arithmetic properties of fractal estimates · arithmetic properties of fractal estimates tommy...

Arithmetic Properties of

Fractal Estimates

Tommy Lofstedt

[email protected]

September 28, 2007

Master’s Thesis in Computing Science, 20 creditsSupervisor at CS-UmU: Fredrik Georgsson

Examiner: Per Lindstrom

Umea University

Department of Computing Science

SE-901 87 UMEA

SWEDEN

Abstract

The ideas of Benoıt Mandelbrot, about the geometrical properties of sets which hecalled fractal, was published as late as 1975. Since then, there have for fractals, andmost commonly the estimation of the fractal dimension, been found uses in the mostdiverse applications. Fractal geometry has been used in information theory, economics,flow dynamics and image analysis, among many different areas.

This thesis investigates how the estimated fractal properties of sets and measuresare affected by arithmetic operations on the fractal set. Operations such as projection,union, intersection and product are investigated.

A literature study was conducted to see how the fractal dimension should be affectedby the above operations. These theoretical results have then been compared to empir-ical simulations to see how the estimation of the fractal dimension is affected by suchoperations.

In the 1990’s, the multifractal geometry emerged and started to be used more andmore in the areas where the fractal dimension was used previously. In this thesis, twodifferent, but equivalent ways to estimate multifractal properties have been investigated.These are the generalized dimensions and the multifractal coarse theory. Simulationshave been made to see how the multifractal properties are change by projections.

The results are positive. The theoretical results which are presented here are almostunanimously confirmed by the empirical investigations.

Aritmetiska egenskaper hos fraktala estimat

Sammanfattning

Sa sent som 1975 publicerades Benoıt Mandelbrots ideer om de geometriska egenskaperhos mangder som han kallade fraktala. Alltsedan dess har det for fraktaler, och da framstfor skattningar av den fraktala dimensionen, funnits anvandningsomraden i de mestskilda tillampningar. Bl.a. har fraktalgeometri anvants i informationsteori, ekonomi,flodesdynamik och bildanalys.

Detta examensarbete undersoker hur skattade fraktala egenskaper hos mangder ochmatt paverkas av aritmetiska operationer pa den fraktala mangden. Operationer somundersoks ar bl.a. projektion, union, snitt och produkt.

En litteraturstudie har genomforts for att se hur den fraktala dimensionen borpaverkas av ovanstaende operationer. Dessa teoretiska resultat har sedan jamfortsmed empiriska undersokningar for att se hur skattningen av den fraktala dimensionenpaverkas av de olika operationerna.

Under 90-talet borjade multifraktal geometri, framst med s.k. multifraktala spektran,anvandas i storre utstrackning inom de omraden dar den fraktala dimensionen anvantstidigare. I detta examensarbete har tva olika satt att skatta det multifraktala spektratundersokts, baserat pa de generella dimensionerna och den multifraktala coarse theory.Undersokningar har har genomforst for att se hur det multifraktala spektrat paverkasav projektioner.

Resultaten ar mycket positiva. De teoretiska resultat som har presenteras bekraftasnastan genomgaende av de genomforda empiriska undersokningarna.

Contents

1 Introduction 1

1.1 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Problem Description 5

2.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.4 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Previous Work 7

4 Fractal Geometry 11

4.1 What is a Fractal? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.2 Fractal Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.2.1 Topological Dimension . . . . . . . . . . . . . . . . . . . . . . . . 13

4.2.2 Similarity Dimension . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2.3 Hausdorff Dimension . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2.4 Box-Counting Dimension . . . . . . . . . . . . . . . . . . . . . . 19

4.2.5 Properties of Dimensions . . . . . . . . . . . . . . . . . . . . . . 22

4.3 Means to Estimate Fractal Dimension . . . . . . . . . . . . . . . . . . . 22

4.3.1 Digital Intensity Images . . . . . . . . . . . . . . . . . . . . . . . 23

4.3.2 Point Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Multifractal Geometry 27

5.1 The Generalized Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.2 Multifractal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.2.1 Measures and Local Dimension . . . . . . . . . . . . . . . . . . . 30

5.2.2 The Fine Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.2.3 The Coarse Spectrum . . . . . . . . . . . . . . . . . . . . . . . . 32

5.3 Estimating the Multifractal Spectra . . . . . . . . . . . . . . . . . . . . 33

5.3.1 Legendre Transformation of Moment Sums . . . . . . . . . . . . 33

iii

iv CONTENTS

5.3.2 A Direct Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.3.3 Finding the Fine Spectrum . . . . . . . . . . . . . . . . . . . . . 40

6 Arithmetics on Fractals 41

6.1 Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6.2 Union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6.3 Intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

7 Projections of Sets 47

7.1 Projecting from R2 to R1 and from R3 to R2 . . . . . . . . . . . . . . . 47

7.2 Projection types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

7.2.1 Orthogonal Projections . . . . . . . . . . . . . . . . . . . . . . . 51

7.2.2 Projection with Foci – Central Projection . . . . . . . . . . . . . 52

7.2.3 X-Ray Simulated Projection . . . . . . . . . . . . . . . . . . . . . 54

7.3 Problems with Projections . . . . . . . . . . . . . . . . . . . . . . . . . . 55

8 Projections of Multifractals 57

8.1 Projections of Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

9 Generating Fractals 61

9.1 Generating Homogeneous Fractals . . . . . . . . . . . . . . . . . . . . . 61

9.1.1 Iterated Function Systems . . . . . . . . . . . . . . . . . . . . . . 61

9.1.2 Custom made Random Fractals . . . . . . . . . . . . . . . . . . . 63

9.2 Generating Multifractals . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

9.2.1 Probabilistic Iterated Function System . . . . . . . . . . . . . . . 64

9.2.2 Measures with Recursive Structures . . . . . . . . . . . . . . . . 66

9.2.3 Dimension of a Measure . . . . . . . . . . . . . . . . . . . . . . . 68

10 Results 71

10.1 Estimating the Box-Counting Dimension . . . . . . . . . . . . . . . . . . 71

10.1.1 Rotational Invariance . . . . . . . . . . . . . . . . . . . . . . . . 74

10.2 Custom made Random Fractals . . . . . . . . . . . . . . . . . . . . . . . 75

10.3 Testing the Fractal Theory . . . . . . . . . . . . . . . . . . . . . . . . . 77

10.3.1 Projections of Fractals . . . . . . . . . . . . . . . . . . . . . . . . 77

10.3.2 Product of fractals . . . . . . . . . . . . . . . . . . . . . . . . . . 81

10.3.3 Union of fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

10.3.4 Intersection of fractals . . . . . . . . . . . . . . . . . . . . . . . . 83

10.4 Equivalence of Projection Types . . . . . . . . . . . . . . . . . . . . . . 90

10.5 Estimating the Spectrum of Generalized Dimensions . . . . . . . . . . . 90

10.5.1 Sensitivity to Noise . . . . . . . . . . . . . . . . . . . . . . . . . . 93


10.6 Estimating the Multifractal Spectrum . . . . . . . . . . . . . . . . . . . 97

CONTENTS v

10.6.1 Estimating the Fine Theory . . . . . . . . . . . . . . . . . . . . . 99

10.6.2 Estimating the Coarse Theory . . . . . . . . . . . . . . . . . . . 99

10.6.3 Sensitivity to Noise . . . . . . . . . . . . . . . . . . . . . . . . . . 100


10.7 Testing the Multifractal Theory – Projections of Measures . . . . . . . . 104

10.7.1 The Generalized Dimensions . . . . . . . . . . . . . . . . . . . . 105

10.7.2 The Multifractal Spectrum . . . . . . . . . . . . . . . . . . . . . 106

11 Conclusions 111

11.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

11.2 Restrictions and Limitations . . . . . . . . . . . . . . . . . . . . . . . . . 112

11.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

12 Acknowledgments 113

A Brief Summary of the Set Theory Involved 115

B The Hough Transform 119

References 123

vi CONTENTS

List of Figures

4.1 The Cantor set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.2 The von Koch curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.3 The Sierpinski triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.4 Lines with Topological dimension 1 . . . . . . . . . . . . . . . . . . . . . 14

4.5 Finding Topological dimension by using covers . . . . . . . . . . . . . . 14

4.6 Definition of Similarity dimension . . . . . . . . . . . . . . . . . . . . . . 15

4.7 Definition of the Hausdorff dimension . . . . . . . . . . . . . . . . . . . 19

4.8 Estimating the Box-dimension . . . . . . . . . . . . . . . . . . . . . . . . 23

4.9 Different box shapes for the Box-counting dimension . . . . . . . . . . . 24

4.10 A Digital Intensity Image . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.11 Box-counting on the box fractal . . . . . . . . . . . . . . . . . . . . . . . 26

5.1 Legendre transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.2 Finding the Legendre spectrum . . . . . . . . . . . . . . . . . . . . . . . 36

6.1 Product of two lines in R . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6.2 Intersection of two lines in R2 . . . . . . . . . . . . . . . . . . . . . . . . 44

7.1 Projections of figures in R3 onto a plane . . . . . . . . . . . . . . . . . . 48

7.2 Projection of a set onto a line . . . . . . . . . . . . . . . . . . . . . . . . 49

7.3 How to deduce the theory of projection onto a line . . . . . . . . . . . . 51

7.4 How to deduce the theory of projection onto a plane . . . . . . . . . . . 52

7.5 Example of central projection . . . . . . . . . . . . . . . . . . . . . . . . 53

7.6 Example of X-ray simulated projection . . . . . . . . . . . . . . . . . . . 55

7.7 A digital sundial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

8.1 Projecting a measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

9.1 IFS generated fractal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

9.2 Custom made, random fractal . . . . . . . . . . . . . . . . . . . . . . . . 64

9.3 Multifractal Sierpinski triangle . . . . . . . . . . . . . . . . . . . . . . . 65

9.4 Recursive Iterated Function Systems . . . . . . . . . . . . . . . . . . . . 67

vii

viii LIST OF FIGURES

9.5 Recursive division of a measure . . . . . . . . . . . . . . . . . . . . . . . 69

10.1 Results for estimating the Box-counting dimension . . . . . . . . . . . . 72

10.2 Results for estimating the Box-counting dimension of custom made ran-

dom fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

10.3 Results for estimating the Box-counting dimension of rotated sets . . . . 75

10.4 Box-counting dimension distribution . . . . . . . . . . . . . . . . . . . . 76

10.5 Distribution of projected points . . . . . . . . . . . . . . . . . . . . . . . 76

10.6 Hough transform of fractals with dimension of about 1.26 . . . . . . . . 78




10.10Results for estimating the Box-counting dimension of rotated and pro-

jected sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

10.11Dimension of product of sets . . . . . . . . . . . . . . . . . . . . . . . . 82

10.12Dimension of union of sets . . . . . . . . . . . . . . . . . . . . . . . . . . 84

10.13The intersection between a line and a set . . . . . . . . . . . . . . . . . 85

10.14The intersection with a line depends on the line thickness . . . . . . . . 86

10.15Normalized dimension distribution of the intersection between a line and

a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

10.16The intersection between a line and sets with different dimension . . . . 87

10.17Translating a set to find many intersections with a second set . . . . . . 88

10.18The intersection between two sets with different dimension . . . . . . . . 88

10.19Normalized distributions of the dimensions of the intersection between

two sets with different dimension . . . . . . . . . . . . . . . . . . . . . . 89

10.20Comparing orthogonal projection and central projection . . . . . . . . . 91

10.21Comparing central projection and x-ray simulated projection . . . . . . 92

10.22Building a recursive measure . . . . . . . . . . . . . . . . . . . . . . . . 94

10.23Testing the generalized dimensions . . . . . . . . . . . . . . . . . . . . . 94

10.24Testing noise sensitivity for Gaussian noise . . . . . . . . . . . . . . . . 95

10.25Testing noise sensitivity for Salt & Pepper noise . . . . . . . . . . . . . . 96

10.26Rotating a measure and estimating its generalized dimensions . . . . . . 97

10.27Multifractal spectrum of a measure . . . . . . . . . . . . . . . . . . . . . 99

10.28Fine multifractal spectrum of a measure . . . . . . . . . . . . . . . . . . 100

10.29Estimated multifractal spectrum of a measure . . . . . . . . . . . . . . . 101

10.30Differing multifractal spectra . . . . . . . . . . . . . . . . . . . . . . . . 101

10.31Testing noise sensitivity for Gaussian noise in the multifractal spectrum 102

10.32Testing noise sensitivity for Salt & Pepper noise in the multifractal spectrum103

10.33Rotating a measure and estimating its multifractal spectrum . . . . . . 104

10.34Projection of the generalized dimensions . . . . . . . . . . . . . . . . . . 106

10.35Projection of the generalized dimensions in R3 . . . . . . . . . . . . . . 106

LIST OF FIGURES ix

10.36Projections of the generalized dimensions of a measure on R2 . . . . . . 107

10.37Projections of the generalized dimensions of a measure on R3 . . . . . . 107

10.38Multifractal spectrum of a projected measure . . . . . . . . . . . . . . . 108

10.39The multifractal spectra of projected measures . . . . . . . . . . . . . . 109

B.1 The Hough transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

B.2 The normal representation of a line . . . . . . . . . . . . . . . . . . . . . 120

B.3 Applying the Hough transform . . . . . . . . . . . . . . . . . . . . . . . 121

x LIST OF FIGURES

List of Tables

10.1 Estimating the Box-counting dimension of self-similar sets . . . . . . . . 72

10.2 Estimating the Box-counting dimension of custom made random fractals 73

10.3 The Box-counting dimension of rotated sets . . . . . . . . . . . . . . . . 74

10.4 The Box-counting dimension of rotated and projected sets in R1 . . . . 80

10.5 The Box-counting dimension of rotated and projected sets in R2 . . . . 80

10.6 Results for set product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

10.7 Results for set union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

10.8 The Box-counting dimension of the intersection of two sets . . . . . . . 87

xi

xii LIST OF TABLES

Chapter 1

Introduction

This report describes the master thesis project “Arithmetic Properties of Fractal Esti-mates” performed at the Department of Computing Science at Umea University.

The history of fractal geometry is filled with work done on nowhere differentiable buteverywhere continuous functions and curves, self-similar sets and fractional dimension.The work and the results was initially seen as anomalies, and any suggestion that i.e.a non-differentiable curve might have some practical application was not at all takenseriously. This situation has been completely reversed today.

The first ideas concerning fractal sets came well over hundred years ago, at the endof the 19th century, but have only received practical use since the 1970’s. The use offractal and multifractal geometry today spans from physics, through economy, biology,medicine, to computer science; among many other areas.

More and more applications of fractal geometry to Computer Science are found. Thecurrent applications are, among others, image compression and enhancement, computergraphics and special effects in movies, music generation and pattern classification.

The last example is the most relevant here. Fractal geometry have been used inseveral research projects to classify breast tissue in mammography images. The resultshave varied, but lately, the interest in multifractal geometry have renewed the hope ofusing fractal theory in classification problems.

The goal of this thesis is to document the usefulness of fractal and multifractaltheory in practice, to see whether other parts of fractal and multifractal geometry canbe considered when i.e. building classifiers.

This thesis describes how the fractal dimension is affected by Cartesian product,union, intersection and three different types of projections, and how well the theoryis followed by simulations of the same. For multifractal geometry has been includeddescriptions of orthogonal projections of multifractals, and tests of how well simulationsfollows the theory.

1.1 Prerequisites

This thesis does not require any previous experience in fractal geometry. All theory willbe built bottom-up, and all graduate students and alike should be able to follow thetext. Some of the set theory might be new to some readers, in which case Appendix Aprobably should be read first. Readers without experience in reading mathematicalproofs should still be able to follow this text without reduced comprehension. In most

1

2 Chapter 1. Introduction

cases, the proofs are merely for completeness; sometimes they are omitted. The onlymathematical prerequisites are basic set theory and a moderate competence in calculus;especially the notion of limits is important.

1.2 Thesis Outline

This thesis is organized as follows. Chapter 1 is this introduction. Chapter 2 states theproblem at hand, i.e. describe the goals and purpose of this thesis. Chapter 3 gives abrief summary of the history of fractal geometry, but also describes some of the scientificand real world work that involve fractal geometry.

Chapter 4 introduces the concept of a fractal. A definition of what a fractal is will begiven. The best, and most general definition there is will be stated, and an explanationof what a fractal is will be given. The concept of fractal dimension will be explained, anddescriptions of a number of different (though, in some cases equivalent) definitions areincluded. A number of properties, which all good definitions of fractal dimension shouldpossess will be stated. Finally, a description on how to estimate the fractal dimensionnumerically is given.

In Chapter 5 we discuss multifractal geometry. We start by introducing the gen-eralized dimensions. A full set of infinitely many fractal dimensions, which for somespecial cases equals the classical definitions we met in Chapter 4. We will explain theidea of local dimension, and then introduce the multifractal spectrum. We will end thischapter with a description of a very efficient and accurate way to numerically estimatethe multifractal spectra.

In Chapter 6 we state known results of uni-dimensional fractal geometry, and howthe arithmetic affects the fractal dimension. Most result will be for both the Hausdorffdimension and the Box-counting dimension, but sometimes results are only known foreither one of them. The arithmetic operations we consider are set product (Cartesianproduct), set union and set intersection.

In Chapter 7 we state the known results for orthogonal projection of a set onto alower-dimensional subspace. We also explain why central projection and a type of x-ray simulated projection (which we explain in the chapter) is equivalent to orthogonalprojection, in the sense of the fractal dimension. We also explain some of the problemswith projections that frequently occurs.

In Chapter 8 we state the known results for projections of measures onto lower-dimensional subspaces. Properties for the dimension of a measure, the generalized di-mensions, local fractal dimension and the multifractal spectrum, when projected, aregiven.

In Chapter 9 we describe how to generate fractals and multifractals. Two methodsfor both are given. For uni-fractals there is the Iterated Function Systems, and some-thing called Custom made Random Fractals, fractal sets with arbitrary dimension. Formultifractals, there is Probabilistic Iterated Function System and a recursive method togenerate true multifractal mesures.

In Chapter 10, the theory of Chapter 6, Chapter 7 and Chapter 8 is simulated andtested to see if it holds up in reality. Results for the fractal dimension estimation, fractalprojection, fractal product, fractal union, fractal intersection, generalized dimensionsestimation, multifractal spectrum estimation and the projection of measures are givenin this chapter.

In Chapter 11 we discuss the results of Chapter 10. Could the tests have been donedifferently? Could other tests have been done? How good or bad are the results? And

1.2. Thesis Outline 3

so on. We also discuss restrictions imposed on the test by design, and give suggestionsfor future work.

In the appendices can be found theory that might be unknown to the reader. Thereader should browse them before embarking the rest of the thesis, just to make sure tohave the correct prerequisites.

4 Chapter 1. Introduction

Chapter 2

Problem Description

In this chapter, the problem statement, the guide to do this master’s thesis is given.Over the course of the work, the original plan was followed quite well, but naturallysome things where added, and some things where removed.

2.1 Problem Statement

How to estimate fractal properties, such as the fractal dimension, is relatively wellunderstood today (at least for subsets of R2 and R3). Research have been done atUmea University, see [NG06], to see how projections of fractal sets affect the estimateddimension of the projected set. The conclusions include upper and lower bounds on theHausdorff dimension of the original set, given the estimated Box-counting dimension(or any equivalent dimension) of the projection. This can be done using mathematicalproperties such as Lipschitz mappings, countable stability, etc. In other research, theperformance of different methods for fractal dimension estimation have been documentedfor sets in R3, see [JG06].

2.2 Goals

The goal of this thesis is to document the practical usefulness of fractal geometry on acomputer; to see whether the mathematical theory holds when simulated in practice.

Simulations of the now classical fractal geometry should be done on either pointsets, intensity images or both. This includes simulations of different operations on andprojections of fractal sets and estimations of the resulting dimensions.

If time be, the same study should be done considering multifractal geometry; inves-tigating how different operations and projections affect the multifractal properties of ameasure, i.e. multifractal spectrum, local dimension, etc.

2.3 Purpose

The purpose of this thesis is to investigate how estimates of fractal dimensions of setsF, G ⊆ R3 is affected by operations such as F ∪ G, F ∩ G, F × G and projΠF , whereprojΠ is the orthogonal projection of F onto a k-dimensional subset Π ⊂ Rn, wherek < n.

5

6 Chapter 2. Problem Description

2.4 Methods

The following steps and methods was suggested:

• Perform a literature study on the mathematical background.

• Perform a shorter literature study on methods for estimating the fractal dimension.

• Pick a good estimation method and know it well.

• Show with theoretical results what happens with fractal sets when different oper-ations are performed on them.

• Verify the theoretical results with computer simulations.

Chapter 3

Previous Work

The study of the special class of sets, which nowdays is known as Fractals, begun alreadyin the 19th century, and in the beginning of the 20th century, the interest in this areaflourished and much literature was written on the subject. The interest subsided how-ever, until it renaissanced in the 1970’s, much thank to Benoıt Mandelbrot’s work, andthe advancement of computers in science. Computers made it possible to draw thesefigures in a way that was never possible before. The Fractal dimension became one ofthe most popular tool with which these sets where described.

The field of mathematics blossomed in the end of the 17th century, when IsaacNewton and Gottfried Leibniz developed calculus. Many ideas came and went duringthe 18th century. By the 19th century, mathematicians thought they had it, by mostpart, figured out. But in 1872, Karl Weierstrass wrote an article, where he provedthat there are functions which are everywhere continuous, but nowhere differentiable,see [Wei72]. This was something completely new, the mathematical community hadassumed that the derivative of a function could be undefined only at isolated points.Much research followed.

In 1904, the Swedish mathematician Helge von Koch wrote an article about a con-tinuous curve, constructed from very elementary geometry, that does not have tangentin any of its points, see [vK04]. This is the curve described in Example 4.1.2. ErnestoCesaro immediately recognized this geometrical figure as being self-similar, and didmuch work on that. The work of Cesaro was taken further in a 1938 paper by PaulLevy, in which he introduces new self-similar curves.

The idea of using a measure to extend the notion of length was used by Georg Cantor,the father of set theory, in 1884 in [Can84], and Emile Borel in 1895 when he studied“pathological” real functions. These ideas where extended by Henri Lebesgue in 1901in [Leb01] by the Lebesgue integral.

The ideas of Lebesgue where developed further by Constantin Caratheodory in 1914,to adapt the theory to lengths in arbitrary spaces. This was later generalized by FelixHausdorff in 1919 in [Hau19] to extend to non integral dimensions. This contribution isthe foundation of the theory of fractional dimensions. The Hausdorff measure provides anatural way of measuring the s-dimensional volume of a set and the Hausdorff dimensionis today generally considered the Fractal dimension of a set. Much of the early workwith the Hausdorff dimension was done by Abram Besicovitch in the 1930’s.

Karl Menger wrote two papers in a communication with the Amsterdam Academy ofSciences, in the beginning of the 20th century. In these papers, he introduced the ideas of

7

8 Chapter 3. Previous Work

a topological dimension. These ideas where new, but other authors had, independentlyof him, the same, or similar, ideas; e.g. Henri Lebesgue.

The definition of the Box-counting dimension dates back to the late 1920’s, and isdue to Georges Bouligand. It has become a very popular definition because of its easeof numerical computation, but also rigorous computation. In his 1928 paper, Bouliganddefined several different variants of the new definition of dimension. In the same paperthere can also be found theory of the dimension of a Cartesian product of two sets beingthe sum of the dimensions of the two sets.

In a 1946 paper, Patrick Moran proves results concerning the Hausdorff dimensionof a Cartesian product of two sets. And in a 1954 paper, J. M. Marstrand proves thegeneral results of this problem for the Hausdorff dimension.

The theoretical results of projections of sets to lower-dimensional subspaces was firstobtained in the plane by J. M. Marstrand in 1954 in [Mar54], and later generalized toarbitrary dimensions by Pertti Mattila in 1975 in [Mat75]. But also investigated furtherby authors like Falconer and Howroyd, among others.

The theory of intersections of sets is a subject of integral theory, and has been thor-oughly researched. Theoretical results for the intersection of fractal sets was introducedby J. M. Marstrand, and can be found in [Mar54]. But more work was done by Jean-Pierre Kahane and Pertti Mattila in the 1980’s.

In the 1960’s, Benoıt Mandelbrot did work on self-similar sets, and in a 1967 paper,see [Man67], he describes the Similarity dimension. Mandelbrot formalizes the work ofLewis Fry Richardson, who noticed that the length of a coast line depends on the unitof measurement used. He also suggests that the theory of fractional dimensions andself-similarity could be used not only in mathematics, but in other branches of scienceas well. Mandelbrot coined the term Fractal in 1975, see [Man75]. Mandelbrot alsosays that self-similar figures seldom are found in nature, but that a statistical form ofself-similarity is ubiquitous. He manifests in his 1982 book [Man82] the idea that fractalgeometry is better at describing the nature than classical Euclidean geometry is.

The study of self-similar sets became one of the main fields of study in the 1980’sand 1990’s. The theory of self-similar sets was formalized by John Hutchinson in a1981 paper, but popularized by Michael Barnsley in his popular 1988 book [Bar88].The iterated function system approach, used to create self-similar sets, was extended tograph-directed constructs by Mauldin and Williams in a 1988 paper.

Measures have always been a fundamental tool in the study of geometrical fractalsets. But because there exists natural fractal measures in many constructions in fractalgeometry, i.e. self-similar measures, fractal properties of measures received increasedattention in the 1980’s and 1990’s. The idea leads to the notion of a dimension of ameasure.

Multifractal analysis, which became one of the most popular topics in geometricmeasure theory in the 1990’s, studies the local structure of measures and provides muchmore detailed information than the uni-dimensional notion that was popular earlier.The multifractal spectra was first explicitly defined by physicists Halsey et al. in 1986in [HJK+86].

In a 1989 paper, [KC89], Keller et al. used the Box-counting dimension and theconcept of lacunarity to discriminate between different textures. They showed that thefractal dimension alone is not enough to classify natural textures.

Recent examples of the use of multifractal analysis is the following: In a 2002 paper,[CMV02], Caron et al. used the multifractal spectrum for texture analysis and objectdetection in images with natural background. In a 2003 paper, [NSM03], Novianto et al.

9

used the local fractal dimension in image segmentation and edge detection.In 2006, Stojic et al., [SRR06], used multifractal analysis for the segmentation of

microcalcifications in digital mammograms. Their method successfully detected micro-calcifications in all test cases.

Nilsson et al., [NG06], investigated projections of sets in R3 onto R2, and found upperbounds for the Hausdorff dimension of the original set in terms of the Box-countingdimension of the projection. They also proved that central projection is equivalent toorthogonal projection.

10 Chapter 3. Previous Work

Chapter 4

Fractal Geometry

When dealing with the class of geometrical objects called Fractals, the classical EuclideanGeometry is not enough to describe their complex nature. In the past decades, a newbranch of geometry, called Fractal Geometry, have grown and received a great deal ofattention from a variety of fields.

This chapter will describe the general theory of fractals and their geometry. First,we will see a short explanation of what a fractal really is. Then some various notions ofdimension, something very important in Fractal Geometry, will be described, followedby a description of some methods to estimate the fractal dimension.

Some notions and theory in this chapter might be new to the reader, in which casewe recommend reading Appendix A first.

4.1 What is a Fractal?

In his founding paper [Man75] Benoıt Mandelbrot coined the term Fractal, and describedit as follows:

A [fractal is a] rough or fragmented geometric shape that can be subdividedin parts, each of which is (at least approximately) a reduced/size copy of thewhole.

The word is derived from the Latin word fractus meaning broken, and is a collectivename for a diverse class of geometrical objects, or sets, holding most of, or all of thefollowing properties [Fal90]:

i. The set has fine structure, it has details on arbitrary scales.

ii. The set is too irregular to be described with classical euclidean geometry, bothlocally and globally.

iii. The set has some form of self-similarity, this could be approximate or statisticalself-similarity.

iv. The Hausdorff dimension of the set is strictly greater than its Topological dimen-sion.

v. The set has a very simple definition, i.e. it can be defined recursively.

11

12 Chapter 4. Fractal Geometry

Property (iv) is Mandelbrot’s original definition of a fractal, however, this propertyhas been proven not to hold for all sets that should be considered fractal. In fact, eachof the above properties have been proven not to hold for at least one fractal. Severalattempts to give a pure Mathematical definition of fractals have been proposed, but allproven unsatisfactory. We will therefore, rather loosely, use the above properties whentalking about fractals [Fal90].

Perhaps a couple of examples are required to get a better understanding of thegeometrical objects we are talking about.

Example 4.1.1: The Cantor set, see Figure 4.1, is created by removing the middlethird segment of a unit line segment, call it E0. We now have two line segments, eachone third of the original line’s length, call this set E1, see Figure 4.1 b). We get E2 byremoving the middle third of the two line segments of E1, see Figure 4.1 c). If we applythe rule (removing the middle third of the line segments) on Ek−1 we obtain Ek, andwhen k tends to infinity, we get the Cantor set in Figure 4.1 d).

Example 4.1.2: The von Koch curve, see Figure 4.2, is created as follows: Start witha unit line segment E0. Remove the middle third of E0 and replace it by two lines, eachof the same length as the removed piece, call it E1. We now obtained an equilateraltriangle (with the base segment gone) as Figure 4.2 b) suggests. E1 now has four linesof equal length, 1

3 of that of E0. We can now create E2 by applying the same procedureas when we created E1, and thus obtains the curve in Figure 4.2 c). Thus, applying therules on Ek−1, we obtain Ek, and when k tends to infinity, we get the von Koch curveof Figure 4.2 d).

Example 4.1.3: The Sierpinski triangle is created by replacing an equilateral triangleof unit size (Figure 4.3 a)), E0, by three triangles of half its size, leaving the middleregion empty, giving E1, see Figure 4.3 b). E2 is created by replacing each of the threetriangles of E1 by three half-sized triangles, leaving the middle region empty as before,see Figure 4.3 c). Thus, as in Example 4.1.2, applying the rules on Ek−1, we obtainEk, and when k tends to infinity, we get the Sierpinski triangle of Figure 4.3 d).

a) b) c) d)

Figure 4.1: a) One, b) two, c) three, and d) several iterations of the Cantor set

a) b) c) d)

Figure 4.2: a) One, b) two, c) three, and d) several iterations of the von Koch curve

Objects in nature often have fractal properties (i.e. a tree has a stem, on whicheach branch is a reduced size copy of the step), and therefore, fractals are used to

4.2. Fractal Dimension 13

a) b) c) d)

Figure 4.3: a) One, b) two, c) three, and d) several iterations of the Sierpinski triangle,or Sierpinski casket

better approximate objects in nature than classical euclidean geometry can do [Man82].Natural objects with fractal properties could be trees, clouds, coast lines, mountains,and lightning bolts.

4.2 Fractal Dimension

An important feature of fractal geometry is that it enables a characterization of irregu-larity at different scales that the classical Euclidean geometry does not allow for. As aresult, many fractal features have been identified, among which the fractal dimension isone of the most important [JOJ95].

The notion of dimension is very important in Fractal Geometry, and also what thisthesis is focusing on. From the works of Euclid, see [Roe03], we know that the dimensionof a point is considered 0, the dimension of a line is 1, the dimension of a square is 2,and that the dimension of a cube is 3. Roughly, we can say that the dimension of a setdescribes how much space the set fills [Fal90].

We will build the theory of fractal dimension from the basic Euclidean definition tothe more mathematically exhaustive definitions of Hausdorff and Box-counting dimen-sions which will be used when empirically estimating the fractal dimension of sets.

One might ask why there are several different definitions of dimension. This is simplybecause a certain definition might be useful for one purpose, but not for another. Inmany cases the definitions are equivalent, but when they are not, it is their particularproperties that makes them more suitable for the task at hand.

4.2.1 Topological Dimension

The intuitive feeling of dimension that was mentioned in the beginning of this section iscalled the Topological dimension. Topology is the study of the geometrical properties ofan object that remains unchanged when continuously transforming the object [Kay94].Thus, the lines in Figure 4.4 both have topological dimension 1, since we could stretchthem both to fit each other, and we know that a line has dimension 1. However, thelines are indeed not very similar, so we would like some way to describe them, otherthan with their topological dimension.

The Topological dimension of a set is always an integer, and is 0 if the set is totallydisconnected. The set should be considered zero-dimensional if it can be covered by


a) b)

Figure 4.4: a) A straight line, and b) a rugged line. Both a) and b) have Topologicaldimension 1.

disjoint sets. The Topological dimension of Rn is n (this can be proven, but we will notdo that now.) Formally, the topological dimension is defined as [Edg90]:

Definition 4.2.1: The Topological dimension of a set A is the smallest integer, n, suchthat every cover C of A has a refinement C′ (i.e. a cover of sets with smaller diameter)in which every point of A occurs in at most n + 1 sets of C′.

In Figure 4.5 you can see, more descriptively, how the topological dimension of theline in Figure 4.4 b) is found. This version of the Topological dimension (there areindeed several different definitions of Topological dimension, but in general they areequivalent) is sometimes called the Covering dimension.

Figure 4.5: A cover on the rugged line from Figure 4.4 b), and its refinement. Notethat every point on the line is an element of at most two subsets of the refinement cover.

Now, imagine measuring the length of a seashore coastline, if we try to approximatethe length of the coastline using a fixed sized length ruler, we would find that the lengthincreases as the length of the ruler decreases, because we are taking finer and finerdetails into account. The ruler is therefore inadequate to describe the complexity of ageographical curve [Man67].

The length of the von Koch curve is a theoretical equivalence of a coastline. We cansee that the length of Ek, in the creation of the curve, is 4

3 of that of Ek−1, thus thelength of Ek is (4

3 )k. This means that the length of the von Koch curve tends to infinityas k tends to infinity (i.e. when we use a shorter ruler). But also, since the von Kochcurve is created from finite line segments, we know that the curve must occupy zero areain the plane. Thus, we cannot use neither length, nor area to describe the size of thecurve. See [vK04] for details.

Mandelbrot suggests in [Man67] that dimension should be considered a continuous


quantity that ranges from 0 to infinity, and in particular that curves could have theirruggedness described by a real number between 1 and 2 that describes their space-fillingability [Kay94]. We will approach a finer definition in the following sections.

4.2.2 Similarity Dimension

A transformation S : Rn → Rn is called a congruence or isometry if it preserves distances(and thus also angles), i.e. |T (x) − T (y)| = |x − y|, for all x, y ∈ Rn. The mapping isthus done without deforming the object [Roe03]. A similar transformation is a mappingT : Rn → Rn where there is a constant c such that |T (x) − T (y)| = c|x − y|, for allx, y ∈ R [Fal90]. We will touch upon this subject again in Chapter 7.

Consider a unit line segment, which thus is 1-dimensional in the classical sense. Ifwe magnify the line segment twice, we will have two connected line segments both ofthe same length as the original one, see Figure 4.6 a). Now we look at a unit square inthe plane, a 2-dimensional object. Magnify it twice (the side lengths, not the area) andwe will have a similar square, made up of four connected unit squares, see Figure 4.6 b).A unit cube, 3-dimensional of course, magnified two times will result in a cube made upof eight cubes of unit size, see Figure 4.6 c).

a) b) c)

Figure 4.6: a) A unit length line magnified two times will make the magnified objectbe made up of two identical unit line segments. b) A unit square magnified two timesgives four cubes of unit size. c) A unit cube magnified two times gives eight unit cubes.

Note that the number of copies of the original object when magnified two times istwo to the power of the dimension. That is

mD = N, (4.1)

where m is the magnification, D is the dimension, and N is the number of copies ofthe original object when magnified m times. Now, if we solve for D in Equation 4.1 weobtain:

D =log N

log m. (4.2)

Of course, this is accurate for the objects in Figure 4.6, because 1 = log 2log 2 , 2 = log 4

log 2 , 3 =log 8log 2 , but what happens with the complex objects of Example 4.1.2 and Example 4.1.3?

The von Koch curve gives, in Ek+1, four copies of Ek of size 13 . Thus, we find D as

log 4log 1/ 1

3

= 1.2618 . . .. It is thus more than 1-dimensional, but less than 2-dimensional.

This agrees with the idea in Section 4.2.1 that neither length nor area can describe thecurve, simple because it is more than a line (the length between any two points on thecurve is infinite), but does not fill the plane either (it has zero area).

The Sierpinski triangle is in Ek+1 made up of three copies of Ek of size 12 , thus the

dimension, D, is log 3log 1/ 1

2

= 1.5849 . . .. All triangles are replaced by smaller triangles, but


the distance between the triangles tends to zero as k tends to infinity. Note that thedimension of the Sierpinski triangle is higher than that of the von Koch curve, we saythat the Sierpinski triangle fills the plane more than the von Koch curve does.

The number obtained in the above way is called the Similarity dimension of a set, andmight appear to be a suitable way to calculate the dimension. However, the Similaritydimension can only be calculated for a small class of strictly self-similar sets. In thefollowing sections we will describe some other definitions of dimension that are muchmore general [Fal90].

4.2.3 Hausdorff Dimension

To be able to understand the more general notions of dimension (there are indeed severaldifferent definitions, which sometimes gives the same value, and sometimes not) we firstneed to set the basics with some elementary Measure Theory.

The Notion of Measure

Before delving into the mathematics of fractal dimensions, we need to briefly look atsome notions of measure. We use measures in the definition of dimension, and in anycase, measures are important in fractal geometry, in some form or another, and we needto set the basics here.

A measure is exactly what the intuitive feeling tells us it is; a way to give a numericalsize to a set such that the sum of the sets in a collection of disjoint subsets have thesame measure as the whole set (the union of the subsets). The numerical size of a setcould e.g. be the mass distribution, or the electrical charge of the set. We define ameasure, µ, on Rn as follows [Fal90]:

Definition 4.2.2: The measure µ assigns a non-negative value, possibly ∞, to subsetsof Rn such that:

(a) µ(∅) = 0; (4.3)

(b) µ(A) ≤ µ(B) if A ⊆ B; (4.4)

(c) If A1, A2, . . . is a countable (or finite) sequence of sets then

µ

(

∞⋃

i=1

Ai

)

≤∞∑

i=1

µ(Ai) (4.5)

with equality in Equation 4.5, i.e.

µ

(

∞⋃

i=1

Ai

)

=

∞∑

i=1

µ(Ai) (4.6)

if the Ai are disjoint Borel sets.

This follows our intuitive feeling that an empty set has no size (a), that a smallerset has smaller size (b), and, as noted above, that the sum of the sizes of the pieces isthe size of the whole, Equation 4.6.

If A ⊃ B, we can express A = B ∪ (A\B), and thus, by Equation 4.6, if A and Bare Borel sets, we have:

µ(A\B) = µ(A) − µ(B). (4.7)


In fact, if X is a set, a set E ⊆ X is called µ-measurable if and only if every set A ⊆ Xsatisfies

µ(A) = µ(A ∩ E) + µ(A\E). (4.8)

A probability measure is a measure µ on a set A, such that µ(A) = 1. A massdistribution is defined as the mass of a bounded set A ⊂ R where 0 < µ(Rn) < ∞.

The Lebesgue measure, L 1, extends the idea of a length to a large collection of subsetsof R. For open and closed intervals, we have L 1(a, b) = L 1[a, b] = b−a. If A =

⋃

i[ai, bi]is a finite or countable union of disjoint intervals we let L 1(A) =

∑

(bi − ai) be thelength of A, which leads to the formal definition of Lebesgue measure [Fal90]:

Definition 4.2.3: The Lebesgue measure, L 1 of an arbitrary set is:

L1(A) = inf

{

∞∑

i=1

(bi − ai) : A ⊂∞⋃

i=1

[ai, bi]

}

(4.9)

This measure follows our intuitive feeling of a length in R, but extends also to areasin R2, volumes in R3, and to the volume of n-dimensional hypercubes in Rn:

voln(A) = Ln(A) = (b1 − a1)(b2 − a2) · · · (bn − an). (4.10)

The n-dimensional Lebesgue measure, L n may be thought of as an extension to the n-dimensional volume for a large collection of sets. By simply extending Definition 4.2.3,we obtain [Fal90]:

Definition 4.2.4: The Lebesgue measure on Rn of an arbitrary set is:

Ln(A) = inf

{

∞∑

i=1

voln(Ai) : A ⊂∞⋃

i=1

Ai

}

(4.11)

We will, however, be concerned mainly with the s-dimensional Hausdorff measure,H s, on subsets of Rn, with 0 ≤ s ≤ n, when we develop the theory of dimensions.The Hausdorff measure is a generalization of the Lebesgue measure to non-integraldimensions. We will talk about this more in the next section.

We will say that a property hold for almost all x, or almost everywhere, if the subsetfor which it does not hold has µ-measure zero. See the following example.

Example 4.2.5: We can say that almost all real numbers are irrational with regardsto the Lebesgue measure. The rational numbers are countable, i.e. Q = {x1, x2, . . .},and thus we can write µ(Q) =

∑∞i=1 µ({xi}) = 0, since every xi is a point and thus

µ({xi}) = 0.

How Measure Relate to Dimension

We have thus far mentioned two different definitions of dimension, which both turnedout to be inadequate. The Hausdorff dimension is the oldest, and probably the mostimportant. As opposed to many of the other definitions of dimension, the Hausdorffdimension is defined for all sets, and is convenient for mathematicians since it is basedon measures. We start with the following definition [Fal90]:


Definition 4.2.6: Let F be a subset of Rn and s ∈ R+, then for any δ > 0 we let:

Hs

δ (F ) = inf

{

∞∑

i=1

|Ui|s : {Ui} is a δ−cover of F

}

(4.12)

We try to minimize the sum of the s-powers of the diameters of the covers of F withdiameter at most δ. When δ decreases, the class of possible covers of F is reduced, andH s

δ (F ) will increase, and thus approach a limit as δ → 0. We can write:

Hs(F ) = lim

δ→0H

sδ (F ). (4.13)

This limit exists for all subsets of Rn, but usually is 0 or ∞. We call H s(F ) thes-dimensional Hausdorff measure of F.

It can be shown that the definition above is invariant of the choice of norm whencalculating the diameter in Equation 4.12 [The90].

It can be shown that H s fulfills the properties of Equation 4.3–4.6, and in particular,the Hausdorff measure generalizes to the intuitive idea of length, area, and volume. Itmay even be shown that for subsets of Rn, the n-dimensional Hausdorff measure is justa constant multiple of the Lebesgue measure. Specifically, if F is a Borel set of Rn, then

Hs(F ) = cnL

n(F ). (4.14)

The scaling properties of normal measures applies to Hausdorff measures as well, andthus, on magnification by a factor λ the length of a curve is multiplied by λ, the area ofa figure in the plane is multiplied by λ2, the volume of an object in space is multipliedby λ3, and in general, the s-dimensional Hausdorff measure scales with a factor λs. Wehave the following proposition [Fal90]:

Proposition 4.2.7 (Scaling property of the Hausdorff measure): If F ⊂ Rn andλ > 0 then

Hs(λF ) = λs

Hs(F ). (4.15)

Proof. If {Ui} is a δ-cover of F then {λUi} is a λδ-cover of λF . Hence

Hs

λδ(λF ) ≤ Σ|λUi|s = λsΣ|Ui|s ≤ λsH

sδ (F ) (4.16)

since this holds for any δ-cover {Ui}. Letting δ → 0 gives that H s(λF ) ≤ λsH s(F ).Replacing λ by 1/λ and F by λF gives the opposite inequality required.

Now, looking at Equation 4.12 again, we note that for any set F and δ < 1, H sδ is

non-increasing with s, and thus H s is non-increasing as well. If t > s and {Ui} is aδ-cover of F , we have

∑

i

|Ui|t ≤ δt−s∑

i

|Ui|s. (4.17)

Thus, taking the infimum of both sides, we get

Ht

δ (F ) ≤ δt−sH

sδ (F ). (4.18)

If we let δ → 0 we see that if H s(F ) < ∞ then H t(F ) = 0 when t > s. Now, if we plotH s(F ) against s we see that there is a critical point at which H s(F ) jumps from ∞ to0, see Figure 4.7. This critical value is known as the Hausdorff dimension (sometimescalled the Hausdorff-Besicovitch dimension) of F , and is denoted dimH F .

Formally, we define the Hausdorff dimension as follows [Fal90]:


∞

0

Hs(F )

0 ndimH Fs

Figure 4.7: A plot of H s(F ) against s for some set F . The Hausdorff dimension ofF is the value s = dimH F at which the graph jumps from ∞ to 0.

Definition 4.2.8: The value

s = inf{s : Hs(F ) = 0} = sup{s : H

s(F ) = ∞} (4.19)

such that

Hs(F ) =

{

∞ if s < dimH F0 if s > dimH F

(4.20)

is called the Hausdorff dimension of the set F , and is denoted dimH F . If s = dimH F ,then H s(F ) may be zero or infinite, or may satisfy

0 < Hs(F ) < ∞.

This jump is easily understood by considering the dimension when measuring i.e.the length of lines, the area of squares and the volume of cubes. We can fit an infinitenumber of points on a line, but the area of a line is zero. The length of a square isinfinite, i.e. we can fit an infinite number of lines (or a curve with infinite length) on thesquare, but the volume of a square is zero. A cube has infinite area in the sense thatwe can fit an infinite number of planes (or a plane curve with infinite area) in the cube.Thus, if we use a too small dimension when measuring a set, the measure is infinite, andif we use a too large dimension when measuring, the measure is zero.

The problem, however, with the Hausdorff dimension is that is is difficult to calculate,or measure in practice (it is not feasible to find the infimum, or the supremum, inEquation 4.19) [Fal90]. Because of this we will need some other general definition ofdimension, that can also easily be calculated in practice. The answer to this problem isthe Box or Box-counting dimension, which we describe in the following section.

4.2.4 Box-Counting Dimension

The Box-counting dimension is one of the most common in practical use. This is mainlybecause it is easy to calculate mathematically and because it is easily estimated empir-ically.

We note that the number of line segments of length δ that are needed to cover a lineof length l is l/δ, that the number of squares with side length δ that are needed to cover


a square with area A is A/δ2, and that the number of cubes with side length δ that areneeded to cover a cube with volume V are V/δ3. The dimension of the object we try tocover is obviously the power of the side length, δ. Now, can we generalize this to findthe dimension of any set using this method?

We let F be a bounded non-empty subset of Rn and Nδ(F ) be the smallest numberof sets of diameter at most δ that covers F . The number of boxes needed to cover theobject is Nδ(F ). Following the discussion above, the number of boxes needed to coverthe object should be proportional to the box size [Fal90]:

Nδ(F ) ∼ C

δs, (4.21)

when δ → 0. Thus, for the constant C we have

limδ→0

Nδ(F )

δ−s= C. (4.22)

Taking the logarithm of both sides gives:

limδ→0

(log Nδ(F ) + s log δ) = log C. (4.23)

We solve for s and get an expression for the dimension as

s = limδ→0

log Nδ(F ) − log C

− log δ= lim

δ→0

log Nδ(F )

− log δ. (4.24)

We have the following definition [Fal97]:

Definition 4.2.9: The lower and upper Box-counting dimensions of a set F are definedas

dimBF = lim infδ→0

log Nδ(F )

− log δ(4.25)

and

dimBF = lim supδ→0

log Nδ(F )

− log δ, (4.26)

respectively. If their values are equal, we refer to the common value as the Box-countingdimension of F

dimB F = limδ→0

log Nδ(F )

− log δ, (4.27)

This says that the least number of sets of diameter δ that can cover F is of theorder δ−s where s = dimB F . The dimension reflects how rapidly the irregularities ofthe object develop as δ → 0 [Fal90].

There are a number of equivalent definitions of the Box-counting dimension. Mainly,the differences concerns the shape of the box used to cover the set. However, the shapeof the box is of no importance, and we can use both squares and circles, and their higherdimensional equivalences. In fact, we can even use general subsets of Rn with diameterδ. In the limit, the shape will not matter [Fal97].

There is a very nice connection between the Lebesgue measure and the Box-countingdimension, that the following proposition establishes [Fal90]:


Proposition 4.2.10: If F ⊂ Rn, then

dimBF = n − lim supδ→0

log Ln(Fδ)

log δ(4.28)

and

dimBF = n − lim infδ→0

log L n(Fδ)

log δ(4.29)

where Fδ is the δ-parallel body to F .

Proof. Omitted, but can be found in [Fal90].

The relationship between the Hausdorff dimension and the Box-counting dimensionis established by the following proposition [Fal90]:

Proposition 4.2.11: The following is true for F ⊂ Rn

dimH F ≤ dimBF ≤ dimBF (4.30)

Proof. If a set F ⊂ Rn can be covered by Nδ(F ) sets of diameter δ, then, by Defini-tion 4.2.6,

Hs

δ (F ) ≤ Nδ(F )δs.

If 1 < Hs(F ) = limδ→0 H

sδ (F ) then taking logarithms gives log Nδ(F ) + s log δ >

log 1 = 0 if δ is sufficiently small. Thus s ≤ lim infδ→0 log Nδ(F )/ − log δ so the propo-sition follows for any F ⊂ Rn because of Definition 4.2.9.

The above proposition does not in general have equality. The Hausdorff and Box-counting dimensions are only equal for reasonably regular sets, but there are severalexamples where the inequality is strict [Fal90].

There is a problem with the Box-counting dimension, that at first might seem ap-pealing, but has undesirable consequences [Fal90].

Proposition 4.2.12: Let F denote the closure of F (i.e. the smallest closed subset ofRn that contains F ). Then

dimBF = dimBF (4.31)

and

dimBF = dimBF. (4.32)

Proof. Let B1, . . . , Bk be a finite collection of closed balls of radii δ. If the closed set∪k

i=1Bi contains F , it also contains F . Hence the smallest number of closed balls ofradius δ that cover F is enough to cover the larger set F . The result follows.

Let F be the countable set of rational numbers between 0 and 1. Then F is theentire interval [0, 1], so that dimBF = dimBF = 1. Thus, countable sets can have non-zero Box-counting dimension. The Box-counting dimension of each rational number isis zero, but the countable union of these points has dimension 1.


4.2.5 Properties of Dimensions

The above definitions for the Hausdorff and upper and lower Box-counting dimensionsfulfill the following important properties [Fal97]:

Monotonicity: If E1 ⊂ E2 then dimE1 ≤ dim E2.

Finite sets: If E is finite then dimE = 0.

Open sets: If E is a (non-empty) open subset of Rn then dimE = n.

Smooth manifolds: If E is a smooth m-dimensional manifold in Rn then dimE = m.

Lipschitz mappings: If f : E → Rm is Lipschitz then dim f(E) ≤ dim E.

Bi-Lipschitz invariance: If f : E → f(E) is bi-Lipschitz then dim f(E) = dimE.

Geometric invariance: If f is a similarity or affine transformation then dim f(E) =dimE.

The Hausdorff and upper Box-dimensions are finitely stable, that is dim∪ki=1Ei =

max1≤i≤k dimEi, for any finite collection of sets {E1, . . . , Ek}. The lower Box-countingdimension is, however, not in general finitely stable. The Hausdorff dimension is alsocountably stable, meaning that dimH ∪∞

i=1Ei = sup1≤i<∞ dimH Ei. This is not true forthe Box-counting dimension, however, as was described above.

Most definitions of dimension will take on values between the Hausdorff dimensionand the upper Box-counting dimension. Thus, if dimH E = dimBE then all normaldefinitions of dimension will take on this value [Fal97].

4.3 Means to Estimate Fractal Dimension

The difficulty with implementing the Hausdorff dimension for numerical applications isthe need for finding the infimum and supremum for all coverings, described in Equa-tion 4.19. When we relax this requirement, and e.g. considers a fixed-size grid instead,we can numerically estimate the Box-counting dimension, as mentioned above, and thusfind an upper bound for the Hausdorff dimension as Proposition 4.2.11 suggests. Butas Theorem 9.1.3 says, the Hausdorff and Box-counting dimensions are equal for mostsets of interest to us [The90].

There are plenty of different ways to estimate the Fractal dimension of a set, andthey all have their pros and cons. We will in this text focus on methods that are basedon the Box-counting dimension.

In this thesis we will focus exclusively on estimating the fractal dimension of pointsets and digital intensity images. There is an obvious problem with doing this; thenumber of points in a constructed set and the number of pixels in an image is finite,and countable, the theoretic dimension of the sets and images we consider is always zero(see Section 4.2.5). However, we only want to estimate the dimension of the underlyingfractal that the image depicts, and this allows itself to be done, with varying results.

Most definitions of dimension is based on the idea of a measurement at scale δ.For each δ, we look at properties of the set, but ignoring irregularities smaller thanδ, and look at how these measurements change when δ → 0 [Fal90]. Remember fromSection 4.2.4 that the number of boxes needed to cover a set is:

Nδ(F ) ∼ C

δs,

4.3. Means to Estimate Fractal Dimension 23

##

##

##

##

##

##

##

##

b

b

b

b

b

log Nδ(F )

log δ

Figure 4.8: Least-squares fit to the points of log Nδ(F ) against log δ.

which gives the dimension as

s = limδ→0

log Nδ(F )

− log δ.

In [Man67], Mandelbrot says that geographers cannot be concerned with minutedetails when they measure the length of a coastline. Simply because below a certainlevel of detail, it is no longer the coastline that is being measured, but other details thataffects the length of the curve. There is no clear cross-over either, so one simply haveto choose a lower limit of geographically interesting features. A similar problem occurswith digital images, but here we have a strictly imposed lower limit of measurementgranularity in the single pixel being the smallest measurable unit. Thus, we cannot letδ → 0, but have to stop when δ = 1. This means we will not get the true value for thedimension this way. However, if we use successively smaller value of δ, say δk, and letδk → 0 when k → ∞, then each value of sk will differ from the correct value by just asmall amount. We can now estimate the value of s by plotting log Nδ(F ) against − log δand take the slope of a least-squares fit to the points as the value of s, see Figure 4.8.The dimension is thus the logarithmic rate at which Nδ(F ) increases as δ → 0, and thebest we can do here is estimate the slope of the points we get in the bi-logarithmic plot.

It doesn’t matter what shape the boxes have when we estimate the Box-countingdimension. Actually, it doesn’t even have to be boxes, any set with diameter δ willdo equally well. In [Fal90], an argument for five different shapes can be found, but asstated, the list could be made longer, and in the end it is the particular application thatdecides which to use. In Figure 4.9 we can see the different methods with which we cancover the set F .

The following subsections describe how we can apply the theory in practice, bothwhen we are working with digital intensity images and when we are working with pointsets.

4.3.1 Digital Intensity Images

A digital intensity image of size M × N is a function f(x, y) with an intensity valuefor every coordinate pair (x, y) of the image. The intensity values range between 0 andsome maximum value, usually 255. We can represent an image in the following compact


a) b) c)

d) e) f)

Figure 4.9: Different ways to find the Box-counting dimension. We have the set Fin a), b) the number of closed balls with radius δ that cover F , c) the least number ofboxes of side δ that cover F , d) the number of δ-mesh cubes that intersect F , e) the leastnumber of sets of diameter at most δ that cover F and f) the greatest number of disjointballs of radius δ with centers in F .

matrix form:

f(x, y) =

f(0, 0) f(0, 1) . . . f(0, N − 1)f(1, 0) f(1, 1) . . . f(1, N − 1)

......

. . ....

f(M − 1, 0) f(M − 1, 1) . . . f(M − 1, N − 1)

. (4.33)

Each element of the above matrix is called a picture element, or pixel. All images aremade up of a finite number of pixels, and it is the varying intensity values of each pixelthat make up the image. See Figure 4.10 for an example of an image and a zoomed-inportion of the same image.

When we have gray scale images, the theory of the Box-counting dimension can beapplied rather straight-forward. We put a mesh grid on the image with side length δk

and count the number of squares with at least one pixel with non-zero intensity in it (welet black pixels be empty and non-black pixels be filled, but in this report we will ofteninvert the colors to make images more suitable for printing). We need not be concernedwith δ equal to the image size, since that will only be one square, empty or not. If it isempty the image contains nothing, and thus have dimension zero, and if it is not empty,the entire fractal will be inside the square, so we gain no information from that. Thuswe start with δk equal to half the image size and successively reduces the grid side by afactor of two as long as δk ≥ 1. Now, plotting log Nδk

against log δk for k = 1 . . . n andfinding the least squares fit to the points will give an approximation of the dimensionof the set, as was illustrated in Figure 4.8.

In Figure 4.11 we have covered the box fractal, which has a Similarity dimension ofD = log 5

log 3 ≈ 1.46, with a mesh with size length 1/3 and also by a mesh of side length

4.3. Means to Estimate Fractal Dimension 25

a) b)

Figure 4.10: The classical Lena image a) and a zoomed-in portion of her right eyeb). In the zoomed-in image, we can clearly see the pixels, and their different intensityvalues.

1/9. For Figure 4.11 a) we have 5 non-empty boxes and a box-size of 1/3, and thus get

D1 = − log 5

log 13

=log 5

log 3= D.

For Figure 4.11 b) we have 25 non-empty boxes and a box-size of 1/9, and thus get

D2 = − log 25

log 19

=log 5

log 3= D.

We can see that every factor of 1/3 of this box size will yield the same quotient, andthus we are confident that our estimation is really the true similarity dimension.

Obviously, we won’t retrieve all information possible from a fractal image if we onlyconsider empty and non-empty boxes. Since each pixel contains more information thanjust whether it is black or not – it contains a full range of possible intensity values.

When dealing with intensity image, we may extract more information than just thedull empty or not empty property that the box-counting dimension considers. The pixelsdo not have the same intensity values, and we may want to deal with that somehow.We can think of the intensity of a pixel as a measure of that pixel, and use a measureµ(F ) on a set F . This is the topic of the next chapter.

4.3.2 Point Sets

Also for point sets, the theory of Box-counting can be applied straight forward. A pointset is a set of coordinates for points in space. The format is as follows:

P =

xi

yi

zi

=

x1 x2 . . . xn

y1 y2 . . . yn

z1 z2 . . . zn

, (4.34)

where xi, yi, zi ∈ [0, 1]. Such a matrix can easily be transformed by linear operations(such as projection) by a simple matrix multiplication.


a) b)

Figure 4.11: Box-counting the box fractal. The fractal has true similarity dimension

of D = log 5log 3 ≈ 1.46 (remember Section 4.2.2). We cover the fractal with a δ-mesh and

count the number of non-empty boxes to estimate its dimension.

When putting a mesh grid on the point set, to see if a box is empty or not, thenumber of points with x, y and z coordinates within the particular box are simplycounted, letting the box side length δk go towards zero. The problem with gray-scaleimages having a smallest side length ∆ = 1 is not a concern in this case. The onlyproblem is if δk is less than the smallest distance between points in the set, then the δk

does not make sense with the current number of points. Since the fractal set generationalgorithms (see Chapter 9) involve contractions, the smallest distance between pointswill decrease as new points are added to the set. Thus, the smallest reasonable δk canbe found, given the number of points in the set. In general, starting with δ1 = 0.5, thenumber of successive reduces of δk by two needed is not more than ten, i.e. the smallestgrid size will not be less than 0.510 ≈ 0.001.

Now, plotting log Nδkagainst log δk for k = 1, . . . , n and finding the least squares fit

to the points will give an approximation of the dimension of the set, as was illustratedin Figure 4.8.

Just as when dealing with intensity image, we may extract more information thanjust the dull empty or not empty property that the box-counting dimension considers.We can find how many points the boxes contain, and thus we can think of the numberof points in a box as a measure, and use a measure µ(F ) on a set F . The idea offinding dimensions of measures is the topic of the next chapter, where we will look atmultifractal geometry, and how we can use the underlying measure of images and pointsets to extract even more information from the sets.

Chapter 5

Multifractal Geometry

For multifractal sets, the single scaling exponent that the fractal dimension representsis not enough to describe the system’s full dynamics. Instead of letting a single dimen-sion describe the behaviour around all points, the interest here will be in local scalingexponents, and the particular behaviours at every single point. A continuous spectrumof scaling exponents, that better describe the fractal, will be found.

A thorough description of this will be given in the following sections. The first sectionintroduces a more general way to define dimensions that will be used in the definition of,and the description of, the multifractal spectrum. The following sections describe themultifractal spectra, how they relate to the generalized dimensions and how to estimatethem.

5.1 The Generalized Dimensions

Although the Hausdorff dimension is a rigorous definition of dimension that has mostproperties we want a dimension to have, it only describes the geometry of the set, andmakes no reference to any underlying measure. In the original article by Hausdorff,see [Hau19], the measure H , mentioned above, is given, but the underlying measure isdefined to be a uniform density over the set, and does not allow for the varying densitieswhich we might be interested in.

The definition of the Box-counting dimension, which is easier to estimate numerically,also lacks the ability to account for the underlying measure. When estimating thedimension by the Box-counting method, one only considers non-empty boxes, and doesnot provision for how many points that actually are inside the box. I.e. the geometricalstructure of the fractal is analyzed, but the underlying measure is ignored [The90].

Until the notion of generalized dimensions was introduced, only three different def-initions of fractal dimensions seems to have been used in practice. These were thesimilarity dimension D, the information dimension σ and the correlation dimensionν [HP83]. The similarity dimension is simply the box-counting dimension defined asabove. The information dimension, σ, is defined as

σ = − limδ→0

S(δ)

log δ(5.1)

27

28 Chapter 5. Multifractal Geometry

where

S(δ) = −N(δ)∑

k=1

pk log pk (5.2)

where N(δ) is the smallest number of non-empty boxes of side-length δ that is neededto cover the set, and pk = µ(Nk)/µ(F ) is the normalized measure of the kth box Nk.Thus pk is an estimate of the probability for a randomly chosen point to be in Nk.

The correlation dimension, ν, is defined as

ν = limδ→0

log C(δ)

log δ(5.3)

where

C(δ) =1

N2

∑

i6=j

θ(δ − |Xi − Xj |), X ⊂ F (5.4)

in which N is the total number of points on the set F , θ is the Heaviside function (whichis one for positive values and zero otherwise), and thus, C(δ) simply counts the numberof pairs of points with distance less than δ.

There have been other definitions of dimension, but they have often been regardedequal to either D or σ. However, it can be shown, see [HP83], that fractals can becharacterized by an infinite number of different (and relevant) generalized dimensions,Dq, and Dq is defined for any q. In fact

limq→0

Dq = D, (5.5)

limq→1

Dq = σ, (5.6)

limq→2

Dq = ν (5.7)

and for q = 3, 4, . . . , n we have higher order correlation integrals.The upper and lower generalized dimensions are defined as follows [HP83, FO99]:

Definition 5.1.1: Cover the set with boxes of side length δ, and let µ be a Borel prob-ability measure. The upper and lower generalized dimensions are defined as

Dq =1

(q − 1)lim inf

δ→0

log∑

pqi

log δ(5.8)

and

Dq =1

(q − 1)lim sup

δ→0

log∑

pqi

log δ, (5.9)

respectively. Where the sum is over the cubes in the set of r-mesh cubes. Note thatDq ≤ Dq.

When the above definitions of Dq and Dq are equal, the common values for each qare the generalized dimensions Dq:

Dq =1

(q − 1)limδ→0

log∑

pqi

log δ, (5.10)

for any q ∈ R, and where pi is the probability of a point being in the ith box.

5.1. The Generalized Dimensions 29

The above definition can be generalized to any Borel probability measure µ that wetake for each cube Ni in the set of r-mesh cubes N , i.e. µ(Ni). For q ≥ 0, q 6= 1 thedefinitions for the lower and upper generalized dimensions are independent of the choiceof r-grids, and give the same values as the continuous generalized dimensions, which wehave chosen not to cover here. The case when q < 0 is dependent on the r-mesh chosen,and is more sensitive to numerical errors than the case when q ≥ 0 [HK97, BGT01].

We have the following proposition regarding the lower and upper generalized dimen-sions on a measure µ [BGT01]:

Proposition 5.1.2: For any q > 1,

0 ≤ Dq ≤ Dq ≤ 1. (5.11)

Proof. Omitted, but can be found in [BGT01].

We see from Equation 5.10 that Dq decreases when q increases, and it can in fact beshown that

Dq ≥ Dq′ (5.12)

for any q′ > q, and we have equality only if the fractal is homogeneous, i.e. all theprobabilities pk = µ(Nk)/µ(F ) are equal.

The minimum dimension, D∞, corresponds to the most-dense points on the set, i.e.the region in the set where the measure is most concentrated. The maximum dimension,D−∞, is associated with the least-dense points on the fractal, i.e. where the measureis most rare. We can show that D∞ and D−∞ are non-trivial numbers, and are foundby [HP83, The90]:

D∞ = limδ→0

log maxi pi

log δ(5.13)

and

D−∞ = limδ→0

log mini pi

log δ. (5.14)

We see that, obviously, if q = 0, the definition is precisely the definition of thebox-counting dimension. I.e.

D0 = limq→0

1

(q − 1)limδ→0

log∑

pqi

log δ= − lim

δ→0

log N(δ)

log δ= D, (5.15)

since for q → 0, pqi = 1 if pi 6= 0 and pq

i = 0 if pi = 0, and thus∑

i pqi is the number of

non-empty boxes on the set.To see that q = 1 equals the information dimension, we rewrite Equation 5.10 as

D1 = limq→1

1

(q − 1)limδ→0

log∑

pqi

log δ(5.16)

= limq→1

1

(q − 1)limδ→0

log∑

pi exp(q − 1) log pi

log δ(5.17)

= limδ→0

∑

pi log pi

log δ= σ. (5.18)

Similar arguments can be made for the correlation dimension as well, including allhigher degree correlations. Thus, we have a more general way of representing the alreadyfamiliar definitions of dimension. If we are running a box-counting algorithm, we can


find all the dimensions Dq at once, by simply calculating Equation 5.10 for all desiredvalues of q.

Mandelbrot have, however, warned against using the term dimension for quantitieslike Dq, since they do not have all the properties that we want a dimension measure tohave (see Section 4.2.5). Because of this, these numbers are referred to as the generalizeddimensions [HP83].

One of the motivations for the generalized dimension was the need to more fullycharacterize fractals with non-uniform measure. These fractals are called multifractals,and we will generate such fractals in Chapter 9. The point is to find a full spectrumof dimension, from D−∞ to D∞, rather than just find one value for the dimension of aset [The90].

5.2 Multifractal Analysis

Actually, the generalized dimensions is also multifractal analysis, but they are put into aseparate category since they will be tested separately in the simulations. Also, multifrac-tal analysis is generally considered the theory of the multifractal spectra, and thereforethey are all considered separately in this section.

5.2.1 Measures and Local Dimension

We already know the fundamentals of measures from Section 4.2.3, but now we will usethe measure theory more extensively than before. We will denote our measures µ anddefine them to be Borel regular measures in line with the following definition.

Definition 5.2.1: If µ is a measure, then µ is a Borel measure if and only if

µ(A ∪ B) = µ(A) + µ(B) (5.19)

when A, B ⊂ X ⊂ Rn and dist(A, B) > 0, i.e. A ∩ B = ∅. The measure, µ, is alsoBorel regular if every subset of X is contained in a Borel set of the same measure, i.e.since A, B ⊂ X then there exists sets C ⊃ A and D ⊃ B such that µ(A) = µ(C) andµ(B) = µ(D).

The support of a measure µ, written sptµ, is the smallest closed set with complementof measure zero. More formally we define the support of a set as [Fal97]:

Definition 5.2.2: The support of a Borel regular measure is

sptµ = X\ ∪ {U : µ(U) = 0} (5.20)

where U is open.

We define the lower and upper local dimensions (sometimes called the pointwisedimension or Holder exponent) of µ at a point x ∈ Rn as:

Definition 5.2.3: The lower and upper local dimensions are defined as

dimlocµ(x) = lim infr→0

log µ(Br(x))

log r(5.21)

5.2. Multifractal Analysis 31

and

dimlocµ(x) = lim supr→0

log µ(Br(x))

log r(5.22)

respectively.

We say that the local dimension exists for x if dimlocµ(x) = dimlocµ(x), and wedenote the common value as dimloc µ(x). Note that in the above definition dimlocµ(x) =dimlocµ(x) = ∞ if µ(Br(x)) = 0 for some r > 0, i.e. the local dimension is infinite ifthe point x is outside of the support of µ and the local dimension is zero if x is an atom(for example a single point), of µ.

The mapping x 7→ dimloc µ(x) is Borel measurable, and thus, sets such as {x :dimloc µ(x) < c} are Borel sets for all c. This means that we could define the dimensionof a measure itself. And in fact, the Hausdorff dimension of a Borel measure µ can bedefined as [Fal97]:

Definition 5.2.4: The Hausdorff dimension of a finite Borel measure µ is

dimH µ = sup{s : dimlocµ(x) ≥ s} (5.23)

for µ-almost all x.

This definition can be rewritten in terms of sets as [Fal97]:

Definition 5.2.5: For a finite Borel measure µ

dimH µ = inf{dimH E : µ(E) > 0} (5.24)

where E is a Borel set.

This definition might be useful in some cases, but should be used with care. TheHausdorff dimension of a measure is only a crude indicator of the measure’s size [O’N00].

The following proposition states the relationship between the local dimension of thepoints of a set E and the global dimension of the set E [Fal97]:

Proposition 5.2.6: Let E ⊂ Rn be a Borel set and let µ be a finite measure.

(a) If dimlocµ(x) ≥ s for all x ∈ E and µ(E) > 0 then dimH E ≥ s.

(b) If dimlocµ(x) ≤ s for all x ∈ E then dimH E ≤ s.

Proof. Omitted.

5.2.2 The Fine Spectrum

In the fine theory, we look at the local limiting behaviour of µ(Br(x)) for decreasing r,and then examine global behaviour of the sets with the required limiting behaviour.

We let µ be a finite Borel regular measure, as before, and define [Fal97]:

Definition 5.2.7: For 0 ≤ α < ∞ we let

Eα = {x ∈ Rn : dimloc µ(x) = α} (5.25)

= {x ∈ Rn : limr→0

log µ(Br(x))

log r= α}. (5.26)


The above definition thus says that Eα is the set for which the local dimension bothexists (since the local dimension is not allowed to be infinite) and equals α.

In the fine theory we now look at the dimension of Eα, i.e. dim Eα. In most examples,Eα will be dense in sptµ, so we will have dimBEα = dimBEα = dimBsptµ. Rememberfrom Section 4.2.4 why this is a problem. Therefore, the Box-counting dimension cannot be used when discriminating between the different Eα. Thus, the Hausdorff dimen-sion will be used instead, and we have the following definition for the fine multifractalspectrum:

Definition 5.2.8: The Hausdorff multifractal spectrum of µ is

fH(α) = dimH Eα (5.27)

whereEα = {x ∈ Rn : dimloc µ(x) = α} (5.28)

for α ≥ 0.

Obviously, Eα is a subset of the support of µ, and thus we have

0 ≤ fH(α) ≤ dimH sptµ (5.29)

for all α ≥ 0. Also, by Proposition 5.2.6, we have that

0 ≤ fH(α) ≤ α (5.30)

for all α.

5.2.3 The Coarse Spectrum

The definition of the coarse spectrum is closely related to the definition of the Box-counting dimension. We work with r-mesh cubes in Rn, i.e. a grid of cubes with sidelength r. We define the coarse multifractal spectrum of µ as [Fal97]:

Definition 5.2.9: Let µ be a finite measure, α ≥ 0 and let Nr(α) be the number ofr-mesh cubes with µ(A) ≥ rα. The Coarse multifractal spectrum of µ is then defined as

fC(α) = limε→0

limr→0

log+(Nr(α + ε) − Nr(α − ε))

− log r(5.31)

for α ≥ 0 if the double limit exists.

In the definition above we have log+ x = max{0, logx}, since we need to havefC(α) ≥ 0. Thus, the definition implies that if η > 0 and ε > 0 is small enough,then

r−fC (α)+η ≤ Nr(α + ε) − Nr(α − ε) ≤ r−fC (α)−η. (5.32)

I.e., −fC(α) is approximately the power law exponent of the number of r-mesh cubesA such that µ(A) ' rα. I.e. we have a local power law, i.e. a local scaling exponent−fC(α) such that

Nr(α + ε) − Nr(α − ε) ∼ r−fC(α), (5.33)

for arbitrarily small η.If the limit of Equation 5.31 does not exist, we define the lower and upper coarse

multifractal spectra of µ as [Fal97]:

5.3. Estimating the Multifractal Spectra 33

Definition 5.2.10: The lower and upper coarse multifractal spectra of µ are

fC

(α) = limε→0

lim infr→0


− log r(5.34)

and

fC(α) = limε→0

lim supr→0


− log r, (5.35)

respectively, for α ≥ 0. If the limit exists, the lower and upper coarse multifractal spectraare equal, and equals the coarse multifractal spectrum fC(α).

The following proposition gives the relationship between the fine and coarse multi-fractal spectra [Fal97]:

Proposition 5.2.11: Let µ be a finite measure on Rn. Then

fH(α) ≤ fC

(α) ≤ fC(α) (5.36)

for all α ≥ 0.


Just as there are many commonly occurring sets that have equal Hausdorff and Box-counting dimension, there are many sets that have equal fine and coarse multifractalspectra. This is so for self-similar measures, which we will generate in Chapter 9.

5.3 Estimating the Multifractal Spectra

It is often difficult to estimate the multifractal spectrum in practice. It seems like itwould be easy to find the spectrum, given the theory in the previous sections. However,it turns out that the fine spectrum is better suited for mathematical analysis thanpractical estimation, just as the Hausdorff dimension is when we want to estimate thefractal dimension of sets. The coarse theory might seem better then, but even thoughit is closely related to Box-counting, it is too tedious to compute in practice [Fal97].

We will therefore use a direct method equivalent to methods based on Legendretransformations and moment sums, described in the following subsection. But as com-parison, we will also try to find the fine spectra using Box-counting methods accordingto the definition in Section 5.2.2, even though the theory states that it should not beworth the while (see Section 5.2.2 for details).

5.3.1 Legendre Transformation of Moment Sums

The method described in this subsection uses the Legendre transformation of certainmoment sums to find the multifractal spectrum. We start off by defining the Legendretransformation, and we do it as follows (the single variable case) [GNS95]:

Definition 5.3.1: Assume that f(x) is a function of the variable x, and is differentiablefor all x, with the differential

f ′(x) = p(x) =df

dx. (5.37)


The Legendre transformation of the function f(x) is then defined as

g(x) = f(x) − xp(x). (5.38)

The Legendre transformation, g(x), is the intersection of the tangent to f at thepoint (x, f(x)) with the y-axis, see Figure 5.1. This is easy to see since the tangent hasthe equation (consider the first order Taylor expansion of the curve at x0)

T (x) = f(x0) + f ′(x0)(x − x0). (5.39)

Now, the intersection with the y-axis occurs at x = 0, thus when g = T (0), i.e.

g(x0) = T (0) = f(x0) + f ′(x0)(0 − x0) = f(x0) − x0f′(x0) = f(x0) − x0p(x0). (5.40)

The Legendre transformation, g(x), is invertible (of course) but note that it is soonly if f ′(x) is strictly monotonic (i.e. strictly increasing or decreasing).

-

6

��

��

��

��

��

��

��

��

��

��

0

f

x0 x

g(x0)

f(x0)

f(x)

T (x)

Figure 5.1: The definition of the Legendre transformation, g(x0) of f(x). The transfor-mation is where the tangent of the point we are transforming, x0, intersects the y-axis.

Now, the coarse spectrum is related to the Legendre transformation of the powerlaw exponents of moment sums. We cover the set by an r-mesh, and consider the q-thpower moment sums of a measure µ, for which µ(A) > 0, as

Mr(q) =∑

µ(A)q , (5.41)

where the sum is over the r-mesh cubes A.


Now, we let a function β : R → R be defined as follows [Fal97]:

β(q) = lim infr→0

log Mr(q)

− log r(5.42)

and

β(q) = lim supr→0

log Mr(q)

− log r. (5.43)

When their values are equal, we denote their common value by

β(q) = limr→0

log Mr(q)

− log r. (5.44)

This is where the generalized dimensions relate to the multifractal spectrum. If wenormalize the measure in Equation 5.41 we get β when we multiply Equation 5.10 by1 − q, i.e.:

β(q) = −(q − 1)Dq. (5.45)

The function β is convex, and thus has a line of support (i.e. a tangent that always iseither less than or larger than the function or set it is support of) Lα with slope −α, fora range of α ∈ [αmin, αmax], and this support line is unique for every α. The Legendretransformation of β is thus a function f : [αmin, αmax] → R given by the intersectionof the line of support Lα and the vertical axis, see Figure 5.2. By Definition 5.3.1 weimmediately see that the Legendre transformation of β is

f(α) = inf−∞<q<∞

{β(q) + αq}, (5.46)

where α = β′(q) is the slope of the line of support Lα, i.e. the tangent to the curve withderivative α.

Now, the lower and upper Legendre spectra of µ are found using the Legendre trans-formation as [Fal97]:

fL(α) = inf

−∞<q<∞{β(q) + αq} (5.47)

and

fL(α) = inf−∞<q<∞

{β(q) + αq}, (5.48)

and when their values are equal, we call the common value the Legendre spectrum of µ:

fL(α) = inf−∞<q<∞

{β(q) + αq}. (5.49)

The following proposition gives the relationship between the coarse and Legendremultifractal spectra [Fal97]:

Proposition 5.3.2: Let µ be a finite measure on Rn. Then for all α ≥ 0

fC

(α) ≤ fL(α) (5.50)

and

fC(α) ≤ fL(α). (5.51)


-

6

aaaaaaaaaaaaaaaaaaaaaaaa

f(α) = β(q0) + αq0

Lα

β(q)

β

q(q0, β(q0))q0

β(q0)

a

a

Figure 5.2: The Legendre transformation of the β function. The transformation is theintersection with the vertical axis, β.

Proof. The moment sums Mr(q) are related to the Nr(α) as

Mr(q) =∑

µ(A)q ≥ rqαNr(α) (5.52)

for all α ≥ 0 if q ≥ 0. This follows directly by the definition of Nr(α). Then, for ε > 0,Equation 5.52 and Equation 5.34 imply that

Mr(q) ≥ rq(α+ε)Nr(α + ε) ≥ rq(α+ε)r−fC

(α)+ε (5.53)

for all r small enough. It follows from 5.42 that

−β(q) ≤ q(α + ε) − fC

(α) + ε, (5.54)

and thusf

C(α) ≤ β(q) + αq = f

L(α) (5.55)

when ε is arbitrarily small. The above argument also holds for q < 0 by a parallelargument. See the details in [Fal97].

The proof for the upper Legendre spectrum is similar, and instead of using Equa-tion 5.53, we use that

Mr(q) ≥ rq(α+ε)Nr(α + ε) ≥ rq(α+ε)r−fC(α)+ε. (5.56)

for small enough r.

There are many measures for which there is equality in the above proposition, andalso the lower and upper Legendre multifractal spectra are often equal, i.e. the case ofself-similar measures. Very often the coarse spectrum is the Legendre transformation ofa function β(q) that can be more explicitly defined, without the limit, and often thereare good reasons for considering the Legendre spectra to be the coarse spectrum [Fal97].

From the above, we can easily deduce the following proposition:


Proposition 5.3.3: If a fractal is not uniform (i.e. it is a multifractal) then the fL(α)curve is concave.

Proof. ForfL(α) = αq + β(q)

we differentiate and get

dfL

dα=

dq

dαα + q +

dβ

dq

dq

dα=

dq

dα

−dβ

dq+ q +

dβ

dq

dq

dα= q.

Thend2fL

dα2=

dq

dα

and since

α = −dβ

dq

we get thatdα

dq= −d2β

dq2< 0

since β is concave up. Since dαdq < 0 then dq

dα < 0 and d2fL

dα2 < 0, and thus the graph of

fL(α) is concave down.

We prove the above only for the Legendre spectrum, but it is in fact true for allmultifractal spectra.

As a parenthesis, we mention that if we can find a suitable function β with a trans-formation that is a good candidate for the multifractal spectrum, then we can also usethe Legendre transformation for finding the fine multifractal spectrum. A continuousanalogue to the mesh cube moment sums is

β(q) = limr→0

log∫

µ(Br(x))q−1dµ(x)

log r(5.57)

(or an upper or lower limit if the limit above fails to exist). For nicely behaved measures,the Legendre transformation of β gives the fine multifractal spectrum [Fal97].

If we define the value of β as the value of the point where a special purpose variantof the Hausdorff measure jumps from ∞ to 0, then we get the following analogue ofProposition 5.3.2 [Fal97]:

fH(α) ≤ inf−∞<q<∞

{β(q) + qα}. (5.58)

These approaches are, however, more suited for mathematical purposes than practicalestimation.

Implementation

The method of Legendre transformation can be readily implemented using the theorydescribed above. Cover the measure with a r-mesh, and find β as Equation 5.44 suggests.Or, equivalently, find the generalized dimension for discrete values of q in the rangeq ∈ [qmin, qmax], and then find β as in Equation 5.45. When the values of β(q) are found,compute f(α) as in Equation 5.49. To estimate the values of the limits in Equations 5.44


or 5.10, find the best linear fit (in a least-squares sense) for log Mr(q) against log r (whenconsidering Equation 5.44) for successively doubled values of r.

There is a problem with this method, however. For negative q, the method is notstable, and if a mesh cube A only covers a small portion of sptµ, the value of µ(A)q

can be very large. We could try to remedy for this by only counting cubes that cover alarge piece of sptµ [Fal97].

5.3.2 A Direct Method

We could calculate β as Equation 5.44 suggests, and then calculate the multifractal spec-trum by using the Legendre transformation as in Equation 5.49. This is the approachthat is suggested in [HJK+86] and [The90]. These articles suggests finding Dq, translat-ing to τ by τ(q) = (q − 1)Dq (note that τ = −β) and use the Legendre transformationto find f(α). The transformation is done by

f = αq − τ (5.59)

where

α =∂τ

∂q. (5.60)

Actually, given f(α) or Dq (or more precisely τ), each of them can easily be translatedinto the other. We translate back to τ by

τ = αq − f (5.61)

where

q =∂f

∂α. (5.62)

In the past, this was the usual approach since the Dq was easily evaluated for mea-sures arising in computer experiments. When using this method, the Dq curve wasfirst smoothed and then Legendre transformed into fL(α). Even though this methodallows itself to be done numerically, it is difficult to apply this method in practice. Mostprominently, the errors from the smoothing makes the estimation of the errors in thedata more difficult. If the f(α) or τ(q) exhibit any discontinuities, the smoothing willremove these, even though they should be part of the transform. But also, the numericalestimate of the derivative of τ , i.e. when we find α, is prone with errors, especially aftersmoothing Dq [CJ89].

Instead, we use a method introduced by Chhabra and Jensen in [CJ89] that directlyfinds f(α), without using the intermediate Legendre transformation. The method ismathematically precise, and can be directly applied to experimental data.

We want to find the Hausdorff dimension of the measure theoretic support of ameasure P . As we stated in Definition 5.2.5, this is the infimum of the dimensions ofthe sets for which the measure is non-zero. There is a special class of measures thatarise from a multiplicative processes, for which there are many theorems that give thedimension of the measure theoretic support of such a measure. The entropy, S, of sucha process is given by Shannons information formula:

S = −∑

i

pi log pi, (5.63)


and the Hausdorff dimension of M, the measure theoretic support of the measure asso-ciated with such a process, can be related to the entropy by [CJ89]:

dimH M = − limN→∞

1

log N

N∑

i=1

pi log pi. (5.64)

If we put a mesh on the fractal and find the probability for a point to be in the ithbin as before by pi(δ), where δ is the bin size, then Equation 5.64 gives the Hausdorffdimension of the set, which is the measure theoretic support of P .

These results can now be used to find f(α) for a multifractal measure P . We firstconstruct a one-parameter family of normalized measures µ(q) where the probabilitiesin the boxes of size δ are [CJ89]:

µi(q, δ) =pi(δ)

q

∑

j pj(δ)q. (5.65)

Just as in the definition of the generalized dimensions (see Section 5.1), the parameterq provides a microscope for examining different regions of the measure. For q > 1, µ(q)amplifies the more singular regions of P . For q < 1, µ(q) accentuates the less singularregions instead. For q = 1 the measure µ(1) equals the original measure. Now, theHausdorff dimension of the measure theoretic support of the measure µ(q) is given byEquation 5.64 as

f(q) = − limN→∞

1

log N

N∑

i=1

µi(q, δ) log µi(q, δ)

= limδ→0

∑

i µi(q, δ) log µi(q, δ)

log δ. (5.66)

Also, we can compute the average value of the singularity strength αi = log pi/ log δwith respect to µ(q) by evaluating

α(q) = − limN→∞

1

log N

N∑

i=1

µi(q, δ) log pi(δ)

= limδ→0

∑

i µi(q, δ) log pi(δ)

log δ. (5.67)

It can be shown, see [MCSW86], that the following relation exists between Equa-tion 5.66 and 5.67:

f(q) ≤ α(q), (5.68)

with equality when q = 1.Equations 5.66 and 5.67 gives the relationship between the Hausdorff dimension f

and the average singularity strength α as implicit functions of q. It can be shown,see [MCSW86], that these definitions of f(q) and α(q) strongly relate to the generalizeddimensions, and in fact that f = qα− τ and α = dτ/dq, as stated in Equation 5.59 andEquation 5.60 respectively. Thus, these implicit functions precisely equal the Legendretransform relations between the generalized dimensions and the singularity spectrumf(α), as stated in Section 5.3.1, and gives an alternative, efficient and highly accurateway to compute the multifractal spectrum.


Implementation

To find the f(α) spectrum we cover the measure with boxes of side-length δ = 2n,for suitable values of n. We then compute the probabilities, pi, for each of the boxes.Then we calculate the one-parameter normalized measures µi(q, δ), as in Equation 5.65.For each value of q we now evaluate the numerators of Equations 5.66 and 5.67, i.e.∑

i µi(q, δ) log µi(q, δ) and∑

i µi(q, δ) log pi(δ) respectively, for decreasing box sizes δ.Then we extract the values of f(q) and α(q) from the slopes of the least-squares linearfit to the graphs of the numerators versus log δ.

This method has to deal with similar problems as the method of Legendre trans-formations of moment sums. For negative q, the method is not stable either, and if acube A only covers a small portion of sptµ, the value of µ(A)q can become very large.We could use the same method to deal with this as before, i.e. we count only cubesthat cover a large piece of sptµ. However, this remedy does not completely remove theproblem, as can be seen in Chapter 10.

5.3.3 Finding the Fine Spectrum

First of all, the local dimension of all points need to be estimated. This is done by im-plementing Definition 5.2.3, i.e. assuming that dimlocµ(x) = dimlocµ(x) and estimating

dimloc µ(x) = limr→0

log µ(Br(x))

log r. (5.69)

That is, we estimated how much the measure in a k × k sized neighbourhood of a givenpoint changes, when the neighbourhood size is decreased towards zero. The local fractaldimension is then found by finding the least-squares fit to the points of log µ(Br(x)) andlog r.

Now, we want to estimate the fine spectra of Definition 5.2.8, i.e. estimate

fH(α) = dimH{x ∈ Rn : dimloc µ(x) = α}. (5.70)

The values of α are discretized to the interval [αmin, αmax] with step (αmin − αmax)/N ,where N is the number of discrete values for αi. Now, all points with local dimensionin the interval [αi−1, αi) are considered equal to αi. Then the Box-counting dimensionsof the set of points with local fractal dimension equal to αi is an estimation of fH(αi).

Chapter 6

Arithmetics on Fractals

In this chapter we will discuss the different arithmetic operations that we have used whenstudying the properties of fractal dimension. The operations are (Cartesian) product,union and intersection. We will describe each operation, and give a summary of whatthe theory have to say about them.

6.1 Product

We are interested in finding out what happens with the dimension of the product of twosets with known dimensions. Remember that the product, or Cartesian product of twosets is the set of points with first coordinate from the first set and second coordinatefrom the second set, i.e.

E × F = {(x, y) : x ∈ E ∧ y ∈ F}. (6.1)

Example 6.1.1: If E is an interval in R, along the x-axis, and F is an interval in R

along the y-axis, then their product is a rectangle in R2, see Figure 6.1.

F

E

Figure 6.1: The product of two lines in R results in a rectangle in R2.

We need the following lemmas to continue [Fal90]:

41

42 Chapter 6. Arithmetics on Fractals

Lemma 6.1.2: If E ⊂ Rn, F ⊂ Rm are Borel sets with H s(E), H t(F ) < ∞, then

Hs+t(E × F ) ≥ cH s(E)H t(F ) (6.2)

where c depends on s and t.


Lemma 6.1.3: Let F be a Borel subset of Rn with H s(F ) = ∞. Then there is acompact set E ⊂ F such that 0 < H s(E) < ∞.

Proof. Omitted, and complicated, but a sketch of the proof can be found in [Fal90].

With the above lemmas at hand we can now give the following theorems about thedimension of the product of sets [Fal90]:

Theorem 6.1.4: If E ⊂ Rn, F ⊂ Rm are any Borel sets then

dimH(E × F ) ≥ dimH E + dimH F. (6.3)

Proof. If s, t are any numbers with s < dimH E and t < dimH F , then H s(E) =H t(F ) = ∞. Lemma 6.1.3 implies that there are Borel sets E0 ⊂ E and F0 ⊂ F with0 < H s(E0), H

t(F0) < ∞, and by Lemma 6.1.2 H s+t(E × F ) ≥ H s+t(E0 × F0) ≥cH s(E0)H

t(F0) > 0. Hence dimH(E × F ) ≥ s + t. By choosing s and t arbitrarilyclose to dimH E and dimH F , the result follows.

Theorem 6.1.5: For any sets E ⊂ Rn and F ⊂ Rm

dimB(E × F ) ≤ dimBE + dimBF. (6.4)

Proof. Omitted, see [Fal90] for details.

Because of Proposition 4.2.11 we know that if we multiply two sets, their dimensionwill always be less than or equal to the sum of the individual dimensions. If we combinethe above theorems (Theorem 6.1.4 and Theorem 6.1.5) we immediately get the followingimportant corollary.

Corollary 6.1.6: If dimH F = dimBF then

dimH(E × F ) = dimH E + dimH F, (6.5)

anddimB(E × F ) = dimB E + dimB F. (6.6)

Proof. We have that

dimH E + dimH F ≤ dimH(E × F ) ≤ dimB(E × F ) ≤ dimBE + dimBF (6.7)

by Proposition 4.2.11, Theorem 6.1.4, and Theorem 6.1.5. Thus, we must have equalitythroughout the inequality since dimH F = dimBF , and Equation 6.5 follows. Equa-tion 6.6 follows directly from Proposition 4.2.11 and Definition 4.2.9.

6.2. Union 43

6.2 Union

We remember from Section 4.2.5 that the Hausdorff and upper Box-dimensions arefinitely stable. We formalize this property in the following theorem [Fal90]:

Theorem 6.2.1: The Hausdorff and upper Box-dimensions are finitely stable, i.e.

dimk⋃

i=1

Fi = max1≤i≤k

dimFi, (6.8)

for any finite collection of sets {F1, . . . , Fk}, and where dim is either dimH or dimB.

Proof. Omitted.

For the Hausdorff dimension we also have the following theorem [Fal90]:

Theorem 6.2.2: If F1, F2, . . . is a countable sequence of sets, then

dimH

∞⋃

i=1

Fi = sup1≤i<∞

dimH Fi. (6.9)

We say that the Hausdorff dimension is countably stable.

Proof. It has to be that dimH ∪∞i=1Fi ≥ dimH Fj , for each j, since the Hausdorff di-

mension is monotonic. But if s > dimH Fi for all i, then H s(Fi) = 0, so thatH s(∪∞

i=1Fi) = 0, and thus we have the opposite inequality.

The above theorem implies that the dimension of any countable set has Hausdorffdimension zero, which the Box-counting dimension does not need to say [Fal97].

Thus, the dimension of a union of two sets should be the highest dimension of thetwo.

6.3 Intersection

To say anything about the intersection of fractals is very difficult, since obviouslydim(F1 ∩ F2) = dimF1 = dimF2 if F1 = F2 and dim(F1 ∩ F2) = 0 if F1 ∩ F2 = ∅(even if F1 and F2 are congruent). The cases we will be interested in, and the caseswhere we have general theory that applies, are illustrated in the following example.

Example 6.3.1: Consider a line segment, F , in R2 and let F1 be a congruent copy ofF . Now, F ∩ F1 will in general be the empty set, but can, in the exceptional case whenF and F1 are collinear and overlapping, be another line segment. If we let F and F1

meet at an angle, then their intersection is a single point. If we now let F2 be anothercongruent copy of F , and F2 is similar to F1, it will also intersect F in a point, and sowill all other similar congruent copies of F . Thus, the case where F and Fn intersect ata point is exceptional, but it occurs frequently (in fact infinitely often.) See Figure 6.2.

There thus seems to be something we can say about the typical case. The followingproposition gives limits for the intersection of two sets when one is translated by a pointx [Fal90].


````````b

bb

bb

bb

bb

bb

bb

bb

bbb

JJ

JJJ

JJJ

F

F1

F

F1

F

F1

F2

a) b) c)

Figure 6.2: The intersection of two lines are in general the empty set a), but in thenormal case it is a point, b) and c).

Proposition 6.3.2: If E, F are Borel subsets of Rn then

dimH(E ∩ (F + x)) ≤ max{0, dimH(E × F ) − n} (6.10)

for almost all x ∈ Rn.

Proof. The proof uses some theorems we do not need to restate here, and thus is omitted.The proof when n = 1 can be found in [Fal90].

In the following proposition we need the definition of codimension, which we stateas follows:

Definition 6.3.3: Let F ⊆ Rn, dimF = d and d ≤ n, then the codimension of the setF is defined as

codimF = n − d. (6.11)

Thus, the codimension of a set is based on the dimension of the set and the set inwhich it exists. I.e. if E is a point in R2, then its codimension is 2− 0 = 2, if F is a linein R2, then its codimension is 2 − 1 = 1, if G is a plane in R3, then its codimension is3 − 2 = 1, and so on.

We use the above definition in the following theorem [BGZ00]:

Theorem 6.3.4: Let E, F ⊆ Rn be Borel sets and write t = dimH E and s = dimH F .Assume that

1. 0 < H t(E) < ∞,

2. lim infr→01rs H s(F ∩ Br(x)) > 0 for all x ∈ F .

If 0 < dimH E < n, 0 < dimH F < n and dimH E + dimH F − n > 0, then

codim H(E ∩ F ) = codim HE + codim HF, (6.12)

and

dimH(E ∩ F ) = dimH(E) + dimH(F ) − n. (6.13)

6.3. Intersection 45

Proof. The proof of Equation 6.12 is omitted, but the proof of Equation 6.13 followsimmediately from Equation 6.12 and Definition 6.3.3 since, by the definition of codi-mension, we have that

codim H(E ∩ F ) = codim HE + codim HF,

equalsn − dimH(E ∩ F ) = (n − dimH(E)) + (n − dimH(F )),

which givesdimH(E ∩ F ) = dimH(E) + dimH(F ) − n.

The theorem says that Equation 6.13 holds for all non-empty intersections of non-empty sets with dimH E +dimH F > n. If this property is not fulfilled, the dimension ofthe intersection is simply empty. However, there exists theory in which it makes perfectsense to talk about negative dimensions, in which this last restriction does not matter.

Consider, for example, the intersection of two lines and the intersection of a line anda plane. Wouldn’t it make sense to say that the intersection of two lines is emptier thanthe intersection of a line and a plane, even though they are both of dimension zero?Consider also a point and the empty set. They are both of dimension zero, but isn’t theempty set emptier than the set of just one point? See [Man90] for more information onnegative dimensions.

Chapter 7

Projections of Sets

In this chapter we will discuss the projection, or shadow of a set in Rn to lower di-mensional subspaces, and how the projection affects the dimension of the projected set.In general, we will work with projections from R3 to R2, but we will build the theorystarting with projections from R2 to R1. Almost all projections of a line from R2 to R1

or from R3 to R2 is a line with dimension 1, and almost all projections of plane figuresor solids from R3 to R2 is a plane figure with dimension 2, see Figure 7.1.

We will also describe the different methods of projection that we have used in thiswork, and how they relate to each other.

7.1 Projecting from R2 to R1 and from R3 to R2

First of all we need to introduce the notion of a Lipschitz mapping, which is a specialcase of the Holder condition [Fal90]:

Definition 7.1.1: Let F ⊂ Rn and f : F → Rm, then a mapping such that

|f(x) − f(y)| ≤ c|x − y|α, (x, y ∈ F ) (7.1)

for constants c > 0 and α > 0, is called a Holder condition of exponent α.

The Holder condition implies that f is continuous. Particularly important is thefollowing special case:

Definition 7.1.2: The case when the Holder condition has exponent α = 1, i.e.

|f(x) − f(y)| ≤ c|x − y|, (7.2)

is called a Lipschitz mapping.

We will need the following lemma, proposition and corollary to continue [Fal90]:

Lemma 7.1.3: Let F ⊂ Rn and f : F → Rm be a mapping that meets a Holdercondition:

|f(x) − f(y)| ≤ c|x − y|α, (x, y ∈ F )

for constants c > 0 and α > 0. Then for each s

Hs/α(f(F )) ≤ cs/α

Hs(F ). (7.3)

47

48 Chapter 7. Projections of Sets

Figure 7.1: The projections of a line, a plane figure, and a cube onto a plane. Theline is for almost all projection angles a line, the plane figure is for almost all angles aplane figure and the cube is for all angles a plane figure.

Proof. If {Ui} is a δ-cover of F , then, since |f(F ∩Ui)| ≤ c|Ui|α it follows that {f(F ∩Ui}is an ε-cover of f(F ), where ε = cδα. Thus

∑

i |f(F ∩ Ui)s/α ≤ cs/α

∑

i |Ui|s, so that

Hs/α

ε (f(F )) ≤ cs/αH sδ (F ). As δ → 0, so ε → 0, and the proposition follows.

Proposition 7.1.4: Let F ⊂ Rn and suppose that f : F → Rm meets a Holder condition

|f(x) − f(y)| ≤ c|x − y|α, (x, y ∈ F )

then dimH f(F ) ≤ 1α dimH F .

Proof. If s > dimH F then by Lemma 7.1.3 H s/α(f(F )) ≤ cs/αH s(F ) = 0, implyingthat dimH f(F ) ≤ s/α for all s > dimH F .

Corollary 7.1.5:

(a) If f : F → Rm is a Lipschitz mapping then dimH f(F ) ≤ dimH F .

(b) If f : F → Rm is a bi-Lipschitz mapping, i.e.

c1|x − y| ≤ |f(x) − f(y)| ≤ c2|x − y|, (x, y ∈ F ) (7.4)

where 0 < c1 ≤ c2 < ∞, then dimH f(F ) = dimH F .

Proof. Part (a) follows directly from Proposition 7.1.4 when α = 1. Using (a) andletting f−1 : f(F ) → F , the opposite inequality needed for (b) follows.

Now, we let Lθ denote a line through the origin of R2 that makes an angle θ withthe horizontal axis, and we let projθ denote the orthogonal projection onto Lθ. Thus, ifF ⊂ R2, then projθF is the projection of F onto the line Lθ, see Figure 7.2.

Obviously, |projθx − projθy| ≤ |x − y| if x, y ∈ R2, and thus is a Lipschitz mapping.We must then have dimH(projθ) ≤ min{dimH F, 1} for any F ⊂ R2 and θ ∈ [0, π) byCorollary 7.1.5 (a), and since the projection is on a line it cannot have dimension greaterthan 1. We have the following important theorem [Fal90]:

7.1. Projecting from R2 to R1 and from R3 to R2 49

�� Lθ

θ

projθF

F

��

m&%'$

��

��

Figure 7.2: Projection of the set F onto the line Lθ, projθF .

Theorem 7.1.6 (The Projection Theorem): Let F ⊂ R2 be a Borel set.

(a) If dimH F ≤ 1 then dimH(projθF ) = dimH F for almost all θ ∈ [0, π).

(b) If dimH F > 1 then projθF has positive length and has dimension 1 for almost allθ ∈ [0, π).

Proof. The proof uses theoretic characterizations of the Hausdorff measure and is verycomplex. Because of this it is omitted here, but can be found in [Fal90].

The above theorem says what we already deduced earlier, that the dimension of aprojection from R2 to R1 is dimH(projθ) ≤ min{dimH F, 1} for all projection angles, butthat we have equality most of the time.

These results are the oldest, and are only for the Hausdorff dimension. If dimH F =dimB F then we get the same results for the Box-counting dimension, but we are notalways that lucky. We can, however, easily deduce similar results for the Box-countingdimension. Since the projection is a Lipschitz mapping, it can be shown that we have

dim(projθ(F )) ≤ min{dimF, 1} (7.5)

in the general case [NG06].The Projection Theorem in Theorem 7.1.6 generalizes to higher dimensions in the

following way. Let Gn,k be a set of k-dimensional subspaces (Gn,k is called the Grassmanmanifold of all k-dimensional subspaces of Rn), or k-dimensional planes through theorigin in Rn. We then write projΠ for the orthogonal projection onto the k-plane Π. Wehave the following theorem [Fal90]:

Theorem 7.1.7 (Higher-Dimensional Projection Theorem): Let F ⊂ Rn be aBorel set.

(a) If dimH F ≤ k then dimH(projΠF ) = dimH F for almost all Π ∈ Gn,k.

(b) If dimH F > k then projΠF has positive k-dimensional measure and so has dimen-sion k for almost all Π ∈ Gn,k.

Proof. The proof of Theorem 7.1.6 is extended to higher dimensions without difficulty.Omitted here though, but can be found in [Fal90].


In particular, Theorem 7.1.7 applies for k = 2, when projecting sets F ⊂ R3 toa 2-dimensional plane through the origin in R3, see Figure 7.1. In analog with projθ,if F ⊂ R3, the projections onto the plane Π, projΠ will in general have dimensionmin{dimH F, 2}. This means that if the dimension of a projection can be found, esti-mated or calculated, and it is less than 2, then we immediately know the dimension ofthe projected object.

It also can be shown that we have a best possible lower bound on the Box-countingdimension of projections of sets, expressed in the dimension of the original set. We havethe following theorem [FH96]:

Theorem 7.1.8: For F ⊆ Rn we have

dimBprojΠF ≥ dimBF

1 + (1/k − 1/n)dimBF(7.6)

for almost all Π ∈ Gn,k, and

dimBprojΠF ≥ dimBF

1 + (1/k − 1/n)dimBF(7.7)

for almost all Π ∈ Gn,k, where Gn,k is the Grassman manifold of all k-dimensionalsubspaces of Rn.

When the upper and lower Box-counting dimensions are equal, Equation 7.6 andEquation 7.7 are valid for the Box-counting dimension, i.e. we have that

dimB projΠF ≥ dimB F

1 + (1/k − 1/n) dimB F(7.8)

Proof. Omitted, but can be found in [FH96]. Equation 7.8 follows directly from Defini-tion 4.2.9, Equation 7.6 and Equation 7.7.

Thus, solving for dimB F in Equation 7.8 we get an upper bound for the Box-countingdimension of the original set, expressed in the dimension of the projected set [NG06]:

dimB(projΠF )

1 − (1/k − 1/n) dimB(projΠF )≥ dimB F. (7.9)

However, using Proposition 4.2.11, we also get an upper bound for the Hausdorffdimension of the original set from the Box-counting dimension of the projected set.I.e., if we can estimate the Box-counting dimensions of the projected set, we know thetheoretical upper limit for the Box-counting and Hausdorff dimensions of the originalset.

7.2 Projection types

In this thesis, we have used three types of projections when doing the experiments. Thefirst is orthogonal projection, sometimes called ideal projection, which is the one withthe strongest theoretical foundations. The second projection is a more similar to x-rayprojection in that it uses a projection foci that sends rays from the foci to each point,and then onto the projection plane. The third one is even more like x-ray projectionsin that it sends rays from a projection foci, and the points along the ray’s path areattenuated with the ray onto the plane.

7.2. Projection types 51

��1

��1

�

BB

BB

BBB

��BB

θ

a

p = ka = aTbaTa

a

b

e = b− p

Figure 7.3: The projection, p, of b onto the line that is constituted by a.

The following three subsections describe the three different projection methods andhow they relate to each other.

To avoid confusion between scalars and vector, we will in this chapter denote vectorsby boldface letters, i.e. v, while scalars are typeset as before, e.g. x. Matrices will alsobe in boldface but will likely not cause any confusion.

7.2.1 Orthogonal Projections

All the above projection theorems apply when we deal with orthogonal projections (thatwas a condition for the theorems). We start by describing how to project a point into aline, and then move on to describe how to project a point onto a plane. Given a vector,a = (a1, . . . , an), that represents the direction of the line, we want to find the point, p,along the line that is closest to the point b = (b1, . . . , bn) we want to project. The keyidea here is that the vector from p to b, namely b − p, is orthogonal to a [Str05]. Thepoint p is a constant multiple times a, i.e. p = ka, the vector b − p is thus b−ka. Now,two vectors are orthogonal if their dot product is zero, thus a·(b−ka) = a·b−ka·a = 0,and solving for k yields

k =a · ba · a =

aTb

aTa, (7.10)

and thus, the projection, p, of b onto the line through a is given by

p =aTb

aTaa = ka = ak = a

aTb

aTa=

aaT

aTab = Pb. (7.11)

The above way is thus how we would project a set in R2 onto a line, see Figure 7.3.Now we go on to describe how to project a set in R3 onto a plane; even though thetheory is generalized to sets in Rn.

Given a set of n vectors a1, . . . ,an that are linearly independent, we want to find thevector p = k1a1 + · · · + knan that is closest to a point b = (b1, . . . , bn), see Figure 7.4.We let the vectors a1, . . . ,an be the columns of a matrix A, and thus we have p = Ak,where k = (k1, . . . , kn). Just as before, we use the fact that the vector from p to b isorthogonal to the subspace spanned by a1, . . . ,an (the line in the 1-dimensional case,when n = 1). That vector is b−Ak, and because p is orthogonal to A, it is orthogonal


BBBBBBBB

��

��

BBBBBBBB��:

��

��

�

@@

@@R

a2

a1

b

e

p = Ak = PbaT

1 e = 0

aT2 e = 0

ATe = AT(b − Ak) = 0

Figure 7.4: The projection p is the nearest point to b in the column space of A. Theperpendicular difference e must be orthogonal to a1 and a2.

to all a1, . . . ,an [Str05]. Thus

aT1 (b − Ak) = 0

...aT

n (b − Ak) = 0

or, more elegantly

AT(b − Ak) = 0

ATAk = ATb. (7.12)

If the vectors a1, . . . ,an are linearly independent, the matrix ATA in Equation 7.12, isinvertible, and thus we find the coefficient vector k as

k = (ATA)−1ATb, (7.13)

which gives the projection of b onto the subspace spanned by the column vectors of Awhen inserted into p = Ak. Thus

p = Ak = A(ATA)−1ATb = Pb (7.14)

is how we find our projection matrix. When we have a set F in R3 that we want toproject onto a plane, we have two linearly independent vectors, a1 and a2 that span theplane. Those two vectors constitute the columns of the matrix A. We simply calculatethe projection matrix P according to Equation 7.14 and multiply the points in F by Ato project them onto the plane spanned by a1 and a2.

7.2.2 Projection with Foci – Central Projection

The orthogonal projection is mainly of theoretical interest, since in practice we will ingeneral not have this kind of projection. Therefore, we are investigating how a morerealistic type of projection works, and how it is related to orthogonal projection.

With central projection, we have a foci point, p, from which photons radiate, andfor every point in the set F , the point is attenuated with the photon and hits the planeon the photon’s way down. See Figure 7.5 for an illustrative example.

7.2. Projection types 53

Figure 7.5: Central projection of a set F onto the image plane.

More formally we define this type of projection, from R3 to R2, as follows. LetF ⊂ R3 be a bounded set above the xy-plane, Π. Let p = (x0, y0, z0) ∈ R3 be a pointfor which

z0 > sup{z : (x, y, z) ∈ F},i.e. the point p is above all points in F .

Now, we denote central projection of a set F onto the plane Π, using p as the centerof the projection, as projΠ,pF . With the above choice of z0, the mapping

f(x, y, z) =

(

x0 +x − x0

z0 − zz0, y0 +

y − y0

z0 − zz0, z

)

(7.15)

is a bi-Lipschitz mapping of F such that projΠ,pF = projΠf(F ) [NG06]. Thus, fromCorollary 7.1.5, we see that

dimH projΠ,pF = dimH projΠf(F ). (7.16)

The above is also true for the Box-counting dimension [NG06].Points are readily projected by this method. Either we can use Equation 7.15 to

transform all points and the simply project them orthogonally by Equation 7.14, or wecan deduce how to find the intersection of a line and a plane. We do both.

Let a plane Π be given by nx = d, where n is the plane’s normal, x is a point in theplane and d is the orthogonal distance from the origin. Let a line segment be given bythe parametric equation L(t) = a+t(b−a), for 0 ≤ t ≤ 1, where a and b are two differentpoints on different sides of the plane. The value of t gives the point of intersection withthe plane, and is found by substituting the parametric equation by the point x in theequation for the plane, and solving for t [Eri04]:

n(a + t(b − a)) = d


na + nt(b − a)) = d

tn(b − a) = d − na

t =d − na

n(b − a).

The expression for t can now be inserted into the parametric equation of the line to findthe intersection point q:

q = a +

[

d − na

n(b − a)

]

(b − a). (7.17)

If the points are not on different sides of the plane, the line segment will not intersectthe plane, and thus t will be smaller than zero or greater than one. If we are interestedof where a line intersects a plane and not only where and if a line segment does, theequation still works, but t will be arbitrary.

The Equation 7.17 obviously does not work in the case where we divide by zero.This happens when the line is parallel to the plane, in which case they do not intersecteach other.

7.2.3 X-Ray Simulated Projection

Central projection is more realistic than orthogonal projection, in that every ray fromthe foci to the plate on which the photons hit a point will attenuate the point to theplane. To test how the attenuation properties of x-rays affect the estimated dimension,we have tested yet another projection type.

X-ray photons are created when electrons are accelerated in a potential field, andhigh energy electrons collide with an anode plate. The plate is a slightly curved surface,that constantly rotates to avoid getting too hot. From the plate, the electrons radiatein all possible directions. To control the direction of the x-ray beam, and to minimizethe dangerous effects of x-ray radiation, a metal plate with a rectangular hole in it issituated beneath the anode. The x-rays will then only radiate through the rectangularhole, and for any practical purpose we can assume the focus area to be a convex regionabove the image plane [NG06].

The idea with this method is to radiate x-rays in many directions, and attenuatepoints that are close to the ray, where close is by means of some function of the distanceto the line.

We divide the plane into an r-mesh, and towards every mesh cube’s midpoint froma foci point, p, we send a ray towards the corners. All points that are close enough tothe ray are attenuated with the ray. See Figure 7.6 for an illustrative example.

However, when the mesh cube side length tends towards zero, i.e. limr→0, we willhave one line for each point in the set F . Thus, this method is identical to central pro-jection in that case. The advantage of this method is that we can apply any attenuationfunction to the ray, and thus make each point affect the attenuation in whatever waywe like. We could, for example, let the x-rays be affected by each point according to aGaussian function with mean in the point and some standard deviation. Thus, all pointwould affect all x-rays, but only the ones close to the x-ray would be significant.

This makes the projection even more realistic, since in real a x-ray system, the raysare scattered when they hit the molecules of the object, and may hit other molecules,molecules not on the straight line path towards the detector, blurring the resulting x-rayimage.

7.3. Problems with Projections 55

Figure 7.6: Rays are sent from the focus onto the image plane. Points that are closeenough (by means of some function of the distance to the ray) are attenuated with theray onto the plane. The points that where attenuated are marked with the orthogonalprojection line onto the ray. Note that a point can be attenuated with several rays.

7.3 Problems with Projections

The theorems in this chapter are all mathematically sophisticated, but cannot give acomplete answer to the question of the dimension of a projected set. A subset F of R2

with Hausdorff dimension 1, need not have dimension 1 when projected; it may not evenhave positive Hausdorff measure. In fact, sets can be constructed that give projections ofwhatever we want in whatever projection angle we want. A practical example is digitalsundials. When the sun is shining from a particular direction, the shadow is a digitalimage of the current time. See Figure 7.7 for a live example.

Figure 7.7: A digital sundial in the Sundial Park in Gent, Belgium.

Chapter 8

Projections of Multifractals

We will investigate the relationship between the multifractal spectrum of a measureon Rn and the multifractal spectrum of its projection onto a lower dimensional sub-space, Rm. We have already covered projections of uniform sets onto lower dimensionalspaces, and one may deduce from the dimension of a measure (see Section 5.2) andthe projection theorems the behaviour of the Hausdorff dimension of a measure underprojection [O’N00].

We are interested in relationships between multifractal properties of µΠ and corre-sponding properties of µ that hold for almost all Π ∈ Gn,k. It turns out that this is notas intuitive and easily deducable as the corresponding cases for uniform fractals.

8.1 Projections of Measures

We denote the projection of µ, a Borel probability measure on Rn supported by acompact set, onto a linear subspace Π by µΠ, and define it by [FH96]:

Definition 8.1.1: The projection onto Π of a measure µ is denoted by

µΠ(A) = µ({x ∈ Rn : projΠx ∈ A}) (8.1)

for all A ⊆ Π.

We have the following proposition for the Hausdorff dimension of the projection ofa measure [FO99]:

Proposition 8.1.2: For µ, a Borel regular probability measure, we have that for almostall Π ∈ Gn,k

dimH µΠ = min{k, dimH µ}. (8.2)

Proof. Omitted.

We may hope that for typical projections of a measure µ onto Π ∈ Gn,k, we haveDq(µΠ) = min{Dq(µ), k}. While this is true under some circumstances, the reality ismore complicated. We can, however, say the following about the generalized dimensionsof a projected measure [FO99]:

57

58 Chapter 8. Projections of Multifractals

Proposition 8.1.3: Let 1 ≤ k ≤ n and µ be a Borel regular probability measure. Forall Π ∈ Gn,k and q > 0

max{0, Dq(µ) − (n − k)} ≤ Dq(µΠ) ≤ min{Dq(µ), k} (8.3)

andmax{0, Dq(µ) − (n − k)} ≤ Dq(µΠ) ≤ min{Dq(µ), k}. (8.4)

Proof. Omitted, but can be found in [FO99].

For a special case of the above, we can give even stronger bounds [FO99]:

Proposition 8.1.4: Let µ be a Borel regular probability measure. For all 1 < q ≤ 2and almost all Π ∈ Gn,k

Dq(µΠ) = min{Dq(µ), k} (8.5)

Proof. Omitted, but can be found in [FO99].

In the case when Dq = Dq we have the following proposition [HK97].

Proposition 8.1.5: Let µ be a Borel probability measure with compact support, and let1 < q ≤ 2. If Dq = Dq, then for almost all Π ∈ Gn,k we have that

Dq(µΠ) = min{Dq(µ), k} (8.6)

Proof. Omitted, but can be found in [HK97].

Remember from Section 5.2.1 that the upper and lower local dimension of a measureat a point in the support of the measure is defined as:

dimlocµ(x) = lim infr→0

log µ(Br(x))

log r(8.7)

and

dimlocµ(x) = lim supr→0

log µ(Br(x))

log r(8.8)

respectively.We can say the following about the local or pointwise dimension [HK97]:

Proposition 8.1.6: Let µ be a Borel probability measure with compact support. Foralmost all Π ∈ Gn,k

dimlocµΠ(projΠx) = min{k, dimlocµ(x)} (8.9)

for almost every x with respect to µ. If dimlocµ(x) = dimlocµ(x) for almost all x, thenfor almost every Π ∈ Gn,k the local dimension exists (i.e. dimlocµΠ = dimlocµΠ) atprojΠx and is given by

dimloc µΠ(projΠx) = min{k, dimloc µ(x)} (8.10)

for almost all x.

Proof. Omitted, but can be found in [HK97].

8.1. Projections of Measures 59

It can be shown that∫

dimlocµ(x)dµ(x) ≤ D1(µ) ≤ D1(µ) ≤∫

dimlocµ(x)dµ(x). (8.11)

Thus, if the local dimension of µ exists almost everywhere, then the information dimen-sion of µ exists and is given by the average local dimension. I.e.

D1(µ) =

∫

dimloc µ(x)dµ(x). (8.12)

From this and Proposition 8.1.6 the following corollary immediately follows [HK97]:

Corollary 8.1.7: Let µ be a Borel probability measure with compact support. If the localdimension dimloc µ(x) exists and does not exceed m for almost all x, then for almost allΠ ∈ Gn,k, the information dimension of µΠ exists and is given by

D1(µΠ) = D1(µ). (8.13)

In fact, the above corollary can be generalized as follows [BB06]:

Proposition 8.1.8: Let 0 < q ≤ 1, and µ be a Borel probability measure with compactsupport. If Dq(µ) ≤ k, then for almost all Π ∈ Gn,k we have

Dq(µΠ) = Dq(µ). (8.14)

We have the following proposition about the multifractal spectrum:

Proposition 8.1.9: Let µ be a Borel probability measure with compact support in Rn.Projecting µ onto the lower dimensional subspace Π ⊂ Rk, k < n, has the consequencethat

maxα

{fµΠ(α}) ≤ max

α{fµ(α)}. (8.15)

Proof. This is a direct consequence of Theorem 7.1.7 and Equation 5.29.

We conclude with a short description of what it means to project a measure. Ameasure is a set function, it operates on a set. The set can of course be manipulatedas usual, with the measure conforming to the new circumstances. Projecting a measuremeans projecting the set it measures, and measuring the projected set. See Figure 8.1.

Sierpinski triangle Random fractal

Figure 8.1: An illustration of what it means to project a measure in R2 onto R1.

60 Chapter 8. Projections of Multifractals

Chapter 9

Generating Fractals

In this chapter we describe two ways to create homogeneous fractal images or point sets.The first method described is called Iterated Function System, IFS, and the second is aprobabilistic method that use properties of the definition of the Box-couting dimensionto create fractals.

We also describe two methods to generate multifractal images or points sets. Bothmethods are based on the idea of Iterated Function Systems.

9.1 Generating Homogeneous Fractals

In this section we describe the two methods to create fractals. The first method generatesfractals from a set of transformations using Iterated Function Systems and the secondmethod created fractals with arbitrary dimension, i.e. whatever dimension we want.

9.1.1 Iterated Function Systems

Many fractals are self-similar, i.e. they are made up of parts that are similar to thewhole, but often scaled and translated. These self-similarities are not only properties ofthe fractals but can in fact be used to generate them [Fal90].

Let F be a subset of Rn, then the mapping S : F → F is called a contraction on Fif there is a number c with 0 < c < 1 such that |S(x) − S(y)| ≤ c|x − y| for x, y ∈ F . If|S(x)− S(y)| = |x− y|, then S is called a similarity. Let S1, . . . , Sn be contractions (oraffine transformations), then we say that the set F is transformation invariant if

F =m⋃

i=1

Si(F ). (9.1)

This is illustrated in Example 9.1.1.

Example 9.1.1: Consider the middle third Cantor set (see Figure 4.1.) Let S1, S2 :R → R be given by S1(x) = 1

3x; S2(x) = 13x + 2

3 . Then S1(F ) and S2(F ) are the twoparts of F , such that F = S1(F )∪S2(F ), i.e. F is invariant for the two transformationsthat make up the Cantor set.

Such a set of contractions is called an Iterated Function Scheme or Iterated FunctionSystem, and a set that can be expressed as a union of sets, each of which is a reduced

61

62 Chapter 9. Generating Fractals

copy of the full set, is called a strictly self-similar set [The90]. We have the followingtheorem [Fal90]:

Theorem 9.1.2: Let S1, . . . , Sm be contractions on D ⊂ Rn such that

|Si(x) − Si(y)| ≤ ci|x − y| (9.2)

with ci < 1 for each i. Then there exists a unique non-empty compact set F that isinvariant for the Si, i.e. which satisfies

F =

m⋃

i=1

Si(F ). (9.3)

Moreover, if we define a transformation S on the class S of non-empty compact sets by

S(E) =m⋃

i=1

Si(E) (9.4)

and write Sk for the kth iterate of S given by S0(E) = E, Sk(E) = S(Sk−1(E)) fork ≥ 1, then

F =

∞⋂

k=1

Sk(E) (9.5)

for any set E in S such that Si(E) ⊂ E for each i.


What the above theorem says is that for a contracting transformation on a set,there is always some set that is invariant of the transformation, and also, the repeatedapplication of the transformations on any set yields better and better approximationsto the set which is invariant to the transformations.

Returning to the Cantor set, we can see that if E = [0, 1], then using S1 and S2

from Example 9.1.1 yields Sk(E) = Ek, where Ek is described in Example 4.1.1, andthe Cantor set is obtained when k tends to infinity, see Figure 4.1 d).

We can now construct fractals by letting an initial set of only one single point x0 betransformed in the above way. Select the initial point x0, let it be any point. Randomlyselect a contraction Si1 from S1, . . . , Sk and let x1 = Si1(x0). Iterate this, choosingrandomly a contraction Sik

from S1, . . . , Sk and letting xk = Sik(xk−1), for k = 1, 2, . . ..

For large k, the points will be indistinguishably close to the fractal. Thus, plotting thepoints x0, x1, . . . , xk will yield an image of the fractal generated. If the point x0 waschosen arbitrarily, the hundredth or so first points may be ignored, since they mightnot have converged to the fractal yet, but if x0 was chosen so that x0 ∈ F from thestart then it will always stay in F , and thus, this is not necessary. See Figure 9.1 forillustrations of how the algorithm works.

We have the following theorem about the dimension of a fractal made up of unionsof similarities of a set F [Fal90]:

Theorem 9.1.3: If there exists a non-empty bounded open set V such that

V ⊃m⋃

i=1

Si(V ) (9.6)

9.1. Generating Homogeneous Fractals 63

a) b) c)

Figure 9.1: For the Sierpinski triangle we have three contractions: S1(x) = (0.5 0.5) ·x, S2(x) = (0.5 0.5) ·x+(0.25 0.5), S3(x) = (0.5 0.5) ·x+(0.5 0.0). a) The first six pointsof the IFS are found. Adjacent points are connected by line segments. b) 100 points ofthe IFS are found, note that only the first few are outside the triangle lines. c) 10,000points of the IFS are found, the fractal structure is apparent.

with the union disjoint, for the similarities Si on Rn with ratios ci (1 ≤ i ≤ m), and Fis the invariant set satisfying

F =

m⋃

i=1

Si(F ) (9.7)

then dimH F = dimB F = s, where s is given by

m∑

i=1

csi = 1. (9.8)

Moreover, for this value of s, 0 < H s(F ) < ∞.


If the property in Equation 9.6 does not hold, it can still be shown that dimH F =dimB F , but this value may be less than s.

The above theorem says that for sets that do not overlap too much, and that aremade up of similar copies of the whole, the Hausdorff and Box-counting dimensions areequal. E.g. this holds for the Cantor set, the Sierpinski triangle, the von Koch curve,and all other self-similar fractals that we work with in this text.

9.1.2 Custom made Random Fractals

The innate regularity of the above fractals become a problem when projecting from R3 toR2. The projection theorems stated above are said to be valid for almost all projectionangles. This limitation is not a limitation in the continuous world of mathematics (sincethe set of problem angles have zero measure), but when we try to estimate the dimensionof a projected set it turns out that the problematic angles appear more often than wewould like them to do. To avoid this, a new set of fractals with arbitrary dimension hasbeen created.


Figure 9.2: A fractal created with the random method described in the text. The fractalis given a finer and finer structure by successively dividing the squares in smaller squares.The above fractal would have a dimension of s = log 12

log 1

5

≈ 1.54 in the limit.

Since we want the dimension to be

s = limδ→0

log Nδ(F )

− log δ

for F ⊂ Rn, and s ∈ (0, n),we can approximate s by s, having an error of at most ε, bythe fraction

s =log d1

− log 1d2

(9.9)

by finding the constants d1, d2 ∈ Z+ such that |s − s| ≤ ε. When we have found theconstants d1, d2, we divide a unit n-dimensional hypercube into dn

2 hypercubes of equallength, and randomly keep d1 of those hypercubes. The d1 hypercubes that we keptare divided into dn

2 hypercubes with equal side length, and d1 of those hypercubes arerandomly selected for the same procedure to be repeated ad infinitum, see Figure 9.2.

This kind of sets, that, when magnified, does not precisely reproduce the entireoriginal set, but have the same qualitative features, are called statistically self-similarsets [The90].

9.2 Generating Multifractals

In this section we will generate multifractal measures. The first method considered isa variant of the Iterated Function System described above. The second method is arecursive algorithm that generates true measures when given a set of measures for eachof the transformations involved.

9.2.1 Probabilistic Iterated Function System

Let {S1, . . . , Sm} be an Iterated Function System on X ⊂ Rn and let p1, . . . , pm beprobabilities with 0 ≤ pi ≤ 1 for all i and

∑mi=1 pi = 1. Recall from Section 9.1.1

that an Iterated Function System is given by a set of transformations, mappings byS : X → X .

In the iterated function systems described in Section 9.1.1, we randomly selected oneof the transformations and applied it to the set. However, all the probabilities whereuniform, i.e. all transformations were selected with the same probability. Now, if we let

9.2. Generating Multifractals 65

a) b) c)

Figure 9.3: The Random Iteration Algorithm has been applied to the Sierpinski trianglewith probabilities p1 = 0.6, p2 = 0.3, p3 = 0.1 for the bottom left, bottom right and toptriangles respectively. In a) the image is created with 5,000 iterations, in b), the imageis created with 25,000 iterations, and in c) the image is created with 100,000 iterations.Note how the triangle becomes more apparent the more iterations we use, and also thatthe parts with higher probabilities are darker, i.e. have more points on them.

different transformations have different probabilities, the different parts of the fractalwill fill in at different rates. We select an initial point x0 ∈ A, and select one of thetransformations S1, . . . , Sk at random. The probability that Si is selected is pi, fori = 1, . . . , k. The selected transformation is applied to x0 and produces x1. Again, weselect a transformation Si with probability pi and apply it to x1 to produce x2. Thisprocess continues for N steps, producing a finite series of points x0, x1, . . . , xN , whichwill all lie on the attracting fractal [Bar88]

The rate at which the figure is filled in depends on the probabilities, and if we iteratefor N steps, then we will have approximately piN points in the part that correspondsto transformation Si. Thus, this process gives us a non-uniform measure on the fractal.Remember that even though the different parts of the fractal fills in at different rates,the attractor is still the same as it would be if we had uniform probabilities for thetransformations. See the following example.

Example 9.2.1: We take the Sierpinski triangle as an example. We use an iteratedfunction system with the following three transformations:

S1(x, y) = (x/2, y/2)

S2(x, y) = (x/2, y/2) + (1/2, 0)

S3(x, y) = (x/2, y/2) + (1/4, 1/2)

Now, if we give these transformations different probabilities, e.g. p1 = 0.6, p2 = 0.3, p3 =0.1 (note that

∑

pi = 1), and run the iterated function system on them, selecting trans-formation Si with probability pi, the fractal will be filled in completely when the numberof iterations tend towards infinity. But before that, the regions will be filled in at dif-ferent rates. See Figure 9.3 for an illustration of this example. The iterated functionsystem is for the Sierpinski triangle, and we have used the above probabilities.

We may define a measure µ on this hierarchy of sets that come from the recursive


calls of Si on X such that

µ(Si1(. . . (Sik(X)) . . .)) = pi1pi2 · · · pik

. (9.10)

This can be extended to a Borel measure, supported by X (see [Fal97]). In fact, we havethe following proposition [Fal97]:

Proposition 9.2.2: Let {S1, . . . , Sm} be an iterated function system on X ⊂ Rn withassociated probabilities {p1, . . . , pm}. Then there exists a unique Borel probability mea-sure µ (i.e., with µ(X) = 1) such that

µ(A) =m∑

i=1

piµ(F−1i (A)) (9.11)

for all Borel sets A, and where F−1i is the inverse mapping of Si. Moreover, sptµ = E

is the attractor of the iterated function system {Si : 1 ≤ i ≤ m and pi 6= 0}.


If we replace A in Equation 9.11 by the chain of transformations Si1(. . . (Sik(X)) . . .),

we get µ(Si1(. . . (Sik(X)) . . .)) = pi1µ(Si2 (. . . (Sik

(X)) . . .)), and thus, if we iterate thiswe will get Equation 9.10.

When we let x ∈ sptµ and iterate under these random sequence of transformations(Si chosen with probability pi), the proportion if iterates lying in a set A will be anapproximation of µ(A).

The measures resulting from similarity transformations, as those described above,are called self-similar measures. Equivalently, measures resulting from affine transfor-mations are called self-affine measures [Fal97].

9.2.2 Measures with Recursive Structures

We stumbled on the fact that Probabilistic Iterated Function Systems would give non-uniform measures on the fractal. The Probabilistic Iterated Function Systems does implythat there is a unique density on the attractor, and the application of the algorithmillustrates this. But the iterative non-deterministic fashion of the above algorithm is notdesirable. We are not really working with measure, we only imply them, and measuresis what multifractals are all about.

We can generate a self-similar or self-affine measure in the following way. We startwith an original region, A, which has measure 1, i.e. µ(A) = 1. Divide the region inpieces Fi, i = 1, 2, . . . , N , each with measure pi and each is transformed by Si. In thenext stage we also divide each piece, Fi, into N pieces, each with a measure reduced bypi, transformed by Si, but denoted by Fij . If we let the measure be 0 ≤ pi ≤ 1 and∑N

i=1 pi = 1, then the piece Fij will have measure pipj. When we transform A by Si,we will get the measure µ(Si(A)) = µ(Fi) = pi, and thus if we transform twice, we getµ(Sj(Si(A))) = µ(Fij) = pipj. See Figure 9.4 for an illustrative example. If we performthis recursively, we will distribute the measure differently in different parts of the object,and obtain a self-affine measure similar to the above.

To illustrate this concept, we reproduce the Sierpinski triangle of Figure 9.3 usingthis method. We let the entire object have measure 1, and the first piece be a rectangle.


F1

F11

F12

F13

F111

F112F113

F121

F122F123

F131

F132 F133

Figure 9.4: Illustrating the idea of the recursiveness of the Iterated Function Systemsdescribed here. In this example we have three affine transformations that we apply in F1

to get F11, F12, and F13. We continue this procedure in the limit.

We transform this rectangle using the same equations as before, namely:

S1(x, y) = (x/2, y/2)

S2(x, y) = (x/2, y/2) + (1/2, 0)

S3(x, y) = (x/2, y/2) + (1/4, 1/2),

with associated probabilities p1 = 0.6, p2 = 0.3, p3 = 0.1 respectively. We will get threenew rectangles, F1, F2, F3, with measures µ(S1(A)) = µ(F1) = 0.6, µ(S2(A)) = µ(F2) =0.3, µ(S3(A)) = µ(F3) = 0.1. See Figure 9.5 b). We transform each of these pieces usingthe same transformations and get three new rectangles for each of Fi, i.e. Fi1, Fi2, Fi2,for i = 1, . . . , 3, see Figure 9.5 c). The measure is defined on finer and finer scales whenwe continuously subdivide the measure. We can say that the mass (i.e. the measure) isredistributed among the smaller cells that come from the transformations. If we continuein this way for N steps, we will get something like in Figure 9.5 d) or e). Note that thesum of the measures, the total measure in all the subdivided cells is still 1 [Bar88].

Note that the triangle in Figure 9.5 e) is the same as the ones in Figure 9.3, if wehad plotted them in gray-scale. However, this method is much faster. The fractal inFigure 9.3 c) was created by iterating 100,000 times, but the measure in Figure 9.5 wascreated from only three transformations in 6,541 recursive calls, i.e. 6, 561 · 3 = 19, 683transformations. That is more than five times less computational effort, and this methodis deterministic!

What we have here is a true measure, a continuous, self-affine invariant measurethat we can use when we do our multifractal analysis. We can generate any figure thatcan be created using affine transformations, and get whatever measure we want on that


figure. Actually, it is possible to work backwards also; to find affine transformation thatcreates a given measure [Bar88].

9.2.3 Dimension of a Measure

If we generate a fractal measure as described above, with transformations Fi and asso-ciated probabilities pi, for i = 1, 2, . . . , N , i.e. translated copies of the original measure,µ, but scaled by the factors p1, . . . , pN , respectively. Now, if the translated copies donot overlap too much, the dimension of µ can be found as [Edg98]:

dimH µ =

∑Ni=1 pi log pi

∑Ni=1 pi log ri

. (9.12)

Otherwise, we define it as before for Borel measures µ by:

dimH µ = sup{s : dimlocµ(x) ≥ s} (9.13)

for µ-almost all x. And also that the above definition can be rewritten in terms of setsas:

dimH µ = inf{dimH E : µ(E) > 0} (9.14)

where E is a Borel set.


1.00

0.60 0.30

0.10

a) b)

0.36 0.18

0.06

0.18 0.09

0.03

0.06 0.03

0.01

c) d) e)

Figure 9.5: a) The original object, a rectangle, with measure 1. b) The measure isdivided in three pieces with different measures, that comes from different probabilities.c) Each piece has been subdivided further, the measure of each piece from b) has beenmultiplied by the measure, or probability, of each transformation. d) The Sierpinskitriangle emerges in the limit. All boxes have been drawn for each subdivision in thisfigure, to make the procedure more clear. e) A density plot of the procedure in the limit.The measure is more concentrated in the lower left region of the figure.

Chapter 10

Results

This chapter presents the results from simulations of the fractal and multifractal theory.The arithmetic theorems of Chapter 6, Chapter 7 and Chapter 8 are tested on a computerto see if they have any practical value.

Results include testing the algorithm for finding the Box-couting dimension; testingthe fractal theory of set product, set union and set intersection; testing how the di-mension is affected by projections. For multifractals are results included that tests thegeneralized dimensions, and the coarse multifractal spectrum.

Included is also an informal argument of why the custom made random fractals ofChapter 9 are better than traditional self-affine fractals and that central projection andx-ray simulated projection is equivalent to orthogonal projection.

10.1 Estimating the Box-Counting Dimension

To see how well the Box-counting estimation algorithm works (see Section 4.3), we haveperformed test runs, where the Box-counting dimension of a number of self-similar setsand custom made random fractals were estimated. The true dimension of the test sets isof course known. When the estimated dimensions are found for all test sets, the valuescan be compared with the known true dimensions and conclusions can be drawn.

Tests were performed on point sets in R1, R2 and R3. The results can be seen inTable 10.1 and Table 10.2. The point sets that were tested on consisted of 50,000 points.The theory of Section 4.3 was used. The grid size was successively reduced seven timesby two, i.e. there were seven points in the estimate if the slope of log Nδ against log δ.

The results of Table 10.1 are illustrated in Figure 10.1, and, as can be seen, theestimation follows the true values rather well. Between dimensions zero and two thereis a very small error, but when the true dimension increases to between two and three,the error in the estimated dimension increases. This is a known problem with thebox-counting algorithm, as noted in i.e. [HLD94], but since, in general, the estimateddimensions increases when the true dimension increases (i.e., there is a one-to-one cor-respondence between the true values and the estimated values), the results still implythat the algorithm can be used in e.g. a segmentation process.

The results for the tests performed on the custom made random fractals described inChapter 9 can be seen in Table 10.2. The point sets have dimensions ranging from about0.1 to 3.0. In Figure 10.2 the results are plotted, and, as can be seen, the estimateddimension follows the true, or theoretical dimension, very well. However, when the

71

72 Chapter 10. Results

Table 10.1: Experimental results for estimating the Box-counting dimension of setswith known dimensions. The results are grouped in 1D, 2D and 3D sets in the table,starting with the 1D sets.

True dimension Estimated dimension Difference0.631 0.703 0.0721.000 1.000 0.000

1.262 1.406 0.1441.465 1.524 0.0591.585 1.567 0.0181.893 1.923 0.0302.000 2.000 0.000

1.893 2.072 0.1792.000 1.994 0.0062.585 2.449 0.1362.727 2.564 0.1633.000 2.661 0.339

0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

True dimension

Est

imat

ed d

imen

sion

Estimating the box−counting dimension

True dimensionEstimated dimensionError

Figure 10.1: Experimental results for estimating the Box-counting dimension of setswith known dimensions. Lines are connecting 1D, 2D and 3D results. The solid line isthe true dimension of the set, the dashed and dotted lines are the estimated dimensions,and the solid line at the bottom is the error of the estimations.

10.1. Estimating the Box-Counting Dimension 73

Table 10.2: Experimental results for estimating the Box-counting dimension of artificialsets with known dimension. The first three lines are the results for 1D sets, the nextthree lines are the results for 2D sets and the bottom three lines are the results for 3Dsets.

True 0.110 0.208 0.301 0.406 0.500 0.602 0.699 0.792 0.898 1.000Est. 0.000 0.200 0.264 0.587 0.696 0.602 0.803 0.725 0.910 1.000

Error 0.110 0.008 0.037 0.181 0.196 0.000 0.104 0.067 0.012 0.000

True 1.107 1.209 1.292 1.404 1.500 1.594 1.699 1.792 1.893 2.000Est. 1.130 1.227 1.351 1.495 1.437 1.672 1.743 1.798 1.928 2.000

Error 0.023 0.018 0.059 0.091 0.063 0.078 0.044 0.006 0.035 0.000

True 2.096 2.196 2.292 2.402 2.500 2.605 2.696 2.807 2.893 3.000Est. 2.101 2.171 2.236 2.262 2.480 2.573 2.689 2.408 2.261 2.399

Error 0.005 0.025 0.057 0.140 0.020 0.032 0.007 0.399 0.632 0.601

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

True dimension

Est

imat

ed d

imen

sion

Estimating the box−counting dimension of custom made random fractals


Figure 10.2: Experimental results for estimating the Box-counting dimension of arti-ficial sets with known dimensions. Lines are connecting 1D, 2D and 3D results respec-tively. The solid line is the true dimension of the set, the dashed and dotted lines are theestimated dimensions, and the solid line at the bottom is the error in the estimations.


dimension exceeds about 2.5, the results start to deviate from the true value. The causeof this is likely because the number of points in the set is not enough to fill it at thosedimensions in R3, giving many empty boxes that should have been filled. Further testingimplies that this is the case.

10.1.1 Rotational Invariance

To see how well the estimation algorithm cope with rotated sets, a number of tests havebeen performed. In theory, the results should be unchanged if the set is rotated orscaled.

The sets were rotated by nine degrees about the necessary axes, i.e. the z-axis forsets in R2, and the x- and y-axes for sets in R3. For every rotation, the set’s dimensionwas estimated. The point sets consisted of 25,000 points, and the grid size was reducedseven times by a factor of two. Tests were made in R2 and R3, with sets of one and twodimensions in R2, and sets of three dimensions in R3.

See Figure 10.3 for an illustrative figure of the results. As can be seen, the dimensionfollows the true values for sets of dimension up to about 2. For almost all sets in R3,i.e. the sets that for the most part have a dimension greater than 2, the estimateddimensions deviate more and more when the true dimension increases. We have seenthis before. Obviously, the number of points in the tested sets (25,000) is not enoughwhen the dimension increases. Further testing imply this.

Still, the results imply a linear relationship between the true dimensions and theestimated dimensions. Thus, rotations of the set to estimate might not be a problem ini.e. a segmentation process, since the estimated dimension is increasing when the truedimension is increasing.

Table 10.3: Experimental results for estimating the Box-counting dimension of setsthat have been rotated. The first two lines are the results for 1D sets, the next five linesare the results for 2D sets and the bottom three lines are the results for 3D sets. Thestandard deviation (stddev) is how much the estimated values varies.

True Max Min Mean Stddev0.631 0.703 0.703 0.703 0.0001.000 1.000 1.000 1.000 0.000

1.262 1.406 1.271 1.326 0.0041.465 1.586 1.430 1.497 0.0041.585 1.626 1.565 1.602 0.0021.893 1.921 1.714 1.783 0.0062.000 2.000 1.779 1.853 0.007

1.893 1.944 1.795 1.853 0.0492.000 1.970 1.884 1.918 0.0202.585 2.235 2.131 2.167 0.0212.727 2.190 2.139 2.167 0.0153.000 2.272 2.207 2.233 0.020

10.2. Custom made Random Fractals 75

0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

True dimension

Est

imat

ed d

imen

sion

Estimating the Box−counting dimension of rotated sets

True dimensionAverage estimated dimensionError

Figure 10.3: Experimental results for estimating the Box-counting dimension of setsthat have been rotated. In theory, the dimension should not change when the sets arerotated. Lines are connecting 1D, 2D and 3D results respectively. The solid line is thetrue dimension of the set, the dashed and dotted lines are the estimated dimensions, andthe solid line at the bottom is the error in the estimations.

10.2 Custom made Random Fractals

This section will give an explanation why the traditional fractals, the self-similar and self-affine fractals, are not ideal for the experiments conducted here. An informal argumentstating why the custom made random fractals are better will also be given.

The problem with the traditional fractals is that the dimension of their projectionvaries substantially when they are rotated. This section investigates how much theestimated dimension varies for strictly self-similar fractals and for custom made fractalswith the same dimension. In Figure 10.4 can be seen the distribution of Box-countingdimension estimates for fractals with a true dimension of about 1.89, 2.00, 2.58, and2.73, and custom made random fractals with the same dimensions (within some errorbound ε), when projected from R3 to R2 using parallel projection. The fractals arerotated a full revolution about two axes, and the dimension of resulting projection hasbeen estimated (and thus, a very large number of projections have been accounted for).As can be seen, the estimated dimensions are less scattered and more evenly distributedfor the custom made fractals than for the strictly self-similar ones.

With the following argument, we will explain why the custom made random fractalsare so much better than the strictly self-similar ones. When the self similar fractals arerotated, there is a high possibility of self-similar parts lining up, so that the projectionis biased by the collinear points of the fractal.

The random fractals does not have this obvious line-up of points, but will (in thegeneral case) have a distribution that is equally distributed in the projection. In Fig-ure 10.5 we have projected two fractals, the Sierpinski triangle, and a custom maderandom fractal with the same dimension (within ε) as the Sierpinski triangle. We can


1.4 1.5 1.6 1.7 1.80

1

2

3

x 104 Box−counting dimension: 1.89

Dimension

Occ

uren

ce

3D Cantor dustRandom fractal

1.6 1.7 1.8 1.90

1

2

3x 10

4 Box−counting dimension: 2.00

Dimension

Occ

uren

ce

TetrahedronRandom fractal

1.8 1.85 1.9 1.950

0.5

1

1.5

2x 10


Dimension

Occ

uren

ce

Octaedron fractalRandom fractal

1.8 1.85 1.9 1.950

0.5

1

1.5

2x 10


Dimension

Occ

uren

ce

Menger spongeRandom fractal

Figure 10.4: Box-counting dimension distribution for strictly self-similar fractals anda custom made random fractals with precision ε = 0.01.

see that the projection of the Sierpinski triangle is strongly biased by the structure ofthe fractal, but also that the custom made random fractal does not have this problem.Note also that the Sierpinski triangle have two very different but equally self similarprojections, but that the two projections of the random fractal is quite random in bothcases. This is the property we wish the fractals to have.

We use the Hough transform (see Appendix B) to estimate the degree of collinearityin a fractal. The Hough transform will give us the number of points that are collinear inthe figure. Thus, if we look at the peaks, i.e. the points in the transform that have the

Sierpinski triangle Random fractal

a) b)

Figure 10.5: Distribution of projected points for the Sierpinski triangle a) with dimen-

sion log 3log 2 ≈ 1.57 and a random fractal b) with dimension 1.57 ± 0.01. The distribution

has an obvious pattern for the Sierpinski triangle, but is much more random and equallyspaced for the random fractal.

10.3. Testing the Fractal Theory 77

highest count (the highest number of collinear points), this will give us a hint whetherthe collinearity will affect the projection or not. In Figure 10.6–10.9 we can see theHough transform for strictly self-similar fractals, and for custom made random fractalsof the same dimension (with an error ε). The colour bar tells us the number of collinearpoints for a certain colour, and we can see that the strictly self-similar fractals havethe highest count of collinear points for all fractals but the Sierpinski carpet. The peakmean is the mean of the ten highest counts for each transform. This value is arbitrary,but gives a hint about how much the points line up in the fractals.

The graphs implies that the strictly self-similar fractals have much more collinearpoints than the custom made random ones in general, which concludes our informalargument. Of course, the above argument applies to higher dimensional fractals as well,and thus we conclude that our three dimensional custom made random fractals are alsomore appropriate to use than the three dimensional strictly self-similar ones.

10.3 Testing the Fractal Theory

In this section we will simulate and test the theory described in Chapter 6 and Chapter 7.A short description of the test environment will be given for each test, as well as theparameters used for the different settings. The results will be listed in tables and plottedin figures, and discussed briefly.

10.3.1 Projections of Fractals

In this section we will demonstrate the results from rotating sets in R2 and R3 andprojecting them to lower-dimensional subspaces, i.e. to R1 and R2 respectively.

Projecting from R2 to R1

The first test that is performed is rotating 1D and 2D sets in R2, and projecting themonto R1, i.e. the x-axis. All tests here will use orthogonal projection, as described inChapter 7. The theory says that the dimension of the projected set should be the smallestof the original set’s dimension and the dimension of the subspace, i.e. dim(projθ(F )) =min{dimF, 2} if F ⊂ R2 is projected. Thus, if we project from R2 to R1 a set whichhas dimension 1.5, the dimension of the projected set should be 1.0.

Point sets with 50,000 points and dimensions ranging from 0.631 to 2.000 was rotatedtwo degrees at the time, and for every rotation projected onto the x-axis. The Box-counting dimensions of the projected sets was then estimated using seven points in thelog-log least squares approximation.

The results can be seen in Table 10.4. The results are unanimous; the estimateddimensions follow the theory very well.

Projecting from R3 to R2

This test is done by rotating 1D, 2D and 3D sets in R3, projecting them onto R2, i.e.the xy-plane, and then estimating the Box-counting dimension of the projection. Alltests here will use orthogonal projection, as described in Chapter 7.

Point sets with 25,000 points and dimensions ranging from 0.631 to 3.000 was rotatedby four degrees at the time, and for every rotation projected onto the xy-plane. The


2D Cantor dust. Dimension: 1.26. Peak mean: 99.4

10

20

30

40

50

60

70

80

90

100

110

Random fractal. Dimension: 1.26. Peak mean: 70.40

10

20

30

40

50

60

70

80

a) b)

Figure 10.6: The Hough transform of a) the 2D Cantor dust, and b) a custom maderandom fractal with the same dimension. The 2D Cantor dust have a higher count ofcollinear points. The top ten highest counts are marked with squares.

Box fractal. Dimension: 1.46. Peak mean: 484.0

50

100

150

200

250

300

350

400

450


20

40

60

80

100

120

140

160

180

200

a) b)

Figure 10.7: The Hough transform of a) the Box fractal, and b) a custom made randomfractal with the same dimension. The Box fractal have a higher count of collinear points.The top ten highest counts are marked with squares.


Sierpinski triangle. Dimension: 1.58. Peak mean: 316.3

50

100

150

200

250

300

350

400

450


50

100

150

200

250

a) b)

Figure 10.8: The Hough transform of a) the Sierpinski triangle, and b) a custom maderandom fractal with the same dimension. The Sierpinski triangle have a higher count ofcollinear points. The top ten highest counts are marked with squares.

Sierpinski carpet. Dimension: 1.89. Peak mean: 583.4

100

200

300

400

500

600


100

200

300

400

500

600

700

a) b)

Figure 10.9: The Hough transform of a) the Sierpinski carpet, and b) a custom maderandom fractal with the same dimension. The Random fractal have a higher count ofcollinear points. The top ten highest counts are marked with squares.


Table 10.4: Experimental results for estimating the Box-counting dimension of setsthat have been rotated, and for every rotation projected. The first two lines are theresults for 1D sets, the next five lines are the results for 2D. The standard deviation(stddev) is how much the estimated values varies.


1.262 1.000 0.703 0.968 0.0041.465 1.000 1.000 1.000 0.0001.585 1.000 1.000 1.000 0.0001.893 1.000 1.000 1.000 0.0002.000 1.000 1.000 1.000 0.000

Table 10.5: Experimental results for estimating the Box-counting dimension of setsthat have been rotated, and for every rotation projected. The sets were all subsets of R3,and projected onto R2 (the xy-plane). The first two lines in the table are the results for1D sets, the next five lines are results for 2D sets and the last five lines are the resultsfor 3D sets. The standard deviation (stddev) is how much the estimated values varies.


1.262 1.412 0.889 1.333 0.0821.465 1.525 1.135 1.465 0.0661.585 1.666 1.110 1.556 0.0881.893 1.916 1.140 1.775 0.1472.000 1.999 1.143 1.837 0.170

1.893 1.830 1.390 1.737 0.0902.000 1.972 1.580 1.757 0.0612.585 1.992 1.775 1.881 0.0572.727 1.983 1.725 1.905 0.0573.000 1.999 1.778 1.905 0.053

Box-counting dimensions of the projected sets was then estimated using seven points inthe log-log least squares approximation to the scaling exponent.

The results can be seen in Table 10.5 and in Figure 10.10. The results are not assatisfying as in the previous simulation, but not at all bad. For 1D and 2D sets withtrue dimension less than or equal to two, the estimated dimensions follow an obviouslinear relationship to the true dimension, and the maximum absolute error is only about0.163. For 3D sets with true dimension less than or equal to two, the absolute error isabout 0.243. For 3D sets with true dimension greater than two, the absolute error is


0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

True dimension

Est

imat

ed d

imen

sion

Estimating the Box−counting dimension of rotated and projected sets

True dimensionAverage estimated dimensionError

Figure 10.10: Experimental results for estimating the Box-counting dimension of setsthat have been rotated and projected. In theory, the dimension should be the least of thedimension of the original set and the dimension of the subset onto which we project.Lines are connecting 1D, 2D and 3D results respectively. The solid line is the truedimension of the set, the dashed and dotted lines are the estimated dimensions, and thesolid line at the bottom is the error in the estimations. The dotted line at y = 2 is onlyfor reference.

only about 0.119, and decreasing with increasing true dimension. Likely, if the pointsets had more points, the results would have been better.

10.3.2 Product of fractals

In this section, the Box-counting dimension for the Cartesian product of a numberof fractal sets have been estimated. In Table 10.6 we can see the results of the esti-mated dimension of the product. Fractals with dimensions in the left-most column weremultiplied by fractals with dimensions in the top-most row, and the resulting object’sdimension was found using Box-counting. The value in the corresponding row-columnis the estimated dimension of the resulting set, and the value in parentheses is the truetheoretical dimension according to Corollary 6.1.6.

The sets used here were points sets created with 3,000 points for each fractal set,giving a resulting set with 9,000,000 points. The Box-counting dimension was thencalculated using the method described in Section 4.3 with a maximum of 128 discretesamples for the number of boxes.

According to Corollary 6.1.6, the dimension of the product of two sets should beequal to the sum of the individual dimensions, if dimH F = dimBF . This property isfulfilled for the sets in these tests, because they were all created from Iterated FunctionSystems.


Table 10.6: Experimental results for fractal set product. The values in parentheses arethe theoretical true dimensions.

0.631 1.000 1.262 1.465 1.585 1.893 2.0000.631 1.406

(1.262)1.703(1.631)

2.079(1.893)

2.145(2.096)

2.264(2.216)

2.459(2.524)

2.454(2.631)

1.000 1.703(1.631)

2.000(2.000)

2.372(2.262)

2.448(2.465)

2.574(2.585)

2.756(2.893)

2.782(3.000)

Figure 10.11: How the sum of the dimensions of two sets relate to the dimension oftheir Cartesian product. The solid line is the sum of the dimensions, and the dashedline is the dimension of the product of the sets.

We can, however, see from Figure 10.11 that the product does not always equalthe sum of the dimensions. We can see that the dimension of the product of two setsincreases as the sum of their dimensions increases. But as the dimensions increases,the dimension of their products does not increase as much. For sums of dimensions ofabout 1.2 to about 1.9 we see a linear correspondence, from between 2.0 to about 2.6 wesee another linear relationship, and from 2.6 to 3.0 there seems to be yet a third linearrelationship.

It is likely that the drop in accuracy is because of the limited number of points inthe sets. The largest set has dimension 2.0, which really is not filled with only 3,000points.

Even though the results seem to decay in accuracy and consistency, an increasednumber of points will likely remedy this (informal experiments imply this), we have amaximum absolute error of only 0.218 and the minimum absolute error of 0.000, and


thus, the results could likely be of experimental interest.

10.3.3 Union of fractals

The same kind of experiment as for the Cartesian product above have been performedhere. The Box-counting dimension for the union of a number of sets have been estimated.In Table 10.7 we can see the results of the estimated dimension of the union of sets withdimensions in the left-most column and the top-most row. Values in parentheses are thetheoretical true dimensions, in accordance with Theorem 6.2.1.

The sets used here were point sets created with 1,000,000 points for each set. Theunion of the sets was found by considering points with distance less than 1/1000000 apartto be equal, and then finding all pairs of points that are equal by this criteria. The Box-counting dimension was then calculated using the method described in Section 4.3 witha maximum of 128 discrete samples for the number of boxes.

Table 10.7: Experimental results for fractal set union. The values in parentheses arethe theoretical true dimensions.

0.631 1.000 1.262 1.465 1.585 1.893 2.000 2.585 2.727 3.0000.631 0.703

(0.631)1.000(1.000)

1.262 1.406(1.262)

1.648(1.465)

1.651(1.585)

1.923(1.893)

2.000(2.000)

1.893 2.109(1.893)

2.200(2.000)

2.492(2.585)

2.786(2.727)

2.994(3.000)

According to Theorem 6.2.1, the Hausdorff and upper Box-counting dimension ofthe union of two sets is equal to the maximum of the individual dimensions. Thus,if dimH F = dimBF , then the Box-counting dimension equals this value. All sets herewere created using Iterated Function Systems, and thus we have that dimH F = dimBF ,and so, the dimension of the union of two sets dim(A∪B) should be max(dim A, dimB).

In Figure 10.12 we can see that the dimension of the union of the sets follow themaximum dimension very well. There seems to be a consistent linear relationship be-tween the dimension of the union of the sets, and the maximum of their dimensions. Wecan see that the least-squares linear fit to the points follow the theoretical true valuevery well. The slope of the line fit is a little lower, but only slightly. Even though themaximum absolute error is 0.2160 (the minimum absolute error is 0.000), there are onlya few points that differ much.

10.3.4 Intersection of fractals

As we could see in Theorem 6.3.4, the Hausdorff dimension of the intersection of two setsA and B should be dimH(A∩B) = dimH A+dimH B−n, when A, B ⊂ Rn. We do nothave any theory for the Box-counting dimension, but assumes that similar relationshipscan be found is that case. For sets that are created from i.e. Iterated Function Systems,dimH F = dimB F , and thus the theory applies directly in that case.

When working with finite point sets in finite precision floating-point arithmetic, wecannot in general find the true intersection between two sets. Instead we can at best


Figure 10.12: How the maximum of the dimensions of two sets relate to the dimensionof their union. The solid line is the maximum of the dimensions, the dashed line is thedimension of the union of the sets, and the dash-dotted line is the least-squares fit to thepoints.

say that two points are equal if the distance between them is no more than δ.

Intersection Between a Set and a Line

The first experiment that was performed was to find the dimension of the intersectionbetween the Sierpinski triangle, with dimension about 1.585, and a straight line, withdimension 1.000, see Figure 10.13 a). We say that all points not further than δ fromthe line lies on the line, project all those points on the line and find the Box-countingdimension of the projected points. The dimension of the intersection should be about1.585 + 1.000− 2.000 = 0.585.

To see what the general intersection dimension is, we let the line move a distance ρalong its normal through the origin, and let the normal rotate θ ∈ [0, 2π] around the x-axis. See Figure 10.13 b) for an illustration of how the line is moved through the fractal.For all rotations and translations of the line we have, we estimate the Box-countingdimension, and save all non-zero values. When we are done, we plot a histogram of thedimension values, and note the most common value.

There is an obvious problem with this method, however. When we take the line tohave a thickness, it is effectively a rectangle, and thus we are finding the intersectionbetween a rectangle and the set, which should give a dimension of about 1.585+2.000−2.000 = 1.585, i.e. no change. What happens then is thus that we project a portion ofthe set onto the line. Remember from Theorem 7.1.6 that the dimension of a projection


θ

ρ

a) b)

Figure 10.13: a) Finding the intersection between a set (the Sierpinski triangle, withdimension about 1.585), and a line, b) the line is rotated and translated to find as manydifferent intersections as possible.

from R2 onto a line, R1, is 1 if the set has a dimension greater than 1. Thus, weshould really expect all dimensions of all intersections with a line which has a thicknessgreater than zero to be 1. But we can see when we look at decreasing thicknesses,that the estimated dimension approaches the theoretical value for the dimension, seeFigure 10.14.

The simulation was done using point sets, with 250,000 points in the sets. Lineswere simulated to have decreasing thicknesses of 0.5, 0.05, 0.005 and 0.0005. The pointsclose enough to the line was projected onto the line using orthogonal projection. TheBox-counting dimension of those points was then estimated using seven points in thelog-log least squares approximation.

We performed the same kind of experiment with other sets than the Sierpinski trian-gle, and got similar results. These results can be seen in Figure 10.15 and Figure 10.16.It seems like the method works best for sets with higher dimensions. The result for theset with dimension of about 1.262 is about 0.400, with an error of about 0.138, while theresult for a set with dimension 2.000 is 0.98, an error of only about 0.02. In Figure 10.16the mean value for the thinnest line is plotted with the theoretical true value.

Intersection Between Two Sets

Just as for the intersection between a set and a line, the intersection between two setswas found. To see what the general intersection dimension is, we let one of the setsmove a distance ρ along a line through the origin, and let the line, and with it the set,rotate θ ∈ [0, 2π] around the x-axis. See Figure 10.17 for an illustration of how the set


Figure 10.14: Normalized distributions of the dimensions of the intersection betweenlines with different thicknesses and a set with dimension about 1.585. The thickness ofthe line affects the estimated dimension. The thicker the line, the higher the estimateddimension.

Figure 10.15: Normalized distributions of the dimensions of the intersection betweenlines with different thicknesses and sets with dimension about 1.262, 1.465, 1.893 and2.000. The thickness of the line affects the estimated dimension. The thicker the line,the higher the estimated dimension.


0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

True dimension

Est

imat

ed d

imen

sion

Estimating the Box−counting dimension of the intersection between sets

Theoretical dimensionEstimated dimensionError

Figure 10.16: The estimated dimension of the intersection between a line and a set.When the theoretical true dimension increases, the error in the estimated dimensiondecreases.

Table 10.8: Experimental results for estimating the Box-counting dimension of the in-tersection of two sets. The dimension of the intersection between sets with dimensions inthe corresponding row and column of the table was estimated. The values in parenthesesare the theoretical true dimensions.

1.46 1.58 1.89 2.001.26 0.88 (0.72) 0.93 (0.84) 1.15 (1.15) 1.23 (1.26)1.46 1.11 (1.04) 1.35 (1.35) 1.47 (1.46)1.58 1.49 (1.47) 1.59 (1.58)1.89 1.86 (1.89)

is moved. For all rotations and translations of the set, we estimate the Box-countingdimension of the set of all points that have a point in the other set not further away thanδ. When done, a histogram of the dimension values is plotted, and the most commonvalue is noted.

The simulation was done using point sets, with 250,000 points each set. The alloweddistance between points was set to δ = 1/250000 and the set was moved along thethought normal a distance of 0.005 at the time. I.e., points with distance less than orequal to δ = 1/250000 was considered equal. The Box-counting dimension of the pointsin the intersection was then estimated using seven points in the log-log least squaresapproximation.

The results can be seen in Table 10.8 and in Figure 10.18. The largest absoluteerror is found for low-dimensional sets, but is still only about 0.160. The smallest


Figure 10.17: Estimating the dimension of the intersection between two sets. One ofthe sets is rotated and translated to find as many different intersections as possible.

0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

True dimension

Est

imat

ed d

imen

sion

Box−counting dimension of fractal set intersections


Figure 10.18: Estimating the dimension of the intersection between two sets. One ofthe sets is translated to find as many different intersections as possible.

absolute error is 0.000. The results are very similar to those for set-line intersection, seeFigure 10.16, and is in general very good.

Normalized distributions for the estimated dimensions of the intersections for all setsin this simulation can be seen in Figure 10.19. The most common value has been markedin the figures. Note that the variance in the distribution for the sets with low dimension


a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

10.88

Dimension of intersection between sets with dimensions of about 1.26 and 1.46

Dimension

Occ

urre

nce

b)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

10.93


Dimension

Occ

urre

nce

c)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

11.15


Dimension

Occ

urre

nce

d)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

11.23


Dimension

Occ

urre

nce

e)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

11.11


Dimension

Occ

urre

nce

f)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

11.35


Dimension

Occ

urre

nce

g)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

11.47


Dimension

Occ

urre

nce

h)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

11.49


Dimension

Occ

urre

nce

i)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

11.59


Dimension

Occ

urre

nce

j)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

11.86


Dimension

Occ

urre

nce

Figure 10.19: Normalized distributions of the estimate of the dimensions of the in-tersection between two sets. One of the sets is translated to find as many differentintersections as possible. The most common value is marked. Theoretical dimensions:a) 0.72, b) 0.84, c) 1.15, d) 1.26, e) 1.04, f) 1.35, g) 1.46, h) 1.47, i) 1.58 and j) 1.89.


is larger than for sets with large dimensions. This is likely related to the fact that theerror is larger for sets with lower dimensions as well.

10.4 Equivalence of Projection Types

Tests have been performed to confirm that the projection types of Chapter 7 are indeedequivalent. The first test performed was to test if orthogonal projection and centralprojection is equivalent. They will be considered equivalent if their projections have thesame dimensions. Thus, sets of different dimensions have been generated, projected,and the Box-counting dimensions of many projections have been estimated. Similar towhat has been done before.

In the first test, a number of sets in R2 was generated, and projected onto R1.The sets had 50,000 points, and dimensions ranging from about 1.262 to 2.000. Thedimensions was estimated using the Box-counting method with seven points in the least-squares approximation. From Section 7.2.2 we know that the estimated dimensionsshould be equal in this case.

The results can be seen in Figure 10.20 and as can be seen from the dimensiondistributions, the projection types of orthogonal and central projection seem to be verysimilar; as they should be according to the theory.

The second test is performed to show that central projection is equivalent to x-ray simulated projection. A number of sets in R3 are generated, and projected ontoR2. The sets has 50,000 points, and dimensions ranging from about 1.262 to 2.000. Thedimensions was estimated using the Box-counting method with seven points in the least-squares approximation. From Section 7.2.3 we know that the estimated x-ray simulatedprojection should be equal to central projection when the discrete sample size tendstowards zero.

The results can be seen in Figure 10.21, and as can be seen from the graphs of thedimension for the projection of x-ray simulation, the dimension estimations convergewhen the number of samples increase.

The results of both tests were the expected ones, and this indicates that there reallyis an equivalence between the projection types. The central projection is equivalent toorthogonal projection, and x-ray simulated projection is equivalent to central projection.Thus, orthogonal projection is also equivalent to x-ray simulated projection.

10.5 Estimating the Spectrum of Generalized Dimen-

sions

In this section we will test the implementation for the generalized dimensions. Testswill be made to see how sensitive the method is for rotations, and to see whethertwo measures can be distinguished from each other using the spectrum of generalizeddimensions.

Remember from Section 5.3.1 that the function β is defined as

β(q) = limr→0

log∑

µ(A)q

− log r(10.1)

where the sum is over the r-mesh cubes A, and that the generalized dimensions are

10.5. Estimating the Spectrum of Generalized Dimensions 91

0.7 0.8 0.9 1 1.1 1.2 1.3 1.40

0.2

0.4

0.6

0.8

11.411.35

Dimension

Occ

urre

nce

Dimension distribution of orthogonal and central projection

0.9 1 1.1 1.2 1.3 1.4 1.50

0.2

0.4

0.6

0.8

11.531.52

Dimension

Occ

urre

nce


a) b)

0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.70

0.2

0.4

0.6

0.8

11.60 1.63

Dimension

Occ

urre

nce


Orthogonal projection

Central projection

c)

0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.90

0.2

0.4

0.6

0.8

11.921.91

Dimension

Occ

urre

nce


1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

12.002.00

Dimension

Occ

urre

nce


d) e)

Figure 10.20: Testing the equivalence of orthogonal projection and central projection.Sets have been rotated and projected, and the Box-counting dimension of the projectionshave been estimated. The distributions in all tests peak at the corresponding value of theother projection type.

defined as

Dq = − 1

(q − 1)limr→0

log∑

µ(A)q

− log r= − β(q)

(q − 1). (10.2)

This is readily implemented, just as the Box-counting dimension was implemented,see Section 4.3, adding the power of q to each measure.

If we create a measure recursively, as described in Section 9.2.2, we can let the


0 50 100 150 200 250 300 350 400 450 5000

0.5

1

1.5Dimensions of exact and attenuation projection

Discrete Samples

Est

imat

ed D

imen

sion

Central projection

X−ray simulated projection

Error in the approximation

0 50 100 150 200 250 300 350 400 450 5000

0.2

0.4

0.6

0.8

1

1.2

1.4


Discrete Samples

Est

imat

ed D

imen

sion

Central projection



a) b)

0 50 100 150 200 250 300 350 400 450 5000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


Discrete Samples

Est

imat

ed D

imen

sion

Central projection



c)

0 50 100 150 200 250 300 350 400 450 5000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Dimensions of exact and attenuation projection

Discrete Samples

Est

imat

ed D

imen

sion

Central projection



0 100 200 300 400 500 6000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Dimensions of exact and attenuation projection

Discrete Samples

Est

imat

ed D

imen

sion

Central projectionX−ray simulated projectionError in the approximation

d) e)

Figure 10.21: Tests that compare central projection with x-ray simulated projection.The dimension of a set was computed after projecting it with central projection. Thenthe dimension of the same set was estimated after using more and more discrete samplesof the x-ray simulated projection. The x-ray simulated projection converges towards thecentral projection for increased number of discrete samples.

measures of the r-mesh cubes, A, be the measures of the affine transformations, pi, usedto create the measure. I.e. we can find a discrete set of values for β by

β(q) =log∑N

i=1 pqi

− log ri(10.3)


for N affine transformations, and ri being the scale factor of the transformation. Im-plementing this gives graphs like the one in Figure 10.23 a). The values for D−∞ andD∞ are calculated according to Equation 5.13 and Equation 5.14.

Now, since we have something to compare with, we can compare an estimated spec-trum to the one described above and thus know how well the method estimates thegeneralized dimensions for self-affine measures. We will generate a set of recursive mea-sures, and compare the estimated dimensions to the theoretical one.

The measures will be built using the method of Section 9.2.2. The process is brieflydescribed by Figure 10.22; the measure is divided into four equal squares, with differentmeasures. Each square is then divided into four squares again, dividing its measurebetween them, just as in the first step.

The measures for each transformation used in Figure 10.22 is p1 = 0.3, p2 = 0.25,p3 = 0.2, p4 = 0.25. The scale factor is, obviously, ri = 0.5 for i = 1, . . . , 4. Findingthe theoretical dimensions for q = −100, . . . , 100 for this measure is done by usingEquation 10.3, and finding the estimated dimensions for q = −100, . . . , 100 is doneusing Equation 10.2. The result can be seen in Figure 10.23 b). The method obviouslyworks very well, and it is very difficult to tell the curves apart.

Another relevant test is how much the curves differ if the parameters to the recursivemeasure algorithm are changed just a little and if they are changed very much. If weuse the measures p1 = (0.30, 0.25, 0.20, 0.25) and p2 = (0.31, 0.25, 0.19, 0.25) and plotthe curves in the same figure, we will see how much a small change in input will do tothe estimation. As can be seen in Figure 10.23 c), the curves differs substantially. Thisis both good and bad. It is good because this suggests that the method can be usedto segment, separate different measures from each other. It is, however, also bad, sinceit suggests that the method might be very sensitive to disturbances. More testing isrequired.

If we instead use the measures p1 = (0.05, 0.10, 0.15, 0.70) and p2 = (0.45, 0.25, 0.10,0.20), which differs much, we have a big difference in the input values. We can see theresult in Figure 10.23 d); the curves differ even more than if we changed the values justa little. A reasonable conclusion is to conjecture that the curves will be far apart if theparameters differ much, i.e. if the measures differ much. As said above, more testing isrequired.

10.5.1 Sensitivity to Noise

To get an indication on the sensitivity of the method, we will test how much the curvechanges when we add noise to the measure. We will add different levels of Gaussiannoise, and Salt & Pepper noise.

The results for Gaussian noise can be found in Figure 10.24. All noise have zero-mean, but increasing variance of 0.0005, 0.005 and 0.05. The estimated generalizeddimensions spectra start differing much even for small noise variance, but only for neg-ative values of q. The curve for the positive values of q only start differing for the 0.05variance noise added. If this is a big problem, in i.e. a segmentation process, perhapsonly the curve for positive q should be used.

The results for Salt & Pepper noise can be seen in Figure 10.25. Noise levels of2.5 %, 5.0 % and 7.5 % have been added to the original measure in Figure 10.25 a) inFigure 10.25 b), Figure 10.25 c) and Figure 10.25 d) respectively. When the noise levelis 7.5 %, the spectrum of generalized dimensions differ very much from the theoreticalspectrum. Obviously, this is too much Salt & Pepper noise for the method. The quirk


Figure 10.22: Building a recursive measure. a) After one recursive call, b) after tworecursive calls, c) after four recursive calls and d) the limiting measure. The colors havebeen inverted to look better in print.

−100 −80 −60 −40 −20 0 20 40 60 80 1001.6

1.7

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5

2.6

q

Dq

The generalized dimensions

D−∞

D∞

−100 −80 −60 −40 −20 0 20 40 60 80 1001.7

1.8

1.9

2

2.1

2.2

2.3

q

Dq

True generalized dimensions

Estimated generalized dimensions

a) b)

−60 −40 −20 0 20 40 60

1.7

1.8

1.9

2

2.1

2.2

2.3

2.4

q

Dq

p1=(0.30, 0.25, 0.20, 0.25)

p2=(0.31, 0.25, 0.19, 0.25)

−60 −40 −20 0 20 40 60

0.5

1

1.5

2

2.5

3

3.5

4

4.5

q

Dq

p1=(0.05, 0.10, 0.15, 0.70)

p2=(0.45, 0.25, 0.10, 0.20)

c) d)

Figure 10.23: a) The theoretical generalized dimensions spectrum. b) The theoreticalgeneralized dimensions and the estimated generalized dimensions plotted in the samefigure for comparison. c) Two measures generated with very similar parameters. d) Twomeasures generated with very different parameters. They give very different spectra.


a)

−60 −40 −20 0 20 40 601.7

1.8

1.9

2

2.1

2.2

2.3

q

Dq

Estimated generalized dimensionsTrue generalized dimensions

b)

−60 −40 −20 0 20 40 601.7

1.8

1.9

2

2.1

2.2

2.3

q

Dq


c)

−60 −40 −20 0 20 40 601.7

1.8

1.9

2

2.1

2.2

2.3

2.4

q

Dq


d)

−60 −40 −20 0 20 40 601.7

1.8

1.9

2

2.1

2.2

2.3

2.4

q

Dq


Figure 10.24: a) The theoretical and the estimated generalized dimension spectrum forthe measure without noise. b) The theoretical and estimated spectra for the measure witha zero-mean Gaussian noise with 0.0005 variance added. The spectra differ only a little.c) The theoretical and estimated spectra for the measure with 0.005 variance Gaussiannoise added. The spectra is differing very much for negative q, but almost nothing forpositive q. d) Theoretical and estimated spectra for the measure. Zero-mean Gaussiannoise with 0.05 variance added. The spectra starts to differ for positive q as well.


a)

−60 −40 −20 0 20 40 601.7

1.8

1.9

2

2.1

2.2

2.3

q

Dq


b)

−60 −40 −20 0 20 40 601.7

1.8

1.9

2

2.1

2.2

2.3

q

Dq


c)

−60 −40 −20 0 20 40 601.7

1.8

1.9

2

2.1

2.2

2.3

q

Dq


d)

−60 −40 −20 0 20 40 601.7

1.8

1.9

2

2.1

2.2

2.3

q

Dq


Figure 10.25: a) The theoretical generalized dimensions spectrum and the estimatedspectrum for the measure without noise. b) The theoretical and estimated spectra forthe measure with 2.5 % Salt & Pepper noise added. The spectra differ only a little. c)The theoretical and estimated spectra for the measure with 5 % Salt & Pepper noiseadded. The spectra start differing much. d) The theoretical and estimated spectra forthe measure with 7.5 % Salt & Pepper noise added. Now the spectra differ too much forany practical use.

10.6. Estimating the Multifractal Spectrum 97

at about −70 in Figure 10.25 c) and d) is likely because of limited precision in thefloating-point arithmetic used. The measure of the r-mesh cubes became too small, andbecause of that, when raised to the power of −q, they became too big. If the measureshave Salt & Pepper noise in them, they should probably be median filtered before thegeneralized dimensions spectra be estimated (see [GW02]).


Just as when finding the Box-counting dimension, the generalized dimensions shouldnot change if the measure is rotated. We can, however, see in Figure 10.26 b) that themethod is not at all rotationally invariant. In Figure 10.26 b) we see the curves for themeasure in Figure 10.26 a) when it has been rotated 0◦, 18◦, 36◦, 54◦, 72◦ and 90◦. Theresult is obviously really bad, and the curves are not at all continuous. Not for negativeq anyway; for positive q, the result is somewhat better, and for q > 5 the resultingcurve does not diverge too much from the reference curve. This is exactly the problemmentioned in Section 5.3, i.e. if an r-mesh cube A only covers a small portion of sptµ,the value of µ(A)q can become very large for negative values of q. We could prevent thisfrom happening by only counting cubes that cover a large portion of sptµ.

−60 −40 −20 0 20 40 601.7

1.8

1.9

2

2.1

2.2

2.3

q

Dq


a) b)

Figure 10.26: The measure in a) have been rotated five times, eighteen degrees at thetime. For every rotation, the generalized dimensions have been calculated. A plot of thegeneralized dimensions for all rotations can be seen in b).

10.6 Estimating the Multifractal Spectrum

To estimate the accuracy of our method to estimate the multifractal spectrum, we needsomething to compare the results with. We will, with the following arguments, generatethe true multifractal spectrum of a given measure, find its asymptotes and its maximumpoint.


Remember from the previous section how the β function was defined, and that themeasures of the affine transformations used to create a recursive measure, as describedin Section 9.2.2, is pi. Remember also that ri is the scale factor for the affine transfor-mations.

The infimum in Equation 5.49 is found when

α = −∂β

∂q, (10.4)

and we can define (see [Fal97]) β = β(q) as the positive number satisfying

N∑

i=1

pqi r

β(q)i = 1. (10.5)

Now, differentiating Equation 10.5 gives [Fal97]:

α =

∑Ni=1 pq

i rqi log pi

∑Ni=1 pq

i rqi log ri

, (10.6)

and we can see upon inspection that

αmin = min1≤i≤N

log pi

log ri(10.7)

and

αmax = max1≤i≤N

log pi

log ri. (10.8)

corresponding to q approaching ∞ and −∞ respectively.Now, since we know α(q) and β(q), we can find f(q) by

f = αq + β, (10.9)

as stated before. If we plot f against α, we now get the true multifractal spectrum ofthe given measure.

The multifractal spectra, f(α), is continuous on [αmin, αmax] (can be proven fromthe geometry of the Legendre transform). But also, if the numbers pi are all different,then f(αmin) = f(αmax) = 0. Also, using Equation 10.4, we get

∂f

∂α= α

∂q

∂α+ q +

∂β

∂q

∂q

∂α= q, (10.10)

and it follows that f is a concave function of α since q decreases when α increases [Fal97].If q = 0 then we get β(q) = dim sptµ. This is easy to see for the Box-counting

dimension, since Equation 10.1 turns into Equation 4.27 when q → 0, but see [Fal97]for a simple and general proof. We also know from Equation 10.10 and the fact thatthe f(α) curve is concave, that the point where q = 0 corresponds to the maximum off(α). I.e. dim sptµ = maxα f(α).

When q = 1, Equation 10.5 implies that β(q) = 0, and thus f(α) = α by

f(α) = −qα + β(q) = −q∂β

∂q+ β(q). (10.11)


Also, ddα (f(α) − α) = q − 1 = 0, so the f(α) curve lies beneath the line f = α, but

touches it at q = 1. Remember also, from above, that

f(q) ≤ α(q), (10.12)

with equality when q = 1.Using this theory, we can generate multifractal spectra from any given parameters

pi and ri, by applying Equation 10.3, Equation 10.6 and Equation 10.9. Using p =(0.4, 0.3, 0.2, 0.1) and r = 0.5, we get the spectrum in Figure 10.27 b). The measurecan be seen in Figure 10.27 a). This is the true spectra with which we will compare theestimated ones.

10.6.1 Estimating the Fine Theory

The theory of Section 5.3.3 can be directly implemented, giving fairly good results. Thespectrum is not continuous, as we will se in the next section that the coarse theoryspectrum will be, but might still be useful.

We will not test the fine theory here, other than showing the result of a implementa-tion of Section 5.3.3 in Figure 10.28. An account on the fine multifractal spectrum canbe found in [Nil07].

10.6.2 Estimating the Coarse Theory

We use the theory of Section 5.3.2 and estimate the coarse multifractal spectra of ameasure. Once again we use the measure stemming from p = (0.4, 0.3, 0.2, 0.1) and r =0.5. The results from estimating the multifractal spectrum can be seen in Figure 10.29.The result is obviously very good, giving a curve that is identical to the theoretical one.

Relevant tests now are, just as with the generalized dimensions, to see how much thecurve changes if the parameters change just a little, and if they change much. We use the

1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

α

f(α)

f(α) spectrum of p = ( 0.40 0.30 0.20 0.10 ), r = 0.5

Figure 10.27: The theoretical multifractal spectrum of a measure. Plotted is also thevalues of αmax, αmin, the line f = α and the line f = 2. Note that the spectrum istangent to the line f = α, and that is touches the line f = 2 at its maximum point.


measures p1 = (0.4, 0.3, 0.2, 0.1) and p2 = (0.41, 0.3, 0.2, 0.09) with a small change, andthe measures p1 = (0.4, 0.3, 0.2, 0.1) and p2 = (0.35, 0.18, 0.22, 0.25) with a big change.The results can be seen in Figure 10.30. The curves in a) differ only a little, while thecurves in b) differ much. This was expected, and this is a satisfying result. However,just as with the spectrum of generalized dimensions, the result for measures that differonly a little might imply that the method is sensitive for noise in the input measures.The next section will investigate this.

10.6.3 Sensitivity to Noise

Just as for the generalized dimensions, we will test how much a measure is changed ifwe disturb the measures with noise. We will investigate how much a curve change whenwe add noise to the measure. We will add different levels of Gaussian noise, and Salt &Pepper noise.

The results for Gaussian noise can be found in Figure 10.31. All noise have zero-mean, but an increasing variance of 0.0005, 0.005 and 0.05. The estimated multifractalspectrum differs a little already with Gaussian noise with a 0.0005 variance, see Fig-ure 10.31 b), but not at all much. It is the right side of the curve that is affected. No-ticeable is that this is the side corresponding to negative q in the generalized dimensions.The difference for the curve when the variance is increased to 0.005, see Figure 10.31 c),is not substantial. However, when the variance is 0.05, see Figure 10.31 d), the curvediffers much, and on both sides of the curve.

The results for Salt & Pepper noise can be seen in Figure 10.32. Noise levels of

1 1.5 2 2.5 30

0.5

1

1.5

2

α

f H(α

)

True spectrumEstimated spectrum

Figure 10.28: The theoretical multifractal spectrum of a measure and the estimatedfine multifractal spectrum. Plotted is also the values of αmax, αmin, the line f = α andthe line max{fH(α)}. Note that the spectrum is almost tangent to the line f = α, andthat is almost touches the line f = 2 at its maximum point.


1.5 2 2.5 3 3.50

0.5

1

1.5

2

α

f(α)

f(α) spectrum of p = ( 0.40 0.30 0.20 0.10 ), r = 0.5

Theoretical spectrumEstimated spectrum

Figure 10.29: The estimated and theoretical multifractal spectrum of a measure. Plot-ted is also the values of αmax, αmin, the line f = α and the line f = 2. Note that thespectrum is tangent to the line f = α, and that is touches the line f = 2 at its maximumpoint.

1 1.5 2 2.5 3 3.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

α

f(α)

Plot of f(α) for p1 = ( 0.40 0.30 0.20 0.10 ) and p

2 = ( 0.41 0.30 0.20 0.09 )

p1 = ( 0.40 0.30 0.20 0.10 )

p2 = ( 0.41 0.30 0.20 0.09 )

1 1.5 2 2.5 3 3.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

α

f(α)

Plot of f(α) for p1 = ( 0.40 0.30 0.20 0.10 ) and p

2 = ( 0.35 0.18 0.22 0.25 )

p1 = ( 0.40 0.30 0.20 0.10 )

p2 = ( 0.35 0.18 0.22 0.25 )

a) b)

Figure 10.30: The estimated multifractal spectrum of two measures that a) differ alittle and b) that differ much.

2.5 %, 5.0 % and 7.5 % have been added to the original measure in Figure 10.32 a) inFigure 10.32 b), Figure 10.32 c) and Figure 10.32 d) respectively. When the noise levelis 7.5 %, the multifractal spectrum does not differ very much. It seems like the methodused to estimate the multifractal spectra is less sensitive to Salt & Pepper noise thanthe method used to estimate the generalized dimensions.


a)

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.50

0.5

1

1.5

2

α

f(α)

Plot of f(α) for p1 = ( 0.31 0.19 0.21 0.29 )

p1 = ( 0.31 0.19 0.21 0.29 )

Spectrum of blurred measure

b)

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.50

0.5

1

1.5

2

α

f(α)

Plot of f(α) for p1 = ( 0.31 0.19 0.21 0.29 )

p1 = ( 0.31 0.19 0.21 0.29 )


c)

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.50

0.5

1

1.5

2

α

f(α)

Plot of f(α) for p1 = ( 0.31 0.19 0.21 0.29 )

p1 = ( 0.31 0.19 0.21 0.29 )


d)

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.50

0.5

1

1.5

2

α

f(α)

Plot of f(α) for p1 = ( 0.31 0.19 0.21 0.29 )

p1 = ( 0.31 0.19 0.21 0.29 )


Figure 10.31: a) The theoretical and the estimated multifractal spectrum for the mea-sure without noise. b) The theoretical and estimated spectra for the measure with azero-mean Gaussian noise with 0.0005 variance added. The spectra differ only a little.c) The theoretical and estimated spectra for the measure with 0.005 variance Gaussiannoise added. The spectra is still not differing much. d) Theoretical and estimated spec-tra for the measure. Zero-mean Gaussian noise with 0.05 variance added. The spectrastarts to differing on both sides, but still not very much.


a)

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.50

0.5

1

1.5

2

α

f(α)

Plot of f(α) for p1 = ( 0.31 0.19 0.21 0.29 )

p1 = ( 0.31 0.19 0.21 0.29 )


b)

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.50

0.5

1

1.5

2

α

f(α)

Plot of f(α) for p1 = ( 0.31 0.19 0.21 0.29 )

p1 = ( 0.31 0.19 0.21 0.29 )


c)

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.50

0.5

1

1.5

2

α

f(α)

Plot of f(α) for p1 = ( 0.31 0.19 0.21 0.29 )

p1 = ( 0.31 0.19 0.21 0.29 )


d)

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.50

0.5

1

1.5

2

α

f(α)

Plot of f(α) for p1 = ( 0.31 0.19 0.21 0.29 )

p1 = ( 0.31 0.19 0.21 0.29 )


Figure 10.32: a) The theoretical and the estimated multifractal spectrum for the mea-sure without noise. b) The theoretical and estimated spectra for the measure with 2.5 %Salt & Pepper noise. The spectra differ only a little, on the right side. c) The theoreticaland estimated spectra for the measure with 5.0 % Salt & Pepper noise added. The spec-tra is still not differing too much. d) Theoretical and estimated spectra for the measure.7.5 % Salt & Pepper noise added. The spectra starts to differing on both sides, but stillnot very much.



Just as when finding the Box-counting dimension and the generalized dimensions, themultifractal spectrum should not change if the measure is rotated. We can, however, seein Figure 10.33 that the method is not at all rotationally invariant. In particular not forthe right side of the curve, i.e. the part corresponding to negative q of the generalizeddimensions. In Figure 10.33 we see the curves for the measure when it has been rotated0◦, 18◦, 36◦, 54◦, 72◦ and 90◦, 108◦, 126◦, 144◦ and 162◦. The result is clearly reallybad, and the curves are not at all continuous. Not for the right side of the curve anyway.On the left side the result is a bit better. Actually, the results does not deviate muchat all from the true spectrum for the left side, i.e. for values corresponding to positive qof the generalized dimensions described earlier. This is exactly the problem mentionedin Section 5.3 and described in Section 10.5, and we could prevent this from happeningby only counting cubes that cover a large portion of sptµ.

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.60

0.5

1

1.5

2

α

f(α)

Plot of f(α) for rotated p = ( 0.31 0.19 0.21 0.29 )

p = ( 0.31 0.19 0.21 0.29 )Spectra of rotated measure

Figure 10.33: The measure in have been rotated ten times, eighteen degrees at thetime. For every rotation, the multifractal spectrum has been calculated. All multifractalspectra are plotted.

10.7 Testing the Multifractal Theory – Projections

of Measures

This section describes tests made on measures when they are projected onto a lower-dimensional subspace. Tests are performed estimating the generalized dimensions andthe multifractal spectrum of the projected measure, comparing them to the originalspectrum.

10.7. Testing the Multifractal Theory – Projections of Measures 105

10.7.1 The Generalized Dimensions

Remember from Chapter 8 that the projection of a measure µ, a Borel probabilitymeasure on Rn supported by a compact set, onto a linear subspace Π, is denoted by µΠ.We know that if Dq = Dq, then for q > 0

max{0, Dq(µ) − (n − k)} ≤ Dq(µΠ) ≤ min{Dq(µ), k}, (10.13)

for 1 < q ≤ 2Dq(µΠ) = min{Dq(µ), k} (10.14)

and for 0 < q ≤ 1Dq(µΠ) = Dq(µ) (10.15)

if Dq(µ) ≤ k.We can now test if this theory holds by projecting measures in R2 and R3 onto R1

and R2 respectively. The given measures are rotated 180 degrees, 18 degrees at the time,and for every rotation, the measure is projected onto the lower dimensional subspaceR1 or R2. For every projection, the generalized dimensions are found and plotted inthe same figure as the theoretical generalized dimensions, the upper and lower limitsimposed by Equation 10.13.

As an example, we create a self-similar measure, µ on R2, as done before, withparameters p = (0.10, 0.20, 0.20, 0.50) and scale factor r = 0.5. In Figure 10.34 canbe seen the result of estimating the generalized dimensions of the projection of µ. Theupper limit is the minimum of 1 and the original generalized dimensions of µ, i.e. Dq(µ).The lower limit is the maximum of 0 and Dq(µ) − (n − k), where n is the space of themeasure and k is the lower dimensional subspace we are projecting onto. Obviously,n = 3 and k = 2, or n = 2 and k = 1. Thus the lower limit is Dq(µ) − 1 in bothcases. As can be seen in Figure 10.34 a), the result follows Equation 10.13 very good.However, zooming in on 0 < q ≤ 2, as in Figure 10.34 b), we see that the curve does notfollow either of Equation 10.14 or Equation 10.15. For negative q, we do not have anytheoretical results. It is reasonable to believe that the generalized spectrum for q < 0should be less than Dq(µ), and that seems to be the case. Actually, we could conjecturethat the generalized dimensions for q < 0 is less than Dq(µ)− (n− k), since we see thatin almost all cases.

Several other tests were performed by projecting other, different, measures and find-ing their generalized dimensions. All results followed Equation 10.13. Some of them canbe seen in Figure 10.36 a)–d).

The problem, however, is that, even though the upper and lower bounds are followed,this test confirms the concern about the method being sensitive to noise, and obviouslyalso rather sensitive to how the object is rotated before it is projected.

For the case when the measure is on R3, projecting onto R2, measure cubes werecreated using the method described in Section 9.2.1, with probabilities p = (0.12, 0.07,0.05, 0.26, 0.02, 0.04, 0.23, 0.21) and scale factor 0.5. The result can be seen in Fig-ure 10.35, and, as can be seen, the result is satisfying for q > 0, or at least q > 1, butfor negative values of q, the results are very bad; the curve even fails to be decreasing.The cause of this is yet unknown.

Several other tests were performed by projecting other, different, measures and find-ing their generalized dimensions, just as for measures on R2. Most results followedEquation 10.13, but some failed for some projections, and a few failed for several projec-tion angles. However, all of the results was less than or equal to min{Dq(µ), k}. Someof the other test’s results can be seen in Figure 10.37 a)–d).


−60 −40 −20 0 20 40 600

0.5

1

1.5

2

2.5

3

3.5p = ( 0.10 0.20 0.20 0.50 )

q

Dq

Theoretical spectrumUpper limit of projectionLower limit of projectionEstimation of projection

0 0.5 1 1.5 2

0.6

0.7

0.8

0.9

1

1.1

p = ( 0.10 0.20 0.20 0.50 )

q

Dq

Theoretical spectrum

Upper limit of projection

Lower limit of projection

Estimation of projection

Figure 10.34: Estimating the generalized dimensions of a projection of a measure. a)Plotted is the generalized dimensions for the original measure, and the upper and lowerbounds on the projection. b) Zooming in on 0 < q ≤ 2 reveals that the curve does notfollow the theory perfectly.

−60 −40 −20 0 20 40 60

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

p = ( 0.12 0.07 0.05 0.26 0.02 0.04 0.23 0.21 )

q

Dq

Theoretical spectrum

Upper limit of projection

Lower limit of projection

Estimation of projection

Figure 10.35: Estimating the generalized dimensions of a projection of a measure onR3. Plotted is the generalized dimensions for the original measure, the upper and lowerbounds on the projection and the generalized dimensions of the projections.

10.7.2 The Multifractal Spectrum

As mentioned in the previous section: a projected measure, µ, onto a lower-dimensionalsubspace, Π, is denoted by µΠ. We know that

maxα

{fµΠ(α}) ≤ max

α{fµ(α)} (10.16)


a)

−60 −40 −20 0 20 40 600

0.5

1

1.5

2

2.5

3

3.5p = ( 0.10 0.30 0.30 0.30 )

q

Dq


b)

−60 −40 −20 0 20 40 600

0.5

1

1.5

2

2.5

3

3.5p = ( 0.40 0.10 0.20 0.30 )

q

Dq


c)

−60 −40 −20 0 20 40 600

0.5

1

1.5

2

2.5

3

3.5p = ( 0.10 0.10 0.40 0.40 )

q

Dq


d)

−60 −40 −20 0 20 40 600

0.5

1

1.5

2

2.5

3

3.5p = ( 0.10 0.10 0.50 0.30 )

q

Dq


Figure 10.36: Estimating the generalized dimensions of a projection of a measure.Plotted is the generalized dimensions for the original measure, the upper and lowerbounds on the projection, and the generalized dimensions of the projection.

a)

−60 −40 −20 0 20 40 600

2

4

6

8

10

p = ( 0.03 0.00 0.22 0.28 0.03 0.11 0.27 0.07 )

q

Dq


b)

−60 −40 −20 0 20 40 601

1.5

2

2.5

3

3.5

4

4.5

5

p = ( 0.14 0.16 0.04 0.18 0.15 0.21 0.04 0.09 )

q

Dq


c)

−60 −40 −20 0 20 40 601

1.5

2

2.5

3

3.5

4

4.5

5

p = ( 0.05 0.20 0.17 0.09 0.23 0.07 0.17 0.03 )

q

Dq


d)

−60 −40 −20 0 20 40 600

0.5

1

1.5

2

2.5

3

3.5

4p = ( 0.06 0.15 0.12 0.17 0.08 0.13 0.12 0.16 )

q

Dq


Figure 10.37: Estimating the generalized dimensions of a projection of a measure onR3. a) and b) are measures that follow the theory. In c), one of the projection anglesgives projections that does not follow the theory. In d), most of the projection anglesgive projections that does not follow the theory.


is true for almost all projections. Actually, we could expect more of the curve than onlybeing lower that the original curve. We know from Equation 8.10 that

dimloc µΠ(projΠx) ≤ dimloc µ(x), (10.17)

and thus, since the spectrum depends on the distribution of local fractal dimensions,it is likely that there will be more points with lower local dimensions in the projectionof the measure. I.e. the set will be more dense for lower α and less dense for high α.Higher density means higher dimension, and thus, the curve is likely to be higher in theleft half, and lower in the right part for the projection, than for the original set.

We can test if this theory holds by projecting measures in R3 onto R2 respectively,and estimate the multifractal spectrum of the projected measure. The given measuresare rotated 180 degrees around two axes, 18 degrees at the time, and for every rotationthe measure is projected onto R2. For every projection, the multifractal spectrum isestimated and plotted in the same figure as the theoretical multifractal spectrum.

As an example, we create a self-similar measure, µ on R3, as done before, with pa-rameters p = (0.12, 0.19, 0.12, 0.11, 0.13, 0.08, 0.16, 0.11) and scale factor r = 0.5. In Fig-ure 10.38 can be seen the result of estimating the multifractal spectrum of the projectionof µ. As can be seen in, the resulting spectra of the projection follows Equation 10.16.Also, the spectra is moved to the left, which was expected.

1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

3

3.5

p = ( 0.12 0.19 0.12 0.11 0.13 0.08 0.16 0.11 )

α

f(α)

Theoretical spectrumEstimation of projection

Figure 10.38: Estimating the multifractal spectrum of a projection of a measure. Plot-ted is the spectra for the original measure, and the spectra for the projected measure.

Several other tests were performed by projecting other, different, measures and find-ing their generalized dimensions. All results followed Equation 10.16. Some of them canbe seen in Figure 10.39 a)–d). Most of the spectra found looked something like thosein Figure 10.39 a) and b). Some were more scattered, as in Figure 10.39 c). A few hadspectra that for some of the projections, the curve was entirely deformed in the righthalf, as in Figure 10.39 d).


The problem, however, is that, even though the bound is followed, this test confirmsthe concern about the method being sensitive to noise, and obviously also rather sensitiveto how the object is rotated before it is projected, since the different projections givevery different spectra.

a)

1.5 2 2.5 3 3.5 4 4.50

0.5

1

1.5

2

2.5

3

3.5p = ( 0.18 0.08 0.17 0.08 0.20 0.09 0.05 0.15 )

α

f(α)


b)

1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5p = ( 0.14 0.18 0.10 0.16 0.25 0.07 0.03 0.06 )

α

f(α)


c)

1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

3.5p = ( 0.06 0.15 0.12 0.17 0.08 0.13 0.12 0.16 )

α

f(α)


d)

1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5p = ( 0.09 0.09 0.13 0.03 0.17 0.12 0.16 0.22 )

α

f(α)


Figure 10.39: Estimating the multifractal spectra of projections of a measure. Plottedis the multifractal spectrum of the original measure and the multifractal spectrum of theprojection.

Chapter 11

Conclusions

In this chapter, we discuss the results of the thesis. We also discuss restrictions imposedon the test by design, and give suggestions for future work.

11.1 Results

The implementation of the algorithm for finding the Box-counting dimension gives satis-fying results, even though its accuracy decreases when the dimension of the set increases.As mentioned in the thesis, this is a known result, and it should be noted that methodshave been developed, that can be found in the literature, to improve the Box-countingmethod.

The results for rotating a set and estimating its Box-counting dimension was ratherdisheartening. The reason for the rather bad results should be investigated.

The results for projections, product and union of sets was fairly satisfying. Theresults for projections was rather good, and supported the theory. The results forset product seems to be satisfying. The reason for the drop in accuracy for higherdimensional sets is likely, as before, because of the Box-counting method losing accuracyin higher dimensions. It should probably be investigated further, with more sets (usethe Custom made Random Fractals for example!) and more points in the sets. Testswith more points in the set could not be done, however, partly because there simply wasnot time for it, but mostly because the calculations took so long to perform with morepoints. The results for set union seems to be very good, but tests should be done withmore sets.

The results for set intersection are very interesting. The experiments give very goodresults, and actually, the results seem to be better for higher dimensional sets!

The generation of the generalized dimensions seems, at first glance anyways, to bevery good, at least for nice measures. The spectrum is not particularly sensitive toGaussian noise, but more so for Salt & Pepper noise. And the spectrum goes totallyhavoc when it is rotated.

The method to find the coarse multifractal spectrum also seems to be very good. Itis not especially sensitive to noise, but does not cope with rotating the sets.

The results for projecting measures follows the theory, both for the generalized di-mensions and for the multifractal spectrum, but the results are a bit scattered. Only ina few cases does the results not follow the theory completely.

111

112 Chapter 11. Conclusions

11.2 Restrictions and Limitations

The literature study phase consumed very much time, giving little time left to do exper-iments. Because of that, the simulations are not as complete as they could have been.We should have tested on many more sets, and using many more points in the point setsused. Also, the results should be more thoroughly examined. There where also no timeto do the simulations with all of the different types of fractals we are able to generate.

Tests where performed to see how many points was actually needed when projectingto fill the subspace. It turned out that the number of points required was many timesgreater than what was possible to simulate with on the available computers. This isindeed the most severe limitation imposed om the simulations.

11.3 Future work

The Custom made Random Fractals seem to give very good results. There was not timeto investigate them further, but it would be interesting to see more tests done withthem.

In general, tests on more sets, with more points would probably give much betterresults. But the more points, the more time consuming is the task. Some of the testsin this thesis took already several weeks to complete.

The results have not been evaluated statistically, which they should have been. Theyare simply presented as is, without any deeper analysis. This is definitely a task forfuture work, to be able to more confidently believe in the results.

It should be interesting to see more studies of set intersection. The results was verygood, but should they have been even better if we used even thinner lines? When find-ing the intersection with a line, we projected all points onto the line, and estimated thedimension on the 1D set found. A reasonable question is to ask whether this methodis better than estimating the dimension from the 2D values that we get from the inter-section of the set with the rectangle. Will we get converging results for that methodalso?

More extensive tests for the simulations on multifractal measures should be done.Systematic testing with automatic correctness estimation will give more confidence tothe results.

Extensive evaluation of the method for finding the coarse multifractal spectrum isa reasonable next step. What does the literature say; can the problems of rotation besolved?

In any case, more testing could probably tell when, where and why the differingresults occur. Can the erroneous cases be constructed? If that is the case, can they beavoided?

Multifractal analysis is the natural next step, and much more scientific work is likelyto be seen in the future.

Chapter 12

Acknowledgments

I would like to thank my supervisor Fredrik Georgsson, for his constant support, en-couragement, good ideas and positive spirit.

I would also like to thank my girlfriend Linda for helping me proofread the thesis.

113

114 Chapter 12. Acknowledgments

Appendix A

Brief Summary of the Set

Theory Involved

This Appendix aims to give the reader a brief review of set theory, and also, perhapsmost importantly, to explain some of the less basic ideas (concerning both set theoryand general mathematics) and notations used in this work. This review is by no meanscomplete, but should give the reader a good enough summary to be able to understandthe general topics in the text.

In almost every area of mathematics, the notion of sets, and ideas from set theory isused. A set is a well-defined collection of objects, where the objects are called members,or elements of the set. Well-defined in the preceding definition means that we are alwaysable to determine whether an element is in a set or not [Gri90].

In this text, the theory we work with is mostly concerned with set of points from then-dimensional Euclidean space, Rn, but all applications are dealing with sets of pointsfrom 1-, 2- and 3-dimensional spaces, R1, R2, and R3 respectively.

We will use upper case letters to denote sets, and lower case letters to denote el-ements. Sometimes we will use the coordinate form of the points in Rn, and denotethem x = (x1, . . . , xn), we then call them vectors. However, it is the object x that is themember of the set Rn. For a set A, we will write e ∈ A to say that e is a member if theset A (e is in A), and e /∈ A to say that e is not a member of A (e is not in A).

We will write {x : condition} to denote the set of all points fulfilling a given condition.E.g., the set of all even numbers are denoted:

{2x : x ∈ Z}.

Remember that Z is the set of integers, Q is the set of rational numbers, R is the setof real numbers, and C is the set of complex numbers. We will use a superscript + todenote only the positive elements of a set, e.g. Z+ is the set of all positive integers.

Vector addition and scalar multiplication are defined as usual, so that x± y = (x1 ±y1, . . . , xn±yn) and λx = (λx1, . . . , λxn), and also so that A+B = {x+y : x ∈ A∧y ∈ B}and λA = {λx : x ∈ A}. Scalar product, or dot product, is defined as follows:

Definition A.0.1: Let v = (v1, . . . , vn) and u = (u1, . . . , un) be vectors in coordinateform, then their dot product is

v · u =

n∑

i=1

viui. (A.1)

115

116 Chapter A. Brief Summary of the Set Theory Involved

We can now use the dot product to define the length of a vector as [Roe03]:

Definition A.0.2: If v ∈ Rn is a vector, we define the length |v| of v as√

v · v, wherewe take the positive square root. If x, y ∈ Rn are vectors, then the distance between themis the length |x − y| of the vector x − y.

In the above definition (and in the following), | · | denotes some norm, e.g. theEuclidean norm,

L2(x) =

√

√

√

√

n∑

i=1

x2i

for x = (x1, . . . , xn), but it could be any norm.We will say that vectors are linearly independent if they fulfill the following crite-

ria [Str05]:

Definition A.0.3: A sequence of vectors, v1, . . . , vn are called linearly independent ifthe only linear combination that results in the zero vector is 0v1 + · · · + 0vn, thus thesequence of vectors are linearly independent if

k1v1 + k2v2 + · · · + knvn = 0, ki = 0 for i = 1, . . . , n (A.2)

We define subsets as follows [Gri90]:

Definition A.0.4: If A, B are sets, we say that A is a subset of B, and write A ⊆ Bor B ⊇ A, if every element in A is also an element of B. Also, if B contains at leastone element that is not in A, we say that A is a proper subset of B, and write A ⊂ B,or B ⊃ A. Thus, if A ⊆ B, then

∀x(x ∈ A ⇒ x ∈ B) (A.3)

If a set is empty, that is, it contains no elements, it is called the null set, or emptyset. It is a unique set denoted by ∅, or {}.

We define union, intersection and difference of sets as follows:

Definition A.0.5: For two sets A, B, we define the following:

a) The union of A and B is: A ∪ B = {x|x ∈ A ∨ x ∈ B}

b) The intersection of A and B is: A ∩ B = {x|x ∈ A ∧ x ∈ B}

b) The difference of A and B is: A\B = A − B = {x|x ∈ A ∧ x /∈ B}

Two sets are called disjoint if A ∩ B = ∅. Remember that A ∪ B = B ∪ A andA ∩ B = B ∩ A, A ∪ (B ∪ C) = (A ∪ B) ∪ C and A ∩ (B ∩ C) = (A ∩ B) ∩ C,A∪ (B ∩C) = (A∪B)∩ (A∪C) and A∩ (B ∪C) = (A∩B) ∪ (A∩C), A∪A = A andA ∩ A = A, A ∪ ∅ = A and A ∩ ∅ = ∅, and A ∩ Rn = A and A ∪ Rn = Rn. Proofs ofthese properties are omitted, but can be found in [Gri90]. The complement of a set isdenoted Rn\A.

The Cartesian product of two sets A and B is the set of all ordered pairs {(a, b) :a ∈ A ∧ b ∈ B}, and is denoted A × B. If A ⊂ Rn and B ⊂ Rm, then A × B ⊂ Rn+m.

The infimum of a set, denoted inf(A), is the greatest element that is smaller than,or equal to, all elements in A, i.e. the greatest m such that m ≤ x for all x ∈ A. If such

117

an element does not exist, we define inf(A) = −∞. If A = ∅, then inf(A) = ∞. Thesupremum of a set, denoted sup(A), is the smallest element that is greater than, or equalto, all elements in A, i.e. the least number m such that x ≤ m for all x ∈ A. If such anelement does not exist, we define sup(A) = ∞. If A = ∅, then sup(A) = −∞. Intuitively,the infimum and supremum can be thought of as the minimum and maximum of a setrespectively, but need not be members of the set themselves, and they always exist.

The diameter of a non-empty set A is defined as |A| = sup{|x − y| : x, y ∈ A}, withthe convention that |∅| = 0. A set is bounded if it has finite diameter, and unboundedotherwise. The distance between two non-empty sets A, B is defined as dist(A, B) =inf{|x − y| : x ∈ A ∧ y ∈ B}. The δ-neighbourhood, or δ-parallel body, Aδ, of a set Ais defined as Aδ = {b : infa∈A |a − b| ≤ δ}, with δ > 0. It is the set of points withindistance δ of A [Fal97].

We define the closed and open balls of center x and radius r by

Br(x) = {y : |y − x| ≤ r}

and

Bor (x) = {y : |y − x| < r}

respectively. Thus, a closed ball contains its bounding sphere. If a, b ∈ R and a < b,we write [a, b] for the closed interval {x : a ≤ x ≤ b} and (a, b) for the open interval{x : a < x < b}. [a, b) denotes the half-open interval {x : a ≤ x < b}. A set is closed ifit contains its boundary points, and open otherwise. A set is said to be compact if it isboth closed and bounded.

The intersection of all closed sets containing a set A is called the closure of A, andis written A. The closure of A is thought of as the smallest set containing A [Fal97].

A cover of a set A is a countable (or finite) collection of sets that cover A. I.e., if{Ui} is a collection of subsets of A, then {Ui} is a cover of X if

⋃

i

Ui = A.

A set is said to be finite, if there is a bijection between it and a set of the form{1, 2, . . . , N}, where N is zero or a positive integer. I.e., if we can list the elements inthe set by enumerating them. An infinite set is a set that is not finite.

An infinite set is said to be countable, if its elements can be listed in the formx1, x2, . . ., where every element has its specific place in the list. Otherwise the set is saidto be uncountable. Remember that Z and Q are countable, but that R is not. See [Fla98]for a good explanation of countability.

The results in this thesis are often said to hold for almost all subsets, or almost allangles, and so on. The meaning of this is the following:

Definition A.0.6: A property that holds for almost all members of any given set willmean all members of the set with the exception of a subset of measure zero.

There is a special class of sets that need to be mentioned [Fal90]:

Definition A.0.7: The class of Borel sets is the smallest collection of subsets of Rn

with the following properties:

(a) every open set and every closed set is a Borel set;

118 Chapter A. Brief Summary of the Set Theory Involved

(b) the union of every finite or countable collection of Borel sets is a Borel set, and theintersection of every finite or countable collection of Borel sets is a Borel set.

The above definition says that Borel sets are sets that can be constructed from openor closed sets by repeatedly taking countable unions and intersections. The Borel setsare measurable. In this text, almost all theory deals with Borel sets.

Appendix B

The Hough Transform

Given n points in space, suppose that we want to find the collinear points, i.e. the pointsthat lie on straight lines. One solution is to find all lines that are (uniquely) determinedby a pair of points, and then find all points that lie on, or close to, the lines. Thisprocedure is computationally infeasible for a large set of points since we first need tofind n(n − 1)/2 ∼ n2 lines, and then perform n(n(n − 1)/2) ∼ n3 comparisons betweenevery point and every line.

In 1962, Paul Hough proposed a different approach, commonly referred to as theHough transform. We look at individual points, (xi, yi) and consider the general equationof a line, i.e. yi = axi + b. There are infinitely many points going through (xi, yi), butthe line with specific parameters for a and b is unique. We rewrite the equation asb = −xia + yi and consider the lines in the ab-plane, also called the parameter space.Thus, we have a single line for each point (xi, yi). Now, if we have a second point,(xj , yj), it also has a line in parameter space associated with it, and this line intersectsthe line associated with the point (xi, yi) at (a′, b′), where a′ is the slope of the line inthe xy-plane. In fact, all lines that intersect at (a′, b′) must be on the same line in thexy-plane. See Figure B.1 for an illustration of the idea [GW02].

The computational complexity of the Hough transform comes from dividing theparameter space into so called accumulator cells. The cell at the coordinates (i, j) cor-responds to the square with parameter space coordinates (ai, bj). The cells are initiallyzero, and for every point, (xk, yk), in the xy-plane, we let the parameter a range from[amin, amax] and solve for b using b = −xka + yk. For every value of a, we increase theaccumulator cell for a and b by one. When the procedure is done, the value in eachaccumulator cell corresponds to the number of lines that are collinear with slope andintercept being the coordinates of the accumulator cell. I.e. the value of the accumu-lator cell is the number of collinear points on the corresponding line. The number of(discrete) values in the range [amin, amax] will determine how close to the line the pointsare [GW02].

If we divide the a-axis in k discrete values, then we will find k values for b for everypoint, and thus perform nk operations. Thus, the Hough transform is linear in n! Ifthe value of k does not exceed n, the procedure is much more attractive than the naıveapproach suggested above.

A problem with this approach is, however, that the slope of y = ax + b is infinitewhen the line is vertical. This problem is solved by using a different representation for

119

120 Chapter B. The Hough Transform

-

?x

y

\\

\\

\\

\\

\\\

cba (xi, yi)

cba (xj , yj)

-

?a

b

bb

bb

bb

bb

bb

bbb"

""

""

""

""

""

""

cba

b′

a′

b = −xia + yi

b = −xja + yj

a) b)

Figure B.1: a) Two points in the xy-plane, and b) the lines intersect in the parameterspace at (a′, b′). Note that the Hough transform generates two lines in the ab-plane fromthe two points.

lines, e.g.x cos θ + y sin θ = ρ. (B.1)

The use of this representation is equivalent to the method described above. We divideθ in discrete samples, and for every value of θ we increase the accumulator cell at (θ, ρ)by one. θ is the angle the normal of the line makes to the x-axis, and ρ is the distancefrom the line to the point of origin, see Figure B.2 [GW02].

-

?x

y

��

��

��

��

��

��

�

JJJ

JJ

JJJ��JJ

θρ

Figure B.2: The normal representation of a line, x cos θ + y sin θ = ρ.

Figure B.3 illustrates the use of the Hough transform. Each point is mapped to theθρ-plane. The θ values range from π to −π and the ρ values range in ±

√2D, where D

is the width (and height) of the sample image. The white points are transformed usingEquation B.1, and are represented by the sinusoidal curves in Figure B.3 b).

The origin in the sample image is the point in the upper left-most corner. Note theintersection of three curves at roughly π/2 and −π/2. These intersections come from

121

a) b)

Figure B.3: The points in a) are Hough transformed using Equation B.1. Everysinusoidal curve in b) represents a point in a).

three collinear points running through the image’s diagonals, and through the image’sorigin. Note that the upper left-most points corresponds to the (almost) horizontal linein b). The line is not completely horizontal since the point is one pixel from the image’sorigin, and thus not at the origin.

We also have four intersections between two curves in b). Two intersections at π (or−π if you will), and two intersection at 0. These corresponds to the lines made by thetwo upper points, the two lower points, the two right-most points and the two left-mostpoints.

Note the curve going the farthest down at about −π/2. This is the point oppositeto the point at (or rather close to) the origin. That point is farthest from the origin,and thus should give the largest absolute ρ.

The Hough transform can thus be used to detect lines in images, but in fact, theHough transform can be generalized and used with any function of the form g(g, c) = 0,where g is a vector of coordinates, and c is a vector of coefficient. For example, thepoints lying on the circle

(x − c1)2 + (y − c2)

2 = c23 (B.2)

can be detected using the same approach as we described above. The only differencein this example is that we have three coefficients and thus get a 3D parameter space.We let c1 and c2 range over some set of discrete values and solve for c3. We thenincrease the accumulator cell at (c1, c2, c3). I.e., the complexity of the Hough transformis proportional to the number of coefficients, and by the number of coordinates (i.e. thedimension of the space we are transforming).

122 Chapter B. The Hough Transform

References

[Bar88] Michael F. Barnsley. Fractals Everywhere. Academic Press, first edition,1988. ISBN 0-12-079062-9.

[BB06] Fadhila Bahroun and Imen Bhouri. Multifractals and projections. ExtractaMathematicae, Vol. 21(1):83–91, 2006.

[BGT01] Jean-Marie Barbaroux, Francois Germinet, and Serguei Tcheremchantsev.Generalized fractal dimensions: Equivalence and basic properties. Journalde Mathematiques Pures et Appliquees, Vol. 80(10):977–1012, 2001.

[BGZ00] Christoph Bandt, Siegfried Graf, and Martina Zahle. Fractal Geometry andStochastics II (Progress in Probability). Springer, 2000. ISBN 3-7643-6215-4.

[Can84] Georg Cantor. De la puissance des ensembles parfait de ponds. Acta Math-ematica, 1884.

[CJ89] Ashvin Chhabra and Roderick V. Jensen. Direct determination of the f(α)singularity spectrum. Phys. Rev. Lett., Vol. 62(12):1327–1330, Mar 1989.

[CMV02] Yves Caron, Pascal Markis, and Nicole Vincent. A method for detectingobjects using legendre transform. In Maghrebian Conference on ComputerScience, pages 219–225, 2002.

[Edg90] Gerald A. Edgar. Measure, Topology, and Fractal Geometry. Springer-Verlag, first edition, 1990. ISBN 0-38797-272-2.

[Edg98] Gerald A. Edgar. Integral, Probability, and Fractal Measures. Springer-Verlag, first edition, 1998. ISBN 0-38798-205-1.

[Eri04] Christer Ericson. Real-Time Collision Detection. Morgan Kaufmann Pub-lishers, first edition, 2004. ISBN 1-55860-732-3.

[Fal90] Kenneth J. Falconer. Fractal Geometry: Mathematical Foundations andApplications. John Wiley & Sons, 1990. ISBN 0-471-92287-0.

[Fal97] Kenneth J. Falconer. Techniques in Fractal Geometry. John Wiley & Sons,1997. ISBN 0-471-95724-0.

[FH96] Kenneth J. Falconer and John D. Howroyd. Projection theorems for box andpacking dimensions. Math. Proc. Cambridge Phil. Soc., Vol. 119:287–295,1996.

123

124 REFERENCES

[Fla98] Gary William Flake. The Computational Beauty of Nature. MIT Press,1998. ISBN 0-262-56127-1.

[FO99] Kenneth J. Falconer and Toby C. O’Neil. Convolutions and the geometry ofmultifractal measures. Mathematische Nachrichten, Vol. 204:61–82, 1999.

[GNS95] Walter Greiner, Ludwig Neise, and Horst Stocker. Thermodynamics andStatistical Mechanics. Springer Verlag, 1995. ISBN 0-387-94299-8.

[Gri90] Ralph P. Grimaldi. Discrete and Combinatorial Mathematics. Pearson Ed-ucation, second edition, 1990. ISBN 0-201-19912-2.

[GW02] Rafael C. Gonzalez and Richard E. Woods. Digital Image Processing.Prenice Hall, second edition, 2002. ISBN 0-13-094650-8.

[Hau19] Felix Hausdorff. Dimension und ausseres Mass. Mathematische Annalen,Vol. 79:157–179, 1919.

[HJK+86] Thomas C. Halsey, Mogens H. Jensen, Leo P. Kadanoff, Itamar Procac-cia, and Boris I. Shraiman. Fractal measures and their singularities: Thecharacterization of strange sets. Phys. Rev. A, Vol. 33:1141–1151, Feb 1986.

[HK97] Brian R. Hunt and Vadim Yu Kaloshin. How projections affect the di-mension spectrum of fractal measures. Nonlinearity, Vol. 10(5):1031–1046,1997.

[HLD94] Qian Huang, Jacob R. Lorch, and Richard C. Dubes. Can the fractal di-mension of images be measured? Pattern Recognition, Vol. 27(3):339–349,1994.

[HP83] H. G. E. Hentschel and Itamar Procaccia. The infinite number of gener-alized dimensions of fractals and strange attractors. Physica D: NonlinearPhenomena, Issue 3, Vol. 8:435–444, September 1983.

[JG06] Stefan Jansson and Fredrik Georgsson. Evaluation of methods for estimat-ing fractal dimensions of intensity images. F. Georgsson, N. Borlin (eds),Proceedings of SSBA06, pages 69–74, 2006.

[JOJ95] X. C. Jin, S. H. Ong, and Jayasooriah. A practical method for estimatingfractal dimension. Pattern Recognition Letters, Vol. 16(5):457–464, May1995.

[Kay94] Brian H. Kaye. A Random Walk Through Fractal Dimensions. VCH, secondedition, 1994. ISBN 3-527-29078-8.

[KC89] James M. Keller and Susan Chen. Texture description and segmentationthrough fractal geometry. Computer Vision, Graphics, and Image Process-ing, Vol. 45:150–166, 1989.

[Leb01] Henri Lebesgue. Sur une generalisation de l’integrale definie. ComptesRendus, 1901.

[Man67] Benoıt B. Mandelbrot. How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension. Science, Vol. 156:636–638, 1967.

REFERENCES 125

[Man75] Benoıt B. Mandelbrot. Les objets fractals, forme, hasard et dimension.Paris: Flammarion, 1975.

[Man82] Benoıt B. Mandelbrot. The Fractal Geometry of Nature. W. H. Freemanand Company, first edition, 1982. ISBN 0-7167-1186-9.

[Man90] Benoıt B. Mandelbrot. Negative Fractal Dimensions and Multifractals.Physica A, Issue 1, Vol. 163:306–315, 1990.

[Mar54] J. M. Marstrand. Some fundamental geometrical properties of plane setsof fractional dimensions. Proceedings of the London Mathematical Society,Vol. 4(3):257–302, 1954.

[Mat75] Pertti Mattila. Hausdorff dimension, orthogonal projections and intersec-tions with planes. Annales Academiæ Scientiarum Fennicæ, A 1:227–244,1975.

[MCSW86] Paul Meakin, Antonio Coniglio, H. Eugene Stanley, and Thomas A. Witten.Scaling properties for the surfaces of fractal and nonfractal objects: Aninfinite hierarchy of critical exponents. Phys. Rev. A, Vol. 34(4):3325–3340,Oct 1986.

[NG06] Anders Nilsson and Fredrik Georgsson. Projective properties of fractal sets.Chaos, Solitons and Fractals, 2006.

[Nil07] Ethel Nilsson. Multifractal-based image analysis with applications in med-ical imaging. Master’s thesis, Umea University, 2007. UMNAD 697/07.

[NSM03] Sonny Novianto, Yukinori Suzuki, and Junji Maeda. Near optimum estima-tion of local fractal dimension for image segmentation. Pattern RecognitionLetters, Vol. 24:365–374, 2003.

[O’N00] Toby C. O’Neil. The multifractal spectra of projected measures in euclideanspaces. Chaos, Solitons and Fractals, Vol. 11:901–921, 2000.

[Roe03] John Roe. Elementary Geometry. Oxford University Press, 2003. ISBN0-19-853456-6.

[SRR06] Tomislav Stojic, Irini Reljin, and Branimir Reljin. Adaptation of multifrac-tal analysis to segmentation of microcalcifications in digital mammograms.Physica A: Statistical Mechanics and its Applications, Vol. 367:494–508,2006.

[Str05] Gilbert Strang. Introduction to Linear Algebra. Wellesley-Cambridge Press,third edition, 2005. ISBN 0-9614088-9-8.

[The90] James Theiler. Estimating Fractal Dimension. Journal of the Optical Societyof America A, Vol. 7:1055–, Issue. 6 June 1990.

[vK04] Helge von Koch. Sur une courbe continue sans tangente obtenue par uneconstruction geometrique elementaire. 1904.

[Wei72] Karl Weierstrass. Uber continuirliche functionen eines reellen arguments,die fur keinen werth des letzeren einen bestimmten differentialquotientenbesitzen. Royal Prussian Academy of Sciences, 1872.

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