Fractal Image Compression de Yuval Fisher

Fractal Image Compression: Theory and Application

Yuval Fisher

What is "Fractal Image Compression," anyway? You will have to read the book to find out everything about it, and if you read the book, you really will find out almost everything that is currently known about it. In a sentence or two: fractal image compression is a method, or class of methods, that allows images to be stored on computers in much less memory than standard ways of storing images. The "fractal" part means that the methods have something to do with fractals, complicated looking sets that arise out of simple algorithms. This book contains a collection of articles on fractal image compression. Beginners will find simple explanations, working C code, and exercises to check their progress. Mathematicians will find a rigorous and detailed development of the subject. Non-mathematicians will find a parallel intuitive discussion that explains what is behind all the "theorem-proofs." Finally, researchers - even researchers in fractal image compression - will find new and exciting results, both theoretical and applied. Here is a brief synopsis of each chapter: Chapter 1 contains a simple introduction aimed at the lay reader. It uses almost no math but explains all the main concepts of a fractal encoding/decoding scheme, so that the interested reader can write his or her own code. Chapter 2 has a rigorous mathematical description of iterated function systems and their gen eralizations for image encoding. An informal presentation of the material is made in parallel in the chapter using sans serif font.

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Cuprins

1 Introduction.- 1.1 What Is Fractal Image Compression?.- 1.2 Self-Similarity in Images.- 1.3 A Special Copying Machine.- 1.4 Encoding Images.- 1.5 Ways to Partition Images.- 1.6 Implementation.- 1.7 Conclusion.- 2 Mathematical Background.- 2.1 Fractals.- 2.2 Iterated Function Systems.- 2.3 Recurrent Iterated Function Systems.- 2.4 Image Models.- 2.5 Affine Transformations.- 2.6 Partitioned Iterated Function Systems.- 2.7 Encoding Images.- 2.8 Other Models.- 3 Fractal Image Compression with Quadtrees.- 3.1 Encoding.- 3.2 Decoding.- 3.3 Sample Results.- 3.4 Remarks.- 3.5 Conclusion.- 4 Archetype Classification in an Iterated Transformation Image Compression Algorithm.- 4.1 Archetype Classification.- 4.2 Results.- 4.3 Discussion.- 5 Hierarchical Interpretation of Fractal Image Coding and Its Applications.- 5.1 Formulation of PIFS Coding/Decoding.- 5.2 Hierarchical Interpretation.- 5.3 Matrix Description of the PIFS Transformation.- 5.4 Fast Decoding.- 5.5 Super-resolution.- 5.6 Different Sampling Methods.- 5.7 Conclusions.- A Proof of Theorem 5.1 (Zoom).- B Proof of Theorem 5.2 (PIFS Embedded Function).- C Proof of Theorem 5.3 (Fractal Dimension of the PIFS Embedded Function).- 6 Fractal Encoding with HV Partitions.- 6.1 The Encoding Method.- 6.2 Efficient Storage.- 6.3 Decoding.- 6.4 Results.- 6.5 More Discussion.- 6.6 Other Work.- 7 A Discrete Framework for Fractal Signal Modeling.- 7.1 Sampled Signals, Pieces, and Piecewise Self-transformability.- 7.2 Self-transformable Objects and Fractal Coding.- 7.3 Eventual Contractivity and Collage Theorems.- 7.4 Affine Transforms.- 7.5 Computation of Contractivity Factors.- 7.6 A Least-squares Method.- 7.7 Conclusion.- A Derivation of Equation (7.9).- 8 A Class of Fractal Image Coders with Fast Decoder Convergence.- 8.1 Affine Mappings on Finite-Dimensional Signals.- 8.2 Conditions for Decoder Convergence.- 8.3 Improving Decoder Convergence.- 8.4 Collage Optimization Revisited.- 8.5 A Generalized Sufficient Condition for Fast Decoding.- 8.6 An Image Example.- 8.7 Conclusion.- 9 Fast Attractor Image Encoding by Adaptive Codebook Clustering.- 9.1 Notation and Problem Statement.- 9.2 Complexity Reduction in the Encoding Step.- 9.3 How to Choose a Block.- 9.4 Initialization.- 9.5 Two Methods for Computing Cluster Centers.- 9.6 Selecting the Number of Clusters.- 9.7 Experimental Results.- 9.8 Possible Improvements.- 9.9 Conclusion.- 10 Orthogonal Basis IFS.- 10.1 Orthonormal Basis Approach.- 10.2 Quantization.- 10.3 Construction of Coders.- 10.4 Comparison of Results.- 10.5 Conclusion.- 11 A Convergence Model.- 11.1 The r Operator.- 11.2 Lp Convergence of the RIFS Model.- 11.3 Almost Everywhere Convergence.- 11.4 Decoding by Matrix Inversion.- 12 Least-Squares Block Coding by Fractal Functions.- 12.1 Fractal Functions.- 12.2 Least-Squares Approximation.- 12.3 Construction of Fractal Approximation.- 12.4 Conclusion.- 13 Inference Algorithms for WFA and Image Compression.- 13.1 Images and Weighted Finite Automata.- 13.2 The Inference Algorithm for WFA.- 13.3 A Fast Decoding Algorithm for WFA.- 13.4 A Recursive Inference Algorithm for WFA.- A Sample Code.- A.l The Enc Manual Page.- A.2 The Dec Manual Page.- A.3 Enc.c.- A.4 Dec.c.- A.5 The Encoding Program.- A.6 The Decoding Program.- A.7 Possible Modifications.- B Exercises.- C Projects.- C.1 Decoding by Matrix Inversion.- C.2 Linear Combinations of Domains.- C.3 Postprocessing: Overlapping, Weighted Ranges, and Tilt.- C.4 Encoding Optimization.- C.5 Theoretical Modeling for Continuous Images.- C.6 Scan-line Fractal Encoding.- C.7 Video Encoding.- C.8Single Encoding of Several Frames.- C.9 Edge-based Partitioning.- C.10 Classification Schemes.- C.l1 From Classification to Multi-dimensional Keys.- C.12 Polygonal Partitioning305.- C.13 Decoding by Pixel Chasing.- C.14 Second Iterate Collaging.- C.15 Rectangular IFS Partitioning.- C.16 Hexagonal Partitioning.- C.17 Parallel Processing.- C.18 Non-contractive IFSs.- D Comparison of Results.- E Original Images.