a more useful structure than the map top ology describing the asso

ciations b etween several Metamap elements Unlike the map top ology the primary top ology

is optional in the sense that the Metamap may consist of single Metamap elements without

any relations

Secondary top ology

The secondary top ology denes relations b etween the top ographic spatial description and

the thematic nonspatial information In other words the secondary top ology is the key to

the duality of geographic information

There are two main approaches to the design of secondary top ology The top ology may

b e indirectly derived from the common geographic entity or it may b e explicitly mo deled

as asso ciations b etween the spatial and nonspatial domain Since the derived secondary

top ology always is available the explicit secondary top ology is optional

An explicit top ology will consist of relations from one spatial ob ject or one or more

thematic ob jects

Tertiary top ology

Despite that the tertiary top ology is of an optional nature a well designed structuring of the

spatial resp ectively nonspatial domain is p otentially one of the most imp ortant challenges in

the development of a Metamap realization

Spatial top ology

The spatial description of a geographic entity yields a geometric ob ject The relations b etween

such ob jects will together form the spatial top ology of a Metamap

The main scop e with top ology in general and spatial top ology in particular is to provide

a utility structure to supp ort a variety of op erations on the ob jects The ma jority of such

op erations like generalization pro cedures computation of volumes areas and distances and

nding shortest paths in a network will probably include some kind of spatial searching or

sorting To sp eed up computations like these prepro cessing algorithms are widely used Such

metho ds are essentially constructing more or less sophisticated top ologies of relations eg

hierarchies and treestructures making it easier and faster to p erform traversals among the

ob jects This has b ecome a discipline of its own computational geometry see eg PS

for an introduction

The relations may not exclusively supp ort computational pro cedures but for example

express dierent constraints

Other asso ciations may b e more directly expressing geometric prop erties eg if an ob ject

inherits some features from another ob ject The MINIMAP development in chapter will

utilize such relations

PART I I I METAMAP

Metamap The Conceptual Mo del

Thematic top ology

As the thematic elements fall outside the scop e of the thesis and are mo deled and implemented

in a primitive manner the same limiting approach is taken when mo deling the tertiary top ol

ogy involving nonspatial entities As a matter of fact there will not b e introduced such

relations at all in the development of MINIMAP thus leaving the nonspatial features as

indep endent ob jects only linked to the spatial entities by the secondary top ology see section

The Metamap

In the Metamap context a map or a Metamap is essentially a collection of Metamap ele

ments as describ ed in section supplied with a set of utility mechanisms for eg presenta

tion purp oses To defend this suspiciously simple approach we try to describ e a traditional

top ographic map as a Metamap

Assume that we have a collection of Metamap elements Let the main element b e the

top ographic surface The other elements p oints curves and areas are resting on this

mo del of the terrain According to a certain pro jection and a view given eg as a rectangle

in geographic co ordinates we pro ject the spatial description of the Metamap elements to a

plane The height contours may b e derived from the top ographic D surface as curves dened

by sectioning the terrain at given constant elevation values

The pro jections are made according to a sp ecied scale resolution a certain moment in

time and an edition This is accomplished since b oth the spatial and nonspatial description

of a Metamap element are Multimo dels

So far only spatial ob jects are included in the map and only as p oints curves and areas

How to draw or present these geometric is not yet given We then supply the Metamap

by a legend which on basis of the thematic classication asso ciated with each spatial entity

gives rules on how to present the ob ject eg that roads of a given category represented in

a certain scale and edition should b e drawn eg as dashed lines with a given thickness and

color The legend should also give similar information on how to print eg names asso ciated

to ob jects ie how to present the nonspatial information

Finally the Metamap should supply information on the formatting of the pro jected ob

jects eg such that it would b e p ossible to exp ort the information according to a standard

say VPF see section

Based on the example a more precise denition of a Metamap is given as follows

Denition A Metamap A Metamap is essential ly a collection of Metamap elements

with the fol lowing additional information and functionality

Information

Topological structures as outlined in section

Map legend ie a description or dictionary on how to present depending on export

format and Multimodel parameters both spatial and nonspatial information

Functionality

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Ob ject mo del

Edit Insert update delete and access elements change topology and so on given

certain Multimodel parameters

Select a part window of the Metamap according to map window and Multimodel

parameters

Present a window of the Metamap according to legend and presentation format

eg a D visualization system or a traditional paper map projection

Export a window of the Metamap according a given export format eg a certain

GIS standard like the VPF

Note that there are no other relations b etween the Metamap elements than that they all

are part of the same unordered set The necessary structures for traversal and retrieval are

provided by top ologies b etween the elements as explained in the next section

Also note that the Multimo del basis of the top ographic and thematic descriptions in each

Metamap element ensures that the Metamap has access to and may b enet from mechanisms

embedded in the Multimo del structure eg data reduction op erators and other generalization

utilities

This is the conceptual denition of the augmented map concept suggested in the thesis

The sections to come will give some more details and eventually in app endix B a mo dest

implementation is carried through based on denition

In the next section Metamap is formulated as an ob ject mo del

Ob ject mo del

Based on the results and discussions in this part we illustrate the Metamap concept with an

ob ject mo del The mo del may b e considered to b e a metamodel since we are able to design a

variety of more detailed and sp ecialized ob ject mo dels These mo dels may vary signicantly

In app endix B we will present one of many p ossible instantiations of the metamo del shown

in gure

The metamo del is quite simple The main class the Metamap is essentially an aggregation

of one or more GeographicElements This ob ject is the most simple and abstract represen

tation of a geographic phenomenon as describ ed in section The aggregation constitutes

the map top ology of the Metamap see section

The various GeographicElements have some sort of relations with a TopographicElement

which basically is related to a Multimo del spatial description Likewise it is asso ciated to

a ThematicElement the nonspatial part of the information which also is based on a Mul

timo del We assume that some sort of Multimo del library is provided eg the generic

MULTIMOD developed in app endix A

Between the thematic and top ographic elements we may optionally have dened a sec

ondary top ology see section This top ology is considered as a utility structure en

hancing traversals and navigation in the Metamap structure Note that we also have an

By Multimo del parameters we refer to sp ecication of which scale edition and momentinterval in time

that is wanted

A map window is a description of the view of the Metamap that is wanted given eg as a D b ox a

rectangle in geographic co ordinates or a description of a vertical section

PART I I I METAMAP

Metamap The Conceptual Mo del

optional_ optional_ topographic_ thematic_ topology MetaMap topology

1+ is_associated_to is_associated_to GeographicElement

optional_primary_topology TopographicElement ThematicElement optional_secondary_topology

information_described_as information_described_as

MultiModel

Figure Metamap as metamo del

implicit secondary top ology as the spatial and nonspatial descriptions b oth are related to

the GeographicElement

Between the TopographicElement ob jects and b etween ThematicElement ob jects we

have optional tertiary top ologies see section ie relations b etween the ob jects to

facilitate op erations such as generalization op erators

We close the thesis by briey evaluating Metamap as an augmented map concept

Metamap as an augmented map concept

In order to ensure compatibility with denition of an Augmented Map Mo del a verication

of the Metamap as dened in is carried through as follows

Realistic representation The Metamap concept supp orts spatial mo deling in D and

time and is able to incorp orate a wide variety of thematic mo dels In this way a

Metamap may b e characterized as fairly realistic However by not taking advantage of

the p otential of the Metamap degenerated mo dels close to the Paper Map Mo del can

b e designed based on the Metamap concept

Decomp osition and duality A Metamap is a collection of Metamap elements which are

unique geographic entities asso ciated with b oth spatial and nonspatial descriptions

Advanced and exible data structures Metamap is indierent to the structures repre

senting top ographic and thematic representation However a well designed Metamap

should obviously take advantage of the state of the art in available technology If not yet

another instance of the Ptolemiac Paradox as outlined in the Introduction is generated

Unique location combined with multiple thematic interpretations A Metamap is based

on geographic entities with uniquely dened spatial features included lo cational de

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Metamap as an augmented map concept

scription and asso ciated with multiple thematic descriptions

Dierent scales moments or intervals in time and editions The Multimo del basis of

the spatial and nonspatial information ensures the ability to handle these variations

Seamless representation The concept of the map window sp ecifying an arbitrary part

of the reality mo del to b e presented clearly implies a seamless representation

Presentation independence One of the characterizations of a Metamap is that it should

b e able to supp ort a multitude of presentations or exp ort formats such as D p er

sp ective views traditional D maps and proles The concept of the map legend hides

much of the formatting information that characterizes a sp ecic presentation adding

yet another dimension of presentation indep endence

As a conclusion we may say that the Metamap complies well with the Augmented Map Mo del

Still degenerated instances can b e derived that may not b e characterized as an AMM but

rather as close to the Paper Map Mo del

In the next chapter we take a step further towards an implementation with the presen

tation of the MINIMAP mo del

PART I I I METAMAP

Metamap The Conceptual Mo del

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Chapter

MINIMAP Towards an

Implementation

In this chapter we give an example of the use of the Metamap concept in designing a more

sp ecic map mo del which we will call MINIMAP We start by prop osing an informal mo deling

metho dology

Metamap mo deling

The design of a map mo del should b e based on a thorough analysis of the actual real world

phenomena to b e mo deled In addition well dened sp ecications of the functionality and

capability of the systems based on the map mo del are of vital imp ortance in the design

pro cess

The pro cess of Metamap mo deling may b e describ ed by an informal metho dology

Identify the geographic phenomena to b e mo deled according to the purp ose of the map

and the world view the application is based up on Sp ecify a set of geographic elements

to represent the real world geographic entities

Design a set of classes of spatial descriptions to cover the selected geographic entities

Outline the various types of thematic descriptions needed to supply nonspatial infor

mation

Investigate and decide what kind of top ological structures that will ensure a certain

level of p erformance ability

Decide the cardinality or dimension of the Multimo del structure eg if the informa

tion should b e allowed to vary according to b oth scale time and edition

Design the Multimo del structures see section

From the system sp ecications op erations should b e dened covering

edition of the Metamap including generalization op erators and

presentation and exp ort routines including sp ecication of how to present the map

the map legend

Based on the informal metho dology we design the very simple map mo del MINIMAP

MINIMAP Towards an Implementation

MINIMAP

We start by presenting the very banal world view which MINIMAP is based up on

Geographic elements

We restrict the MINIMAP to mo del basically three classes of geographic phenomena the

terrain or the surface of earth the o cean and buildings Note that on this level we do not

say anything ab out the spatial or nonspatial characteristics of these classes of ob jects

Some details of the spatial descriptions are given in the next section

Topographic elements

The main idea b ehind the spatial structures in the Metamap of this thesis is that the re

ality may b e decomp osed into a main ob ject a unique topographic surface representing the

spherical surface of the Earth and a collection of ob jects the buildings that are scattered

throughout this surface The surface may b e said to support all the other ob jects

The surface is understo o d to b e the solid ground of the Earth This term has several

geological interpretations referring to various layers of ro cks and sediments In this approach

the surface is dened to b e the top ography after removing

Man made features such as buildings and roads

vegetation eg forests and

water b o dies ie lakes glaciers rivers canals and the entire o cean

Further for the case of simplicity the top ography is assumed to b e an explicit surface The

top ography may b e expressed as a bivariate function in spherical co ordinates S IR IR

such that

The value of the function is equivalent to the elevation at the geographic p oint

corresp onds to latitudes in degrees in the northern hemisphere and

corresp onds to latitudes in the southern hemisphere Corresp ondingly is the

longitude in degrees east of the Greenwich meridian and is the longitude west of

the zero meridian

Attached to this naked top ographic surface various spatial features may b e identied In

the thesis the selection of such ob jects is limited to features of two dierent classes The

surface may b e said to support these ob jects

The ocean

The feature o cean O will simply b e characterized by a constant radius z IR thus

neglecting tidal variations It will b e assumed that every p oint of the top ography b elow

this level is part of the sea o or

If the earth is dened as the volume E f j S z g IR and dene O

as the volume limited by the o cean level O f j S z g IR we have

the b o dy of the o cean is implicitly dened as the volume b etween the sea level and

the sea o or dened as

IR n E O

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

MINIMAP

Boxes

The b ox ob ject is ment to symbolize man made features such as buildings A b ox

B is parametrized by four D p oints representing the ground o or as an arbitrary

quadrilateral and a parameter h representing the height of the building

B hp p p p hi p IR h

1 2 3 4 i

Note that this gives a unique D description of the b ox assuming the walls and ceiling

to b e p erp endicular resp ectively parallel to the ground o or

However the b ox is dened on a zero level plane To give the b ox a correct elevation

we need a simple relation to the supp orting surface The b ox B is supp osed to derive

the elevation z from the surface S in the following manner

z min S p

i=1

assuming the p oints are in the interval

The b ox may b e considered as a consistent representation with resp ect to changes in

the surface

Note that the spatial elements should b e multiply mo deled as describ ed in Part II This implies

that within one ob ject several variants of essentially the same spatial ob ject is handled The

variants may dier according to scale time and edition

Thematic elements

According to the scop e of the thesis there will b e paid relatively mo dest attention to the rich

domain of nonspatial information Still to make the Metamap conceptually comprehensive

we include one class of thematic element

Text

A text is dened to b e an arbitrary collection of characters assuming they are dened

in one of the standard character sets

As with the top ographic elements the text is assumed to b e a Multimo del

Topological structures

We have in fact already dened a tertiary top ological relation in describing how the b ox

derives its elevation from the surface

We will not design any primary top ologies ie relations b etween geographic elements

neither any secondary relations ie top ology b etween thematic and top ographic elements

Still we have an implicit secondary top ology in the sense that a geographic element have

b oth a thematic and a top ographic description

We introduce one more tertiary top ology b etween the spatial ob jects by the concept of

supporting and supported elements

PART I I I METAMAP

MINIMAP Towards an Implementation

An top ographic element is said to supp ort a collection of other elements if a generalization

of the elements should eect the supp orted elements in a sp ecied manner

An example of such a top ology could b e that a translation generalization of a dened area

of the surface should imply a corresp onding translation of ob jects supp orted by the area say

a group of buildings

This sp ecial kind of dep endencies are frequently used in D and D illustration applica

tions grouping and ungrouping of ob jects and is a well established concept in computer

graphics in general see eg FDFH chapter

In our case the surface of the Earth is default supp orting all other spatial ob jects In fact

the derivation of the elevation of the Box class is an example of a sp ecied relation b etween

supp ortingsupp orted elements

We close the chapter with an ob ject mo del formulation of MINIMAP

Ob ject mo del

By extending and sp ecializing the metamo del in gure according to the discussion in this

section we get the following ob ject mo del

supports MetaMap

GeographicElement

TopographicElement ThematicElement

Box

1+ MultiModel MultiModel

supports

DigMod Surface Text

Figure MINIMAP an ob ject mo del

Note that we do not represent the o cean explicitly but implicitly as a given z value For

more details on the ob ject mo del see the MINIMAP implementation in app endix B

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

MINIMAP

PART I I I METAMAP

MINIMAP Towards an Implementation

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Summary

In this part we have accomplished to develop an example of an augmented map concept as

prop osed in Part I denition

We emphasized the imp ortance of the information integration asp ect in mo deling geo

graphic information It was claimed that the use of the top ology concept is an ecient to ol

in the pro cess of interconnecting the various spatial and nonspatial ob jects

The Metamap was designed as an augmented map concept basically dened to b e a set

of Metamap elements with some additional functionality eg to p ermit various visualizations

of a certain view of the Metamap

The Metamap element is the representation of the abstract geographic entity the neutral

thing includin g a unique spatial description and one or several thematic interpretations The

two latter elements are b oth assumed to b e Multimo dels as describ ed in Part II thus allowing

the Metamap element to b e represented in a multitude of scales editions and moments or

intervals in time

The Metamap was shown to b e compliant with denition of the augmented map concept

We prop osed a Metamap mo deling metho dology in order to design a particular map mo del

We then used the metho dology to develop MINIMAP as a limited example of a Metamap

MINIMAP will b e implemented to some extent in app endix B

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Part IV

CONCLUSIONS AND

IMPLEMENTATIONS

Chapter

Results

The main achievement of the thesis is that we have shown that it is p ossible to augment the

traditional map concept to meet new challenges provided by computer aided management of

spatiotemp oral information

We have accomplished to develop an augmented map concept starting on a conceptual

level and resulting in a computer implementation as a conceptually comprehensive mo del

We have followed an original approach in the sense that similar frameworks to our knowledge

are not suggested in contemporary literature

In this chapter we summarize some of the results achieved during the pro cess

Cartography and GIS

We identied and dened the Paper Map Mo del which has its ro ots in ancient cartography

and showed that it is the core in any GI system to day We claimed that this Ptolemaic

paradox represents a b ottleneck in computer based handling of geographic information

In contrast to some trends in GI science which tend to drift away from cartography and

into the realms of database theory primitive geometric mo deling and low level algorithmic

optimization we claim that GI science may b enet from not discarding the notion of the map

By regarding the long and rich traditions of cartography as a rm foundation we prop osed

to augment the Paper Map Mo del as a step towards a new core mo del of the real world for

use in GI applications

The concept of cartographic generalization inherently generates multiple representations

of geographic ob jects varying according to scale edition and moments or intervals in time

This problem has b een only partially addressed in GI research frequently with emphasis on

multiscale representations of simple spatial structures We suggested a more general and

integrated approach by introducing the Multimo del

Multimo dels

The Multimo del is basically a high level conceptual mechanism for managing collections of

mo del variants in a homogeneous manner ie that we may p erform the same op erations on

the various sets without paying attention to details concerning the actual representation of

the ob jects

Results

We designed a few sp ecialized Multimo dels diering basically due to the prop erties of the

mo dels they could handle Key op erations asso ciated with the Multimo dels were algorith

mically outlined The dierent Multimo dels was shown to represent the variants in various

degrees of compactness and consistency

An implementation of a generic Multimo del customized to handle generalized geographic

information was carried through We dened Multimo dels to some detail for piecewise linear

curves and piecewise linear surfaces

The Multimo del concept should not b e restricted to the domain of spatiotemp oral infor

mation We may exp erience a spino eect due to the versatility of our approach in the sense

that the Multimo del concept may b e adopted and adapted in other application areas such

as computer aided geometric design CAGD and general information systems

Metamap

Metamap was developed as one of many p ossible realizations of the augmented map concept

The two main ideas b ehind the Metamap concept are

To provide structures for integration of the duality of geographic information ie that

a thing has b oth a spatial and nonspatial description

To manage the multitude of generalized variants

The integration of spatial and nonspatial information is maintained by a set of top ologies

or relations b etween various ob jects The top ologies may also b e used to imp ose constraints

and to facilitate navigation in collections of geographic elements

Metamap is incorp orating the Multimo del concept in the sense that b oth the spatial and

the nonspatial ob jects are assumed to b e Multimo deled

We presented an informal metho dology of Metamap mo deling and applied it in the de

sign of a limited Metamap called MINIMAP MINIMAP was rudimentary implemented to

demonstrate key features of the augmented map mo del

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Chapter

Future Research

The interdisciplinary nature of cartography and GI science is the background for the somewhat

grand scop e of the thesis

We have accomplished to design and develop structures which we have used to augment

the traditional map concept However even if the framework is conceptually comprehensive

the lack of details in several areas is obvious

We realize the substantial amount of research needed for developing the ideas in the

thesis into well functioning to ols in a GI setting In this chapter we will try to sort out

some main areas of interest in p ossible future research on augmented map concepts Due to

the wide span of research areas we see the advantages of integrated and co ordinated eorts

A research strategy implying a number of single indep endent pro jects is likely to yield yet

another instance of the Ptolemaic paradox

Cartography and GIS

In Part I we briey investigated cartographic generalization We claimed that this is a

fundamental issue in geographic mo deling resulting in variants of dierent kinds However we

did not go into detail concerning dierent generalization op erators A thorough understanding

of the nature of the generalization pro cess is of vital imp ortance in GI mo deling A successful

development of the Multimo del concept as describ ed in Part II is eg quite dep endent of

such knowledge

There are several promising ongoing research pro jects in this eld see eg BM for an

overview of contemporary trends The structures and functionality of generalization should

supply some of the main premisses in development of the Multimo del concept which would

b enet from incorp orating results from this area

Multimo dels

In Part II we prop osed the Multimo del as a mechanism for management of mo del variants

and claimed that the various sp ecialized Multimo dels provided certain degrees of consistency

and compactness However we did not carry through any empirical exp eriments to supp ort

or verify these assertions Such investigations should b e of central interest in further research

on Multimo dels

Future Research

We only made realistic implementations of multirecords multicurves PLCs and mul

tisurfaces PLSs Clearly to investigate the versatility of the Multimo del concept a wide

variety of digital mo dels should b e implemented The thesis have a geometric approach

and sp ecial attention should therefore b e paid to validate the Multimo del approach in the

thematic domain of geographic mo deling

Such implementations may also reveal other classes of Multimo dels than those prop osed

in the thesis Certainly detailed algorithms covering a complete set of op erations should b e

the result of more extensive research in Multimo deling

A detailed survey of related approaches should b e carried out Adaptations has to b e

made to incorp orate existing techniques such as approximation metho ds within CAGD

Questions concerning database issues will so oner or later have to b e answered Such

considerations are absent in the thesis and ecient Multimo del implementations will to a

large extent b e dep endent of the database implementations

We have o ccasionally claimed that the Multimo del approach may b ecome useful in other

areas than GI science A survey and study of alternative application areas like facelifting

techniques in CAGD see KW should b e of interest

Metamap

The prop osed realization of the augmented map concept Metamap was rudimentary imple

mented in app endix B and key asp ects of the augmented map concept was demonstrated

We b elieve that many alternative formulations of the AMM may b e successfully carried

through and such alternatives should b e investigated However it will require extensive

research to reveal the strengths and weak p oints of an augmented map mo del

An investigation of Metamap should imply an extensive exploration of a wide range of

real world cases Such exp eriments would b e extremely useful in the pro cess of rening and

adjusting the concept

The notion of top ology as an information integration to ol would p erhaps represent the

most challenging part of a Metamap study The design of such top ological structures implies

a large degree of freedom and assumes thorough knowledge in b oth the application areas and

their p erformance demands data mo deling in general and algorithmic design in particular

An imp ortant issue to consider is how to integrate the Metamap approach with existing

concepts and metho ds in GI science Sp ecial attention should b e paid to investigate the

compliance with selected GI standards The Metamap concept should also b e adjusted to

make smo oth integration with other areas p ossible like visualization and computer graphics

At last one should not exclude the p ossibility of applying the Metamap concept in other

areas than GIS

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Future Research

PART IV CONCLUSIONS AND IMPLEMENTATIONS

Future Research

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Chapter

Epilogue

In chapter we asked some questions which initiated the quest resulting in the concepts of

the Multimo del Part II and the Metamap Part III In Part IV we carried through some

rudimentary implementations of the concepts and p erformed some exp eriments to highlight

some selected asp ects of Metamap and Multimo dels

The initial questions in chapter have b een an inspiration throughout the thesis We now

close the thesis by giving some direct remarks on how we have managed to solve the problems

outlined by the questions and

Answer Crossing contours

No direct solution of this problem has been given However an implicit answer may be

found in the D capability of an augmented map concept If we represent the topography or

terrain as an explicit surface we may derive the height contours by horizontal ly sectioning

the surface Since the terrain may be represented as a Multimodel with varying resolution

or scale and the surface is explicit the sectioning process wil l guarantee that crossing of

contours wil l not occur Figure shows a contour map of the terrain visualized in gures

B and B after reducing the number of points in the triangulation from to We do

not observe any crossing contours such as displayed in gure and

Answer Dislo cation

The dislocation problem as il lustrated in gure may be approached with the help of the

topology mechanisms outlined in By dening a tertiary topology as a relation constraining

the road object to be located inside the polygon representing the island we may prevent such

dislocations

Answer Multi scale structures

We have indeed made an attempt to suggest a solution to the problem outlined in question

In Part II we introduced the Multimodel concept to address this and related problems

Implementations and experiments were performed in appendix A to highlight selected features

of the Multimodel and we showed that under certain assumptions the Multimodel is able to

integrate a set of variants in a compact and consistent manner

However the contours are not optimally shap ed from a cartographic p oint of view Thus to design

algorithms for data reduction of PLSs yielding satisfactory contours remains a challenge

Epilogue

Figure Contour map after heavy data reduction of terrain

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Epilogue

Answer Augmentation of the map concept

In Part I we claimed that the traditional map model called the Paper Map Model PMM

is inadequate as a basis in GI systems

An augmentation of the PMM was proposed and in Part III we developed such an aug

mented map concept called Metamap Later in appendix B we carried out a minor imple

mentation that indicated the potential of the concept as a core map model in GI systems

In the thesis a lofty framework has b een developed and some mechanisms have b een

sp ecied Within the framework some relatively limited and simple problems have actually

b een solved We may clearly have found solutions by more straight forward and simple

approaches Still we b elieve that the more comprehensive approach will pay o when the

complexity of the problems increases

However we realize that b oth the Multimo del and the Metamap concepts need substan

tial amounts of renements enhancements and corrections to b ecome useful in realworld

applications Still we hop e we have accomplished to establish a starting p oint from where it

could b e p ossible to develop an augmented map mo del that may b ecome the core of future

GI systems

Epilogue

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

Bibli og raphy

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Time Applications PhD thesis Norwegian Institute of Technology

AD Erlend Arge and Morten Dhlen Simplication of piecewise linear curves Tech

nical rep ort Center of Industrial Research Oslo ISBN

AD Erlend Arge and Morten Dhlen On Generalization of Geometric Ob jects In

Neste Generasjon Geograske Informasjonssystemer pages Trond

heim Norway Desember Institutt for Kart og Oppmaling Universitetet i

Trondheim Norges Tekniske Hgskole

ADWM E Arge M Dhlen G Westgaard and G Misund Ecient numerical line

generalization Technical rep ort Center of Industrial Research Oslo February

ISBN

Ans R W Anson editor Basic cartography volume ICA International Carto

graphic Asso ciation ISBN

AS ystein Andersen and Kari Strande editors Kart og Kartbruk Universitetsfor

laget AS edition ISBN In Norwegian

AS Jostein Amlien and Leif Srb el Automatisert generalisering av geomorfologiske

temakart Meddelelser fra Geogrask Institutt Naturgeogrask serie Rapp ort

nr Geogrask Institutt University of Oslo In Norwegian

Ass ICA International Cartographic Asso ciation editor Basic cartography vol

ume ICA International Cartographic Asso ciation ISBN

ATB Khaled K AlTaha and Renato Barrera Temporal Data and GIS An Overview

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ATSS Khaled K AlThala RT Sno dgrass and MD So o Bibliography on spatiotem

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BK Bud P Bruegger and Werner Kuhn Multiple Topological Representations Tech

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BM Barbara P Butteneld and Rob ert McMaster editors Map generalization Mak

ing rules for know ledge representation Longman Scientic and Technical

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Bro A Lloyd Brown The Story of MAPS Little Brown and Company

BS Olav Mork Bjrnas and David Skogan MultiMo dels and spatiotemp oral mo del

ing in Ob jectOriented GIS Masters thesis Norwegian Institute of Technology

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Bur PA Burrough Are GIS data structures to o simple minded Computer

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Com Comite Europeen de Normalisation TC WG Philosophy pap er revised

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Cur James P Curran editor Compendium of Cartographic Techniques Elsevier

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Des JV Deshpande Introduction to Topology Tate McGrawHill Publishing Com

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DL Morten Dhlen and Tom Lyche Decomp osition of Splines In Tom Lyche and

Larry L Schumaker editors Mathematical Methods in CAGD and Image Pro

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DLR Nira Dyn David Levin and Samuel Rippa Data dep endent triangulations for

piecewise linear interpolation IMA Journal of Numerical Analysis

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DP D H Douglas and T K Peucker Algorithms for the reduction of the number

of p oints required to represent a digitized line or its caricature The Canadian

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Far Gerald Farin Curves and Surfaces for Computer Aided Geometric Design Aca

demic Press Inc ISBN

FDFH JD Foley van A Dam SK Feiner and JF Hughes Computer Graphics

Principles and Practice AddisonWesley second edition ISBN

Flo Leila De Floriani A pyramidal data structure for trianglebased surface descrip

tion IEEE Computer Graphics and Aplications pages March

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FM A U Frank and D M Mark Language Issues for GIS In David J Maguire

Michael F Go o dchild and David W Rhind editors Geographic Information

Systems Vol PRINCIPLES chapter pages Longman Wiley

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Fra AU Frank Spatial concepts geometric data mo dels and geometric data struc

tures Computer Geosciences

Gat A C Gatrell Concepts of Space and Geographical Data In David J Maguire

Michael F Go o dchild and David W Rhind editors Geographic Information

Systems Vol PRINCIPLES chapter pages Longman Wiley

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Go o MF Go o dchild Geographical data mo deling Computer Geosciences

GS Michael F Go o dchild and Yang Shiren A Hierarchical Spatial Data Structure

for Global Geographic Information Systems Graphical Models and Image Pro

cessing

Gup Stephen C Guptill Multiple representations of geographic entities through space

and time In Proceedings of the th International Symposium on Spatial Data

Hand ling volume pages

HB Fred Holroyd and Sarah B M Bell Raster GIS mo dels of raster enco ding

Computers Geosciences

Her I N Herstein Topics in Algebra John Wiley Sons nd edition

Int International Hydrographic Organization COE Working Group Provisional

sp ecications for chart content and display of ECDIS Sp ecial Publication No

International Hydrographic Bureau Monaco May

JA Cristopher B Jones and Ian M Abraham Design considerations for a scale

indep endent cartographic database In Proceedings of the th International Sym

posium on Spatial Data Hand ling Seattle WA

Kot Ciord A Kottman Some Questions and Answers Ab out Digital Geographic

Exchange Standards Technical rep ort Intergraph Corp oration November

KW Johannes Kaasa and Geir Westgaard Mo deling of CAD Mo dels from Measured

Data Technical rep ort Center of Industrial Research Oslo ISBN

LS DT Lee and BJ Schachter Two algorithms for constructing a Delaunay trian

gulation International Journal of Computer and Information Sciences

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Mag David J Maguire An Overview and Denition of GIS In David J Maguire

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Mag David J Maguire The Raster GIS Design Mo del A Prole of Erdas Computers

Geosciences

McM Rob ert B McMaster Automated line generalization Cartographica

McM Rob ert McMaster Map generalization Making rules for know ledge representa

tion chapter Conseptual frameworks for geographical knowledge Longman

Scientic and Technical ISBN

McoCI E Meynen chairman of Comission I I International Cartographic Asso ciation

editor The Multilingual Dictionary of Technical Terms in cartography Franz

Steiner Verlag

MGR David J Maguire Michael F Go o dchild and David W Rhind editors Geo

graphic Information Systems Vol Applications Longman Wiley

MS Gunnar Misund and Bjrn Skjellaug MetaMap en mo dell for handtering

av stedfestet tidsvarierende informasjon SINTEFSI rapp ort STF A

SINTEFSI Juni ISBN Preliminary version In Norwegian

Mul JC Muller Generalization of Spatial Databases In David J Maguire Michael F

Go o dchild and David W Rhind editors Geographic Information Systems Vol

PRINCIPLES chapter pages Longman Wiley ISBN

Nel Mark Nelson The Data Compression Book M T Publishing Inc ISBN

PS Franco T Preparata and Michael Ian Shamos Computational Geometry An

Introduction SpringerVerlag New York Inc ISBN

Rai Erwin Raisz General Cartography McGrawHill Bo ok Company Inc edition

RBP James Rumbaugh Michael Blaha William Premerlani Frederick Eddy and

William Lorensen ObjectOriented Modeling and Design PrenticeHall Inc

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RGM David W Rhind Michael F Go o dchild and David Maguire Language Issues for

GIS In David J Maguire Michael F Go o dchild and David W Rhind editors

Geographic Information Systems Vol APPLICATIONS chapter Epilogue

pages Longman Wiley ISBN

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RM Jonathan F Rap er and David J Maguire Design mo dels and functionality in

GIS Computers Geosciences

RSM Arthur Robinson Randall Sale and Jo el Morrison Elements of cartography

John Wiley and Sons Inc edition ISBN

Sch Larry L Schumaker Triangulation Metho ds In Topics in Multivariate Approx

imation pages Academic Press Inc ISBN

sch The Schlumberger Data Mo del version Schlumberger Technology Corp

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She I D H Shepherd Information Integration in GIS In David J Maguire

Michael F Go o dchild and David W Rhind editors Geographic Information

Systems Vol PRINCIPLES chapter pages Longman Wiley

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Sta William Stallings Data and Computer Communications chapter Lo cal Net

works Macmillian Publishing Company second edition ISBN

Str Bjarne Stroustrup The C Programming Language AddisonWesley second

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TB RV Tooley and C Bricker editors Landmarks of Mapmaking Phaidon Press

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MILSTD

BIBLIOGRAPHY

MULTIMODELS AND METAMAP TOWARDS AN AUGMENTED MAP CONCEPT

App endix A

MULTIMOD A Simple

Multimo del Library

Based on the ob ject mo del describ ed in section and the metho dology and results from

chapter we will in this app endix carry out an implementation using the ob jectoriented

programming language C For details on the syntax and semantics of C see eg

Str We call the implementation MULTIMOD

The goal is to establish a generic class library By generic we mean that the classes should

b e of a general nature able to act as a basis for many types of Multimo dels This implies that

some details has to b e added when it comes to use of MULTIMOD in a sp ecic application

context

We will only present the class denitions and their public op erations Data structures and

op erations that are private or protected will not b e shown following a common tradition in

ob jectoriented development

The implementation is academic in scop e we just want to suggest one of many p ossible

directions to follow and to demonstrate some key asp ects of the Multimo dels Thus minimal

attention is paid to robustness and optimalization of the co de

The co de including the data dep endent triangulation routines is written by the author

A The generic library

We will rst present the three main classes in MULTIMOD and then give some details on

sp ecializations of them

Main structure

The heart of the library is the abstract DigMod class implementing the notion of a digital

mo del according to denition in section

However some core subroutines in the ADapproximation of linear curves are provided by Arge and

Dhlen and the Delauney triangulation is based on a public domain Fortran version of the ClineRenka metho d

MULTIMOD A Simple Multimo del Library

class DigMod

public AppFunc

public

DigModvoid

Access of private data

void setAttNo const int

int getAttNo void const

void setType const TransType

TransType getType void const

virtual void setAtt void const int

virtual void getAtt const int const

char whatAmI void const

The op erations enables us to manipulate the attribute vector and the transformation that

maps the attributes to a real world mo del

DigMod is a sub class of the AppFunc We observe that this class initially is empty By

adding customized op erations dep ending on the application context AppFunc act as an inter

face b etween the Multimo del and the application This will b e shown later in the chapter

class AppFunc

public

AppFuncvoid

The Multimodel is basically a collection of ob jects which are generalized by the DigMod

class

class Multimodel

public

Multimodelvoid

Index EDITIONS

virtual DigMod reconstruct const int

virtual void insert DigMod

virtual void update DigMod const int

virtual void dump void

The class should provide all the necessary op erations to maintain the ordered set of vari

ants In our case we only have paid attention to op erations as describ ed in section

We have also include a dump pro cedure which is supp osed to reveal the internal structure

of the Multimo del esp ecially how the variants are represented Note that Multimodel is an

abstract class

A THE GENERIC LIBRARY

Having established the main structure of MULTIMOD as illustrated with the ob ject

mo del in gure in section we will now design some sp ecializations of the DigMod class

Sp ecializa ti ons of DigMod

The sub classes prop osed in this section follow the categorization of digital mo dels as outlined

in section We design the classes with sp ecial attention to p ossible op erations on sets of

mo dels All following classes are abstract and need sp ecialization b efore taken in use in an

application

Class DigMod represents the most primitive digital mo del with no op erations asso ciated

From this class we successively derive sub classes with increasing degree of complexity and

supplied op erations We start with class PseudoDiff with the op erator called sub

a copyoperator and two op erations for comparing attributes in two mo dels These op erations

will b e needed in the the algorithms and which will b e used in the Multimo del

asso ciated to the PseudoDiff mo dels We choose to use functions when implementing arith

metic op erations we think that the alternative of overloading existing op erators may confuse

the reader

class PseudoDiff

public DigMod

public

PseudoDiffvoid

virtual void copy void

virtual void sub void void

virtual MmBool equalAtt PseudoDiff const int PseudoDiff const int

virtual MmBool equalZero PseudoDiff const int

The next class DiffMod will b e furnished with an addition op erator and inherits all the

op erations in PseudoDiff The DiffMod mo dels are assumed to form an ab elian group see

denition

class DiffMod

public PseudoDiff

public

DiffModvoid

virtual void add void void

The class ApproxMod is supplied with an approximation op erator and we assume that a

metric see denition is available in the sp ecializations of this class such that we will b e

able to build true multiresolution structures

MULTIMOD A Simple Multimo del Library

class ApproxMod

public DiffMod

public

ApproxModvoid

virtual void approx ApproxMod const double

We end the design of mo del classes with RefineMod which adds a renement op erator to

all the pro cedures supplied by the sup erclasses In addition we have designed a set of utility

pro cedures

class RefineMod

public ApproxMod

public

RefineModvoid

virtual void refine RefineMod int int

void setParamNo int

int getParamList void

int getParamNo void

void setParamListNo int int

In the next section we design a suit of Multimo dels capable of handling the various cate

gories of digital mo dels

Sp ecializa ti ons of Multimodel

We start with the most simple Multimo del the TrivialMM This is however a useful Multi

mo del allowing collections of arbitrary kinds of digital mo dels to b e handled in a homogeneous

manner We observe that the virtual op erations from class Multimodel now is implemented

and thus it is p ossible to instantiate ob jects of the class TrivialMM

class TrivialMM

public Multimodel

public

TrivialMM void

TrivialMM DigMod

DigMod reconstruct const int

void insert DigMod

void update DigMod const int

A CUSTOMIZING MULTIMOD

The PseudoMM is an integration of variants of ob jects with the common sup erclass PseudoDiff

Note that the public part of the class is almost identical to the class TrivialMM The main

dierences are hidden in the private parts of the class Of this reason we will not display the

denitions of the classes DifferenceMM SelectionMM and DecomposedMM which all are quite

similar in terms of public op erations

class PseudoMM

public Multimodel

public

PseudoMMvoid

PseudoMMPseudoDiff PseudoDiff

DigMod reconstruct const int

void insert DigMod

void update DigMod const int

The generic part of MULTIMOD is then completed and complies well with the ob ject

mo del in gure in section We will now take a lo ok at a p ossible customization of the

library for use in a primitive GI application

A Customizing MULTIMOD

The rst thing to do is to design the op erations that we want all mo dels in the application

to supp ort

Application functionality

We assume that our application is going to handle geographic information We want to

b e able to pro duce printed maps ie planar pro jections and supply the interface class

AppFunc with the the virtual pro cedure printMap MmPoint max MmPoint min which

p erforms a planar pro jection of the mo del given a rectangular map window see denition

of the augmented map concept In addition we want to b e able to generalize the ob ject

see denition of cartographic generalization The virtual op eration generalize

p erforms a generalization sp ecied as an ane transformation andor sp ecied by a given

tolerance scale

class AppFunc

public

AppFuncvoid

virtual void printMap const

virtual void generalize

MULTIMOD A Simple Multimo del Library

We are now ensured that every mo del in our system supp ort these two op erations at

least as a dummy pro cedure if the op eration do not have any meaningful interpretation in a

certain mo del

In the next sections we implement a variety of classes of ob jects that will b e needed in

our application

Arbitrary text

In our application we want a class for the management of thematic ob jects of type arbitrary

text It is not p ossible to imp ose any structure of these ob jects which may vary in contents

and length and our only choice is to implement the class as a derivation of the plain DigMod

class

class Text

public DigMod

public

Textvoid

Textchar tt

Text operator const Text

void printMap const

void generalize

We are now able to instantiate our rst Multimo del by the following co de segment

Text tThis is Call the Text constructor

Text tour first

Text tMlutiModlle

Text tHello world

TrivialMM tmmt Initialize a TrivialMM with t

tmminsertt Insert models

tmminsertt

tmmreconstructprintMap max min Access models and

tmmreconstructprintMap max min print to map

tmmreconstructprintMap max min

Running the co de yields

Printing model of type Text to map file This is

Printing model of type Text to map file our first

Printing model of type Text to map file MlutiModlle

Printing model of type Text to map file Hello world

The example may seem a little far out in a GIS setting but according to the academic scop e we allow

ourselves this kind of freedom

A CUSTOMIZING MULTIMOD

Observing an error in the fourth variant we correct it by an up date and check the result

Text correctionMultimodel Construct new model

tmmupdatecorrection Update old model

tmmreconstructprintMap max min

The output of the application is as

Printing model of type Text to map file This is

Printing model of type Text to map file our first

Printing model of type Text to map file Multimodel

Printing model of type Text to map file Hello world

Encouraged by this minor achievement we carry on with the slightly more advanced

record class

Records

We mo del the record type as a number of arbitrary words This class should comply with

the denition of the abstract PseudoDiff class and we design the sp ecialization Record

This implements the subtraction op erator dened as a virtual pro cedure in the sup erclass

PseudoDiff in addition to utility op erations inherited from other sup erclasses

class Record

public PseudoDiff

public

Record void

Record const char

Record operator const Record

void copy void

void sub void void

void printMap const

void generalize

MmBool equalAtt PseudoDiff const int PseudoDiff const int

MmBool equalZero PseudoDiff const int

void setAtt void const int

void getAtt const int const

We will now make a Multimo del of the four vewords records written to the les

MULTIMOD A Simple Multimo del Library

recorddta This is a long paper

recorddta This is a boring paper

recorddta This is a long thesis

recorddta This is my master thesis

We run a little example by constructing the records and collect them in a PseudoMM

Multimo del We then dump the representations of the variants to highlight the internal

structure of the Multimo del At last we call the printMap op eration for all the variants to

check if the reconstruction pro cedure works as it should

Record recrecorddta Call the Record constructor

Record recrecorddta

Record utilrecrecorddta

PseudoMM pmmrec utilrec Initialize a PseudoMM

pmminsertrec Insert variants

pmminsertrec

pmmdump Dump representations

pmmreconstructprintMap max min Reconstruct and print to map

pmmreconstructprintMap max min

Executing the co de we rst get a dump of the representations

Dumping representation of variant nr This is a long paper

Dumping representation of variant nr ZERO ZERO ZERO boring ZERO

Dumping representation of variant nr ZERO ZERO ZERO long thesis

Dumping representation of variant nr ZERO ZERO my master ZERO

We observe the ZEROs where no change has taken place relative to the last variant As

assumed in section we may store zeros or sequences of zeros more compact than an

explicit representation The mo dels are reconstructed correctly as

Printing model of type Record to map file This is a long paper

Printing model of type Record to map file This is a boring paper

Printing model of type Record to map file This is a long thesis

Printing model of type Record to map file This is my master thesis

In the two next sections we will lo ok at Multimo dels of piecewise linear curves

Piecewise linear curves DPapproximation

In section we presented a formulation of the piecewise linear curve PLC as two kinds

of digital mo dels diering in their approximation op erator The DouglasPeucker op erator

gave rise to the very compact selection Multimo del Based on the results from section we

design a class PlDpCurve as a sub class of ApproxMod The class implement op erations derived

from the sup erclasses and some utility pro cedures The class is aggregating an instance of

the class DataD to hold the p oints but we nd that this only includes uninteresting details

and omit the description of this class

A CUSTOMIZING MULTIMOD

class PlDpCurve

public ApproxMod

public

PlDpCurve void

PlDpCurve char fname

PlDpCurve const int pntno

PlDpCurve operator const PlDpCurve

void copy void

void add void void

void sub void void

void approx ApproxMod const double

void printMap const

void generalize

MmBool equalAtt ApproxMod const int ApproxMod const int

void setAtt void const int

void getAtt const int const

void setNo const int

DataD getCurve void const

void setCurve DataD

We make some tests with the PlDpCurve class to demonstrate some features of the selection

Multimo del

We use the sinelike curve we studied in section represented by eleven p oints as

illustrated in gure A The curve is approximated according to a set of three tolerances

We then dump the representation ie the initial explicit represented curve and the b o ok

keeping vector The curves are then printed in a map format

PlDpCurve dpcrv f Initialize original

PlDpCurve utilcrvf and utility curve

double tol Set tolerances

tol

SelectionMM smmdpcrv utilcrv tol Construct Multimodel

including approximants

smmdump Dump representation

smmreconstructprintMap max min Reconstruct and

smmreconstructprintMap max min print to map

smmreconstructprintMap max min all variants

smmreconstructprintMap max min

This co de generates the following output in addition to map les containing the curve data

MULTIMOD A Simple Multimo del Library

3 'Curve,initial' 2.5

1.5

Y 0.5

-0.5

-1

-1.5

-2 0 5 10 15 20 25

Figure A Initial PLC curve

Running Douglas Peucker approximation scheme

PlDpCurve approx DP stat oldno newno

Running Douglas Peucker approximation scheme

PlDpCurve approx DP stat oldno newno

Running Douglas Peucker approximation scheme

PlDpCurve approx DP stat oldno newno

Dumping representations of variants

Attribute vector of initial model attributes

Beta vector

Printing PlDpCurve to map file

Printing model of type DataD of points to map file

Printing PlDpCurve to map file

Printing model of type DataD of points to map file

Printing PlDpCurve to map file

Printing model of type DataD of points to map file

Printing PlDpCurve to map file

Printing model of type DataD of points to map file

We see that the approximations yields curves of and of the original p oints By

checking the vector with original attribute vector and the approximants showed in A we

see that this representation is correct

A CUSTOMIZING MULTIMOD

3 'Approximant,tolerance=1.0,4points' 'Approximant,tolerance=0.25,7points' 2.5 'Approximant,tolerance=0.1,9points'

1.5

Y 0.5

-0.5

-1

-1.5

-2 0 5 10 15 20 25

Figure A Approximants of initial PLC

MULTIMOD A Simple Multimo del Library

Piecewise linear curves ADapproximation

In section we designed a decomp osed Multimo del of PLCs based on another approximation

op erator the ArgeDhlen algorithm Basically this metho d allows minor p erturbations of

the p oints in order to optimize the data reduction p erformance The class PlAdCurve is

derived from the RefineMod class which basically add a renement op erator to the sup erclass

ApproxMod

class PlAdCurve

public RefineMod

public

PlAdCurve void

PlAdCurve char fname

PlAdCurve const int pntno

PlAdCurve operator const PlAdCurve

void copy void

void add void void

void sub void void

void approx ApproxMod const double

void refine RefineMod int int

void printMap const

void generalize

MmBool equalAtt ApproxMod const int ApproxMod const int

void setAtt void const int

void getAtt const int const

void setNo const int

DataD getCurve void const

void setCurve DataD

We then integrate a set of PlAdCurves in a DecomposedMM class in order to study some

asp ects of this Multimo del We will not display any co de for these examples since it is

essentially the same statements as in the DPapproximation example However we include

an output from the reconstruction of the original curve

Reconstructing variant no

Refining PlAdCurve from to points

Adding PlAdCurves

Refining PlAdCurve from to points

Adding PlAdCurves

Refining PlAdCurve from to points

Adding PlAdCurves

Printing PlAdCurve to map file

Printing model of type DataD of points to map file

A CUSTOMIZING MULTIMOD

3 'AdCurve,initial' 2.5

1.5

Y 0.5

-0.5

-1

-1.5

-2 0 5 10 15 20 25

Figure A Approximants of initial PLC

We observe that the pro cedure follows algorithm with stepwise renement and addi

tion of dierences

We use again the sinefunction A but now represented with p oints in the original

curve see gure A We use the tolerances and which generates curves of

and p oints resp ectively The set of approximants are shown in gure A

To highlight the ability of ADapproximation to move p oints in order to increase the data

reduction rate an enlargement of the original curve an two of the approximants are shown in

In gure A we have plotted the representation of the decomp osed Multimo del We

observe the coarsest approximation as an explicit representation while the three other variants

are given as successively dierences thus clustering around origo

To study the the dierence vectors in more detail we have made two closeups in gure

A and A

We observe that the magnitudes of the p oints representing the dierences are small By

insp ection we see that the ma jority of the p oints in the dierence according to the original

curve is in the interval which is not unexp ected since the tolerance

of the rst approximation was indeed By the use of standard compression techniques

see eg Nel this phenomenon may yield very compact representations using signicantly

less storage than compared to an explicit representation

MULTIMOD A Simple Multimo del Library

3 'AdApproximant,tolerance=0.5,4points' 'AdApproximant,tolerance=0.05,15points' 2.5 'AdApproximant,tolerance=0.01,31points'

1.5

Y 0.5

-0.5

-1

-1.5

-2 0 5 10 15 20 25

Figure A Approximants of initial PLC

'AdCurve,initial' 2.54 'AdApproximant,tolerance=0.05,15points' 'AdApproximant,tolerance=0.01,31points'

2.52

2.5

2.48 Y

2.46

2.44

2.42

2.4 5 5.2 5.4 5.6 5.8 6 6.2 6.4 6.6 6.8 7

Figure A Detail of AD approximation

A CUSTOMIZING MULTIMOD

3 'Decomposed,tolerance=0.5,4points' 'Decomposed,tolerance=0.05,15points' 2.5 'Decomposed,tolerance=0.01,31points' 'Decomposed,original,65points'

1.5

Y 0.5

-0.5

-1

-1.5

-2 0 5 10 15 20 25

Figure A Decomp osed delta representation of PLC

0.6 'Decomposed,tolerance=0.05,15points' 'Decomposed,tolerance=0.01,31points' 'Decomposed,original,65points'

0.4

0.2

Y 0

-0.2

-0.4

-0.6 -0.5 0 0.5 1 1.5

Figure A Decomp osed delta representation of PLC detail I

MULTIMOD A Simple Multimo del Library

0.01 'Decomposed,tolerance=0.05,15points' 'Decomposed,tolerance=0.01,31points' 'Decomposed,original,65points'

0.005

Y 0

-0.005

-0.01 -0.01 -0.005 0 0.005 0.01

Figure A Decomp osed delta representation of PLC detail II

A CUSTOMIZING MULTIMOD

Piecewise linear surfaces

In section we briey discussed piecewise linear surfaces dened over triangulations We

did not prop ose any implicit storage scheme for multiscale PLSs due to certain problems

asso ciated to this task Still we make a trivial Multimo del representation based on the

class PlSurface The denition of the class is not displayed here since it is very similar

to the PlDpCurve The approximation op erator implements the datadep endent pro cedure

mentioned in

Figure A Original Delauney triangulation

We make a test based on p oint samples The initial triangulation shown as XY plot

in gure A without any data reduction is a Delauney triangulation The following output

gives some statistics on the various triangulations

Points in triangulation

Edges in triangulation

Triangles in triangulation

Tolerance

Points in original triangulation

Reduction in percent of original

Nr of edges swapped

LOP criterion Delauney

Points in triangulation

Edges in triangulation

Triangles in triangulation

Tolerance

Points in original triangulation

Reduction in percent of original

The D and D surface visualiza tion s in the thesis is generated by software written by Per yvind Hvid

steen SINTEF SI

MULTIMOD A Simple Multimo del Library

Nr of edges swapped

Nr of edges rejected to swap

LOP criterion Angle between normals

Points in triangulation

Edges in triangulation

Triangles in triangulation

Tolerance

Points in original triangulation

Reduction in percent of original

Nr of edges swapped

Nr of edges rejected to swap

LOP criterion Angle between normals

The approximated triangulations are illustrated in gures A and A The p oints

omitted in the triangulations are marked as single dots We observe esp ecially in A the

long and thin triangles that are characteristic for datadep endent triangulations in contrast

to the more wellformed triangles of the Delauney triangulation in A

Figure A Reduced triangulation with p oints

The surfaces dened over the triangulations are shown in gures A A and A

The surfaces are rendered to visualize the triangle patches structure We observe that the

approximation with the largest tolerance is quite distorted compared to the original

This should not b e surprising since the zvalues of the surface vary b etween and

A CUSTOMIZING MULTIMOD

Figure A Reduced triangulation with p oints

Figure A Surface dened over p oints

MULTIMOD A Simple Multimo del Library

Figure A Surface dened over p oints

A CUSTOMIZING MULTIMOD

Chalk and cheese

To study the homogeneous asp ects of the Multimo del we consider the following examples

Multimodel divMM Make an array of two Multimodel

of arbitrary kind

divMM tmm Initialize the array with the trivial Multimodel

divMM pmm of text and the pseudo multi model of records

which were constructed in earlier examples

Reconstruct and generate the fourth element in

each of the Multimodels

divMMreconstructprintMap max min

We have two basically dierent Multimo dels a selection mo del and a trivial mo del con

taining dierent types of digital mo dels We see that the Multimo dels are handled on a su

p erclass level thus indep endent on which sp ecic sp ecializations the application deals with

Since the sup erclass AppFunc ensures us that the op eration printMap is implemented in

all sub classes of DigMod and that the pro cedure reconstructindex is an abstract op era

tion in Multimodel we are able to make the reconstructionprinting call without any further

considerations The co de pro duces the following printout

Printing model of type Text to map file Hello world

Printing model of type Record to map file This is my masters thesis

We just exp erienced an example of integrating dierent Multimo dels in a homogeneous

manner We now take a lo ok on how dierent types of digital mo dels are integrated in the

very same Multimo del To achieve this we have to resort to the trivial Multimo del

Initialize the chalkandcheese Multimodel with

a variant from the Text Multimodel from the last example

TrivMM chalkandcheese divMMreconstruct

Insert a variant from the Record Multimodel from the last example

chalkandcheeseinsert divMMreconstruct

Generate and print the two variants in the Multimodel

chalkandcheesereconstructprintMap max min

We should exp ect the output to b e identical of the last example and to our fortune it is

Printing model of type Text to map file Hello world

Printing model of type Record to map file This is my masters thesis

We summarize the elab oration of the generic Multimo del library with an ob ject mo del

Final ob ject mo del

We close the MULTIMOD development with an ob ject mo del of the customization of the

generic library see gure A A

Some minor deviations in the class names and op erations may o ccur

MULTIMOD A Simple Multimo del Library

ApplicationFunctionality

print2Dmap(...) {abstract} TrivialMM DigitalModel Text generalize(...) {abstract}

PseudoDiffMM PseudoDiffModel PlSurface

MultiModel DifferenceMM DifferenceModel Record

SelectionMM ApproximationModel PlDpCurve

DecomposedMM RefinementModel PlAdCurve

Figure A Customization of MULTIMOD

App endix B

MINIMAP A Simple Metamap

Library

In this chapter we make a limited implementation of the MINIMAP as it is describ ed in

section We start with the development of a library based on the metamo del in gure

We use the library to implement the Metamap outlined in the ob ject mo del given in gure

The chapter is closed with some test examples

Like the MULTIMOD development in app endix A the scop e of the implementation is

indeed academic We just want to demonstrate some selected high level mechanisms in a

Metamap and from a GIS p oint of view the examples are far from realistic

The entire co de is written by the author

B The library

Following the ob ject mo del given in gure we start the implementation with the main

class the MINIMAP

class MINIMAP

public

MINIMAP void

MINIMAP GeographicElement

MINIMAP const char double const char

void printMap const int MmPoint MmPoint

void printMap const int const int

MmPoint MmPoint

void generalize const double const double

const int const int

GeographicElement acessGeoElement const int

void addGeoElement GeographicElement const int

APPENDIX B MINIMAP A SIMPLE METAMAP LIBRARY

The class aggregates a set of ob jects of the class GeographicElement The constructors

initialize the ob ject either based on a GeographicElement which is assumed to b e the surface

of the terrain in our map or by a le containing the surface description and a text string

representing the thematic information We have two printMap op erations one printing the

entire map according to a given Multimo del parameter the other printing a selected ob ject

and all its supp orted ob jects see section for the discussion of top ologies of MINIMAP

The printMap pro duces b oth D and D presentations We also implement a very simple

generalization op erator that p erforms translations of spatial ob jects

To maintain the the set of GeographicElements we have supplied the class with the op

erators accessGeoElement which returns a certain geographic element and addGeoElement

which app ends a geographic element to the Metamap and set the supp orttop ology according

to a given parameter

The denition of the class GeographicElement is given as follows

class GeographicElement

public

GeographicElement void

GeographicElement TopographicElement ThematicElement

GeographicElement TopographicElement const char

void printMap const int MmPoint MmPoint

void generalize const double const double

const int

GeographicElement getSupported const int const

void setSupported GeographicElement

int getSupportedNo void

TopographicElement getTopo void

ThematicElement getTheme void

void setZ PlSurface const int

We observe that the class aggregates ob jects of the classes TopographicElement and

ThematicElement The constructor takes two such ob jects as input or alternatively the

thematic element given as a plain text string We nd the printing and generalization op er

ators corresp onding to those given in the MINIMAP class In addition we have a set of utility

pro cedures for maintaining the supp orttop ology

The two classes TopographicElement and TopographicElement are quite identically de

ned and we only display the TopographicElement

B EXAMPLES

class TopographicElement

public

TopographicElement void

TopographicElement Multimodel

void printMap const int MmPoint MmPoint

DigMod accessVariant const int

The class is constructed based on a Multimo del input and we only have two additional

op erations The print pro cedure generates D and D presentations and accessVariant

reconstructs a given variant in the Multimo del

We use the framework to pro duce a few examples

B Examples

According to the academic scop e of the implementation we do not attempt to simulate a

realistic GIS application We will only fo cus on asp ects of the Metamap Issues concerning

Multimo dels will not b e illustrated such examples are given app endix A

Initializa ti on of the MINIMAP

We start our exp eriments by initializi ng a simple MINIMAP consisting of a surface and a

single b ox

Construct a MINIMAP based on a file with a

surface description and a text string

The surface is initially approximated according to

the given tolerance

MINIMAP mapworlddta This is our world

Initialize a box building

Box box

Make a trivial Multimodel of the box

TrivMM mmbbox

Construct a topographic element based on the Multimodel

TopographicElement bmmb

Construct a geographic element based on the topographic object

and a text string

GeographicElement gbox b box

Add the complete description of the building to the map

and let the surface support it

mapaddGeoElement gbox

Print the the complete map represented by the first and

initial variants in the Multimodels here only one variant

mapprintMap max min

The implementation of the Box class is quite trivial an is not explained in any detail

APPENDIX B MINIMAP A SIMPLE METAMAP LIBRARY

Presentation indep endence

By executing the co de we rst get some information on the construction of the surface

Points in triangulation

Edges in triangulation

Triangles in triangulation

Tolerance

Points in original triangulation

Reduction in percent of original

Nr of edges swapped

Nr of edges rejected to swap

LOP criterion Angle between normals

The printMap pro cedure gives the following output

Starts printing MINIMAP with elements

Printing geoelement no

Printing model of type Piecewise linear surface to map file

Printing model of type Text to map file This is our world

Printing geoelement no

Printing model of type Box to map file

Printing model of type Text to map file box

The spatial output of the printMap pro cedure is a D contour plot shown in gure B

and the D rendering in gure B This is an example of the presentation indep endence of a

Metamap Note that the o cean is not explicitly mo deled but merely hardco ded as a certain

zlevel

Generalization

To demonstrate the simple translation generalization we move the b ox with the following

statement

mapgeneralize

The call implies a translation of all spatial ob jects supp orted by the surface geographic

element no The change is p erformed in the initial Multimo del variant no The result

is shown in gure B Observe that the b ox automatically derives the elevation given by the

surface

Grouping of ob jects

We now want to group a collection of b oxes such that a generalization of the supp orting

element automatically propagates to the collection of supp orted ob jects For this purp ose

we construct an empty geographic element only having the mission to aggregate a set of

supp orted b oxes

Construct MINIMAP

MINIMAP mapworlddta This is our world

Construct box elements

B EXAMPLES

Figure B Contour plot of the Metamap

Figure B D visualization of the Metamap

APPENDIX B MINIMAP A SIMPLE METAMAP LIBRARY

Figure B Simple generalization

Box box

TrivMM mmbbox

TopographicElement bmmb

GeographicElement gbox b box

GeographicElement gbox

Construct the empty geographic element

GeographicElement emptysupport

Add the empty to the surface

mapaddGeoElement emptysupport

Add the boxes to the supporting empty element

mapaddGeoElement gbox

Print the map

mapprintMap max min

Figure B shows the new scene with four b oxes

By p erforming a generalization of the empty ob ject which is indexed as all the sup

p orted b oxes should b e aected

mapgeneralize

Comparing the original scene to the generalized version in gure B we observe that the

whole group of b oxes has b een translated as it was supp osed to

B EXAMPLES

Figure B Group of elements

Figure B Propagation of generalization

APPENDIX B MINIMAP A SIMPLE METAMAP LIBRARY