Fractal Image Compression

by Barnsley, Michael F.; Hurd, Lyman P.; Anson, Louisa F.

• ISBN13: 9781568810003
• ISBN10: 1568810008
• Format: Hardcover
• Copyright: 1993-01-01
• Publisher: A K Peters Ltd

Table of Contents

Acknowledgments
Introduction	p. 1
Formulation of Mathematical Models for Real World Images	p. 5
Description and Properties of Real World Images	p. 6
Mathematical Models for Real World Images	p. 25
Scanning and Digitizing	p. 33
Quantization, Scan Order, and Color	p. 38
Mathematical Foundations for Fractal Image Compression I	p. 47
Spaces, Mappings, and Transformations	p. 48
Affine Transformations in R	p. 49
Construction of the Classical Cantor Set Using Two Affine Transformations on R	p. 50
Affine Transformations in the Euclidean Plane	p. 51
Affine Transformations in Three-Dimensional Real Space	p. 60
Norms on Linear Transformations on R[superscript 2]	p. 63
Topological Properties of Metric Spaces and Transformations	p. 63
Contraction Mapping Theorem - Key to Fractal Image Compression	p. 70
Fractal Image Compression I: IFS Fractals	p. 75
Spaces of Images - the Hausdorff Space H	p. 76
Contraction Mappings on the Space H	p. 78
Iterated Function Systems	p. 79
Iterated Function Systems of Affine Transformations in R[superscript 2]	p. 82
The Photocopy Machine Algorithm for Computing the Attractor of an IFS	p. 89
C Source Code for Computing the Attractor of an IFS	p. 93
The Collage Theorem	p. 100
Fractal Image Compression Using IFS Fractals	p. 100
Measures and IFS's with Probabilities for Grayscale Images	p. 105
Grayscale Photocopy Algorithm	p. 108
Fractal Image Compression Using the Collage Theorem for Measures	p. 113
Dudbridge's Fractal Image Compression Method	p. 115
Mathematical Foundations for Fractal Image Compression II	p. 119
Information Sources and Zero-Order Markov Sources	p. 120
Codes	p. 122
Kraft-McMillan Inequality	p. 128
C Source Code for Illustrating Kraft's Theorem	p. 130
Entropy	p. 134
Shannon-Fano Codes	p. 138
Extensions of Sources	p. 140
Higher-Order Markov Sources	p. 142
Huffman Codes for Compression	p. 150
C Source Code Illustration of a Huffman Code	p. 155
Addresses on Fractals	p. 159
Arithmetic Compression and IFS Fractals	p. 162
C Source Code Illustration for Arithmetic Encoding and Decoding	p. 166
Fractal Image Compression II: The Fractal Transform	p. 173
General Description of Fractal Image Compression Methodology	p. 174
Local Iterated Function Systems	p. 176
The Collage Theorem for a Local IFS	p. 179
Calculation of Binary Attractors of Local IFS Using the Escape Time Algorithm	p. 181
The Black and White Fractal Transform	p. 182
The Grayscale Fractal Transform	p. 186
A Local IFS Associated with the Fractal Transform Operator	p. 187
Simple Examples of Grayscale Fractal Transforms	p. 188
C Source Code Implementation	p. 198
Illustrations of Fractal Transform Compression	p. 215
A JPEG Image Compression	p. 219
A.2. Discrete Cosine Transform (DCT)	p. 223
A.3. Quantization	p. 224
A.4. Runlength Encoding	p. 226
A.5. Entropy Encoding	p. 227
A.6. Interchange Format	p. 227
A.7. C Source Code Illustrating JPEG Compression	p. 228
Index	p. 241
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