Fractal Image Encoding and Analysis:A NATO ASI Series Book, Yuval Fisher (ed.), Springer Verlag, New York, 1998.

This book contains contributed articles by many of the leading researchers in fractal image encoding and analysis.

Eroneously omitted from the book, but part of the official proceedings is the following article by Zachi Baharav, Hagai Krupnik, David Malah, and Ehud Karnin: A Multi-Resolution Framework for Fractal Image Representation and its Applications.

Table of contents

Part I Fractal Image Encoding

Chapter 1. Why Fractal Block Coders Work
Geoffrey Davis

1.1 Introduction
1.2 Linear Transform Coders
1.2.1 Entropy Coding
1.2.2 The Karhunen-Loeve Transform
1.2.3 Image Model for Transform Coders
1.3 Fractal Block Coders
1.3.1 Image Model for Fractal Coders
1.3.2 Mechanics of Fractal Block Coders
1.4 A Wavelet Framework
1.4.1 The discrete wavelet transform Discrete Wavelet Transform
1.4.2 A Wavelet Analog of Fractal Block Coding
1.4.3 Convergence Criterion
1.4.4 Why Fractal Block Coders Work
1.5 Self-Quantization of Subtrees
1.5.1 Extrapolation in Scale
1.5.2 Convergence of Fractal Schemes
1.5.3 Results
1.6 Conclusion

Chapter 2. On Fractal Compression and Vector Quantization
Skjalg Lepsy, Paolo Carlini, Geir Egil Oien

2.1 Two coders to compare
2.1.1 Product code vector quantization
2.1.2 A mean-removed gain-shape VQ
2.1.3 A fractal coder
2.2 Methods of codebook population
2.2.1 Population of the codebook in the fractal coder
2.2.2 Population of the codebook in the M-VQ
2.3 Cross-transformability
2.3.1 Details on the experiment
2.3.2 Results and remarks
2.4 Trained codebooks versus self-transformation
2.4.1 Details on the experiment
2.4.2 Results and remarks
2.4.3 Characteristics of the two techniques
2.4.4 Conclusion
2.5 Discussion
2.5.1 Other segmentation schemes
2.5.2 Position-dependent codebooks
2.6 Conclusion
Acknowledgements

Chapter 3. On the Use of Subsampling in Fractal Image Compression
Lars Lundheim

3.1 Introduction
3.2 Notation and terminology
3.3 Decimation by Subsampling
3.4 The family graph
3.5 An efficient synthesis algorithm
3.6 Complexity
3.7 An alternative algorithm
3.8 Empirical Results
3.9 Contractivity
3.9.1 Test of eventual contractivity
3.10 Conclusion

Chapter 4. On the Benefits of Basis Orthogonalization in Fractal Compression
Geir E. Oien and Skjalg Lepsoy

4.1 Introduction
4.2 Introducing basis orthogonalization in the fractal signal model
4.3 Decoder convergence
4.3.1 Convergence speed
4.3.2 Invariance of attractor
4.3.3 Similarity between collage and attractor
4.3.4 Pyramid decoder structure
4.4 Fractal model coefficients
4.4.1 Coefficient optimization
4.4.2 Coefficient and offset subspace properties
4.4.3 Image coding: Practical considerations and compromises
4.5 Encoder complexity reduction
4.5.1 Simplified coefficient computation
4.5.2 Simplified distortion minimization
4.5.3 Directed domain pool search
4.6 Comparisons to other techniques
4.6.1 Product code vector quantization
4.6.2 Other lossy compression techniques
4.7 Conclusion

Chapter 5. On the Dimension of Fractally Encoded Images
Yuval Fisher, Frank Dudbridge, Ben Bielefeld

5.1 Introduction
5.2 The Fractal Dimension of Image Blocks
5.3 Results
5.4 Conclusion
5.5 Acknowledgments

Chapter 6. Fractal Image Compression via nearest neighbor search Nearest Neighbor Search
Dietmar Saupe

6.1 Introduction
6.1.1 The search problem in fractal image encoding
6.1.2 Previous work
6.2 A formula for the least squares error based on projections
6.3 Searching in fractal image compression is nearest neighbor search nearest neighbor search
6.4 Practical considerations
6.5 Implementation
6.6 Results
6.6.1 Choosing parameters for the approximate nearest neighbor search nearest neighbor search
6.6.2 Fast nearest neighbor search nearest neighbor search with classification
6.6.3 Fast nearest neighbor search nearest neighbor search without loss in fidelity
6.6.4 On the use of orthonormally transformed feature space
6.6.5 Quantization issues
6.7 Other work related to feature vectors
6.7.1 Straight feature vectors
6.7.2 Invariant moments
6.7.3 Tree structured methods
6.8 Conclusion

Chapter 7. Fractal Image Coding: Some Mathematical Remarks on Its Limits and Its Prospects
F.M. Dekking

7.1 Introduction
7.2 Basics
7.3 Coding of 1-d Black and White Images
7.4 Grey Value Coding: The Collage and the Attractor
7.5 Pyramids and Martingales in Grey Value Image Coding
7.6 Martingales for Electrical Engineers

Chapter 8. Linear Time Fractal Quadtree Coder
F. Dudbridge

8.1 Introduction
8.2 A model for fractal images
8.3 Image coding
8.4 Reconstruction
8.5 Implementation

Chapter 9. Fractal Encoding of Video Sequences
Yuval Fisher

9.1 Introduction and Notation
9.2 Video Sequences
9.3 Implementation
9.4 Results
9.5 Discussion
9.6 Conclusion

Chapter 10. Theory of Generalized Fractal Transforms
Bruno Forte and Edward R. Vrscay

10.1 Introduction
10.2 Generalized Fractal Transforms
10.3 From IFS to IFSM Fractal Transforms
10.3.1 IFS
10.3.2 IFZS
10.3.3 IFSM
``Place-Dependent'' IFSM
10.3.4 IFSP
10.4 IFS on the Space of Distributions $ \fam \tw@ D ^\prime (X)$
10.4.1 Affine IFSD and the Connection with IFSP and IFSM
10.4.2 Integrals Involving Affine IFSD

Chapter 11. Inverse Problem Methods for Generalized Fractal Transforms
Bruno Forte and Edward R. Vrscay

11.1 Introduction
11.2 Inverse Problem for IFSM
11.2.1 Collage Theorem for IFSM in $ \fam \tw@ L ^2(X,\mu )$
11.2.2 Formal Solution to the IFSM Inverse Problem
11.2.3 ``Local IFSM'' on $ \fam \tw@ L ^p (X , m) )$
11.2.4 Inverse Problem With Place-Dependent IFSM
11.2.5 Application of IFSM Methods to Images
11.3 Approximation of Measures Using IFSP
11.3.1 Approximation by ``Moment Matching''
11.3.2 Collage Theorem for Moments
11.3.3 Formal Solution to Inverse Problem
11.3.4 Some Numerical Results
11.4 IFSM and Fractal Wavelet Compression
11.5 Collage Theorem for Fourier Transforms

Chapter 12. Fractal Compression of ECG Signals
Geir E. Oien and Geir Narstad

12.1 Introduction to ECG signals
12.1.1 ECG measurement and characteristics
12.1.2 Digitization and storage of ECG signals
12.1.3 History of ECG compression -- a brief review
12.2 Fractal compression and ECG signals
12.2.1 Redundancy and irrelevance
12.2.2 Why should fractal compression work?
12.2.3 The fractal signal model
12.2.4 Parameter choices
12.2.5 Parameter optimization
12.2.6 The coefficient quantization scheme
12.2.7 The decoding scheme
12.3 Results and discussion
12.3.1 Quality criteria
12.3.2 Bit allocation experiments
12.3.3 Quantizer characteristics
12.3.4 Coding experiments
12.3.5 Comparisons to other techniques
12.4 Conclusion

Part II Fractal Image Analysis

Chapter 13. Dimensions of Fractals and Multifractals
Kenneth Falconer

13.1 Dimensions of Fractals
13.1.1 Hausdorff and Packing Dimensions
13.1.2 Calculation of Dimensions
13.1.3 Iterated Function Systems
13.1.4 Self-similar Sets
13.1.5 Self-affine Sets
13.1.6 Related Constructions
13.2 Fractal Measures
13.2.1 Local Dimensions and Multifractal Spectra
13.2.2 Invariant Measures
13.2.3 The Multifractal Spectrum of Self-similar Measures
13.2.4 Other Examples
13.2.5 Other Approaches to Multifractal Theory

Chapter 14. Velocity, Length, Dimension
Claude Tricot

14.1 Introduction
14.2 Finite Length
14.3 Infinite Length: A Geometrical Approach
14.4 The Metric Set of Intervals: Preliminary Results
14.5 A Generalized Notion of Length
14.6 Infinite Length: An Analytical Approach
14.7 The Compass Method
14.8 The Method of Constant Breadth
14.9 Comparison of Lengths
14.10 Fractal Dimension of Curves
14.11 A Few Typical Curves
14.11.1 Rectifiable curves
14.11.2 Spirals
14.11.3 Self-similarity
14.11.4 Self-affine graphs
14.11.5 Graphs of continuous functions
14.11.6 Other curve with uniform breadth

Chapter 15. A Local Multiscale Characterization of Edges Applying the Wavelet Transform
Carl J.G. Evertsz, Kathrin Berkner, Wilhelm Berghorn

15.1 Introduction
15.2 Gaussian Kernel Smoothing
15.3 Edge Detection and Filtering with Wavelets
15.3.1 The Wavelet Transform
15.3.2 Holder Exponents of Singularities and Edges
15.3.3 Edge-Parameters
15.4 Numerical Implementation
15.4.1 Finding Modulus-Maxima Lines
15.4.2 Estimating Holder Exponents and Other Edge-Parameters
15.4.3 Noise
15.5 Applications
15.5.1 Bone Tumors
15.5.2 Liver Tumor
15.5.3 Lena
15.5.4 Conclusions and Summary

Chapter 16. Local Connected Fractal Dimension Analysis of Early Chinese Landscape Paintings and X-Ray Mammograms
Richard F. Voss

16.1 Fractals: From Self-Similarity to Self-Affinity and Multifractals
16.2 Measures and Mass Dimension
16.3 Local Image Gradient and the Statistics of Natural Images
16.4 Local Fractal Mass Dimension
16.5 Classification of Early Chinese landscape drawings Chinese Landscape Drawings
16.6 Local Connected Fractal Dimension
16.7 Preliminary Classification of X-ray Mammograms
16.8 Discussion

Chapter 17. Introduction to the Multifractal Analysis of Images
Jacques Levy-Vehel

17.1 Introduction
17.2 A quick review on some ``classical" methods
17.2.1 Canny's edge detector
17.2.2 Mathematical Morphology
17.2.3 Image Multiscale Analysis
17.3 The multifractal approach
17.3.1 Overview
17.3.2 Choquet capacities
17.3.3 Multifractal analysis of sequences of Choquet capacities
17.3.4 An example: the binomial measure
17.3.5 Multifractal correlations
17.4 Application to image analysis
17.4.1 Introduction
17.4.2 Numerical estimation of $f_g$
17.4.3 Image enhancement
17.4.4 Change detection
17.4.5 Edge detection

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