Fractal Geometry in Digital Imaging

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ISBN: 9780127039701 | 0127039708
Cover: Paperback
Copyright: 6/23/1998

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This book presents the analysis of textured images using fractal geometry, and discusses its application to imaging science and computer vision when modeling natural objects. The authors explore the methods which can be used to simulate, analyze, and interpret coherent images, and demonstrate a new approach which segments each image into regions of similarity that can be characterized by a random fractal with a given fractal dimension. Fractal Geometry in Digital Imaging is based on a research project, but has been written with a broad coverage and user friendly math to make the book accessible to a wider audience. It includes real world experiences and applications using the techniques described.

Preface

(4)

Contents

(8)

Notation and Glossary

xvii

1 Introduction

(12)

1.1 Some Thoughts on Geometry and Images

(6)

1.2 About this Book

(2)

1.3 The Internet and other Resources

(4)

Part I: Fractal Geometry

(66)

2 Fractal and Texture Analysis

(10)

2.1 The Image Segmentation Problem

(4)

2.1.1 Image Segmentation

(1)

2.2 Texture Definitions

(4)

2.2.1 What is Texture?

(3)

2.2.2 Texture in SAR Images

(1)

2.3 Texture and Fractal Geometry

(2)

3 Description of Fractal Geometry

(34)

3.1 Von Koch Curve

(2)

3.2 Hausdorff-Besicovitch Dimension

(3)

3.3 D(H) for the Von Koch Curve

(1)

3.4 Similarity Dimensions

(7)

3.4.1 Self-similar Deterministic Fractals

(5)

3.5 Methods of Computing Fractal Dimensions

(12)

3.5.1 The Least Squares Approximation

(2)

3.5.2 Method 1: Walking-Divider

(1)

3.5.3 Method 2: Box Counting

(2)

3.5.4 Method 3: Prism Counting

(1)

3.5.5 Method 4: Epsilon-Blanket

(1)

3.5.6 Method 5: Perimeter Area Relationship

(1)

3.5.7 Method 6: Fractional Brownian Motion

(1)

3.5.8 Method 7: Power Spectrum

(3)

3.5.9 Hybrid Methods

(1)

3.6 Other Fractal Properties

(6)

3.6.1 The Fractal Signature

(1)

3.6.2 The Correlation Dimension and Signature

(1)

3.6.3 Information Dimension

(1)

3.6.4 Lyapunov Dimension

(1)

3.6.5 Lacunarity

(2)

3.6.6 Higher Order Fractals and Dimensions

(1)

3.7 Types of Methods and Types of Definitions

(3)

4 Random Scaling Fractals

(20)

4.1 Stochastic Differential Equations of Fractional Order

(5)

4.1.1 Inverse Solution

(1)

4.1.2 Fractal Images or Surfaces

(2)

4.2 Linking the Dimensions

(4)

4.2.1 Holder Continuity

(1)

4.2.2 Fourier Power Spectrum Dimension and the Box Counting Dimension

(1)

4.2.3 Fractional Derivative and Fourier Dimension

(1)

4.2.4 Algorithm for Fractal Differentiation

(1)

4.3 Real Life Fractional Brownian Motion

(5)

4.4 Summary of Fractal Types

(6)

Part II: Fractals to Segment Images

(82)

5 Object Segmentation

(32)

5.1 Texture Segmentation

(1)

5.2 Edge Detection

(25)

5.2.1 Nine Dimensional Operators

(2)

5.2.2 The Canny Edge Detector

(1)

5.2.3 Second Derivative Operations

(2)

5.2.4 Hierarchical Data Structures

(5)

5.2.5 Pixel Clustering

(2)

5.2.6 Extra Clustering Tools

(9)

5.3 Texture Parameters

106

(6)

5.3.1 Statistical Moments

106

(2)

5.3.2 Generalised Texture Measures

108

(2)

5.3.3 The Problems of Masks

110

(2)

5.4 Overview of the Segmentation Process

112

(1)

6 Fractal Segmentation

113

(36)

6.1 Extraction of a Fractal Dimension D

113

(6)

6.1.1 Description of the Spectral Algorithm on 1-D Signals

114

(4)

6.1.2 2-D Algorithm Using the 1-D Method

118

(1)

6.1.3 Direct 2-D Power Spectrum Algorithm

118

(1)

6.2 Signal Results

119

(15)

6.2.1 Simulated Segmentation

120

(1)

6.2.2 Real Life SAR Image Segmentation

121

(8)

6.2.3 Fractal Measures of Hubble Images

129

(5)

6.3 Multifractal Analysis

134

(5)

6.4 Overview of Fractal Segmentation

139

(2)

6.5 Generalisation of the RSF Model

141

(8)

6.5.1 Basic Properties

142

(2)

6.5.2 Analysis

144

(5)

7 Machine Vision

149

(12)

7.1 What is Vision?

149

(1)

7.2 What is Machine Vision in Practice?

150

(2)

7.2.1 Edge-based, 2-D, Binary Machine Vision

151

(1)

7.3 Texture

152

(1)

7.4 Natural Vision

153

(2)

7.5 What Can Fractal Geometry Contribute?

155

(1)

7.6 Applications of Fractal Geometry to Machine Vision

156

(5)

Part III: Fractals to Create Images

161

(42)

8 Random Noise Image Creation

163

(24)

8.1 Noise Generation, from White to Black

164

(8)

8.1.1 Linear Congruence Method

164

(3)

8.1.2 The Normal Distribution

167

(1)

8.1.3 Statistical Tests

168

(4)

8.2 Synthetic Power Spectrum Fractals

172

(14)

8.2.1 A Case Study: SAR Simulation

176

(4)

8.2.2 Fractal Signal Camouflage

180

(2)

8.2.3 Fractal Camouflage Texture

182

(1)

8.2.4 Deterministic Information

182

(2)

8.2.5 Random Synthetic Bumpmaps

184

(2)

8.3 Overview of Random Fractal Imagery

186

(1)

9 Lindermayer Systems

187

(16)

9.1 A Language for Self-Similarity

187

(11)

9.1.1 L-system Command Language

190

(6)

9.1.2 The Structure of Branching

196

(2)

9.2 Fractal Created Objects

198

(5)

Part IV: Image to Create Fractals

203

(50)

10 Image Representation

205

(16)

10.1 The Reasons for Compression

206

(1)

10.2 Lossless Coding Methods

207

(8)

10.2.1 Probability or Statistical Coding

208

(4)

10.2.2 Fractal IFS Analysis of AC

212

(1)

10.2.3 Dictionary or Substitution Coding

213

(2)

10.3 Lossy Coding Methods

215

(4)

10.3.1 JPEG: Joint Photographic Expert Group

216

(1)

10.3.2 Segmentation Based Compression

217

(2)

10.4 Overview of Compression Coders

219

(2)

11 Fractal Image Compression

221

(32)

11.1 The Contractive Mapping

221

(2)

11.1.1 Hausdroff Distance Functions

223

(1)

11.2 IFS and the Collage Theorem

223

(4)

11.3 PIFS Compression System

227

(5)

11.3.1 Range-Domain Algorithm

228

(1)

11.3.2 Choice of Partitioning Strategies

229

(1)

11.3.3 Choice of Affine Transformation, W=Uwi

230

(1)

11.3.4 Searching Strategy -- Speed-ups

231

(1)

11.3.5 Final Low Level Coding Strategy

231

(1)

11.4 Properties and Features

232

(7)

11.4.1 Knee Points

234

(1)

11.4.2 Stability of the Attractor

235

(1)

11.4.3 Self-similarity-Resolution Independence

235

(3)

11.4.4 Shape Comparison

238

(1)

11.5 Improved Fractal Compression

239

(8)

11.5.1 Colour Considerations

243

(2)

11.5.2 Video Considerations

245

(2)

11.6 Compression Conscious Operations

247

(2)

11.6.1 Fractal Texture Maps

248

(1)

11.7 Summary of Fractal Compression

249

(4)

Part V: Conclusions and Appendices

253

(40)

12 Conclusions, Discussions and Ideas for the Future

255

(8)

12.1 General Overview of the Book

255

(4)

12.2 Food for Thought

259

(4)

Appendixes

263

(30)

A Optical Diffraction Theory and Image Formation

263

(24)

A.1 Scalar Diffraction Theory

264

(1)

A.2 Kirchhoff Diffraction Theory

265

(2)

A.3 Fraunhofer Diffraction

267

(3)

A.3.1 Well Known examples of Fraunhofer Diffraction

270

(1)

A.4 Fresnel Diffraction

271

(2)

A.4.1 Examples of Fresnel Diffraction

273

(2)

A.5 On the Fourier Transforming Properties of a Lens

275

(2)

A.5.1 Basic Principles of Fourier Optics

277

(1)

A.5.2 Optical Filtering

277

(1)

A.6 Optical System and Image Formation

277

(3)

A.6.1 Incoherent Image Formation

280

(3)

A.6.2 Coherent Image Formation

283

(4)

B n-Dimensional Fourier Transform

287

(6)

B.1 Evaluation of the n-Dimensional Fourier Transform

287

(6)

Bibliography

293

(16)

Index

309

(10)

Images of the Book

319

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