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Fractal Image Compression Technique FRACTAL IMAGE COMPRESSION A thesis submitted to the Faculty of Graduate Studies and Research in partial mrnent of the requirernents for the degree of Master of Science Information and S ystems Science Department of Mathematics and Statistics Carleton University Ottawa, Ontario April, 1998 National Library Bibliothèque nationale I*B of Canada du Canada Acquisitions and Acquisitions et Bibliographie Services services bibliographiques 395 Wellington Street 395. rue Weltington Ottawa ON KI A ON4 Ottawa ON KIA ON4 Canada Canada Yow fik Votre referenco Our Ne Notre rëftirence The author has granted a non- L'auteur a accorde une licence non exclusive licence allowing the exclusive permettant à la National Library of Canada to Bibliothèque nationale du Canada de reproduce, loan, distriiute or sell reproduire, prêter, distribuer ou copies of this thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. la fome de microfiche/nlm, de reproduction sur papier ou sur fomat électronique. The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts fiom it Ni la thèse ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation. ACKNOWLEDGEMENTS I would iike to express my sincere t hanks to my supe~so~~Professor Bnan h[ortimrr, for providing me with the rïght balance of freedom and ,guidance throughout this research. My thanks are extended to the Department of Mat hematics and Statistics at Carleton University for providing a constructive environment for graduate work, and hancial assistance. 1 nish to express my gratitude to mu wife; Martine, for her constant encouragement, patience and support during the course of my graduate stu&es. ABSTRACT This thesis examines the theory of kactal image compression and gives a survey of such techniques for stitl images and vide0 sequences. Particular attention is given to partition iterated function systerns (pif), but recurrent iterated function system (ri/)are also dis- cussed. We begïn Wth a discussion of iterated hction systems and there applications in image compression and then go on to provide the theoretical basis for such systems. as well as for pif and rif- In Chapter 3 we discuss different image models as well as the encoding of images using pi/ and rif- Ditferent mo~cationsto the brute force pif are then discussed in Chapter 4, and we conclude the thesis by presenting four methods which extend the theory of pif to video sequences. TABLE OF CONTENTS CHAPTER PAGE 1 INTRODUCTION ....................................................................... 1 1.1 Why Image Compression .............................................................. 1 1.2 The Brute Force Fractal Image Compression Technique ............................ -4 1.3 The Ongins of Fractal Image Compression ............................................ 7 1.4 W'hat we dlcover in this Dissertation ................................................ 8 1.5 Conclusion ................................................................................. 8 2 PTERATED FUNCTION SYSTEMS AND THE CONTRACTNE MAPPING FIXED POINT THEOREM ........................................ 9 2.1 Iterated Function Systems .............................................................. -9 2.2 Analysis of ifs and the Generalized Collage Theorem for Zs ....................... -13 2.3 Recurrent Iterated Function Systems, rifs ........................................... -18 2.4 Partitioned Iterated Function Systems, pifs ......................................... -20 2.5 Conclusion ............................................................................... 20 IMAGE ENCODING USING IFS. RIFS AND PIFS .................... -21 3 -1 Image Models ............................................................................ 21 3.1.1 Pixelized Data .................................................................... -21 3.1.2 Functions on 8" .................................................................. 22 3.2 Colour Images ........................................................................... 23 3.3 IFS Representation of Images ......................................................... -26 3.4 Image Encoding mith rifs and pifs .................................................... -26 3.4.1 Encoding Images with Recurrent Iterated Function Systems ................ 28 3.4.2 Encoding Images nrith Partitioned Iterated Function Systems .............. -28 3.5 Conclusion ............................................................................... 29 4 IMFROVED PARTITIOhmG AND SEARCHING METHODS .... 31 4.1 Block Shapes ......... ................................................................... -31 4.1.1 Variable Sized Square Blocks using Quadtrees ............................... -32 4.1 -2 Rectangular Block Shapes using HV-Partitioning ............................ -35 4.1 -3 Region Based Coding wit h a Heuris tic Search .................................37 4.1.4 Other Partitioning Methods ..................................................... 40 4.2 Improved Searching Methods ......................................................... -41 4.2.1 Lean Domain Pools ............................................................... 41 4.2.2 Points in an Abstract Space with the FFIC Algorithm ...................... -45 4.3 Conclusion ............................................................................... 48 5 FRACTAL COMPRESSION OF VIDE0 SEQUENCES ................ -49 5.1 Inter/lntraham Fractal Video Encoding ............................................. -49 5.1.1 Simple Motion Compensation ............................................... .. .-49 5-12 Low Rate Video Coding .......................................................... 50 5 -2Shree Dimensional Iterated Function Systems ....................................... 84 5.2.1 3D Fractal Block Coding of Video Sequences .................................. 54 5.2 -2 Improved Searching and Classification Methods for Fractal Volume Compression ....................................................................... 58 5.3 Conclusion .............................................................................. -60 LIST OF FIGURES CHAPTER PAGE 1.1 The orio$nal image of Lena ................................................................. 2 1.2 The different steps in the construction of the Koche cuve ............................ -3 1-3 The Koche sno&ake ........................................................................ 3 1.4 The self-similar parts of Lena ............................................................. -4 1.5 The initial image (top left) , the &st iterate (top right), the second iterate (bottom leR) and the tenth iterate (bottom right) ...................................... 6 2.1 The hst thtee iterates of the system in example 2.1, for three different initial images ....................................................................................... 10 2.2 The original image x, and its iterate F(x) ............................................... 11 2.3 Barnsley's fern ..............................................................................12 2 -4 Two different examples of how to calculate the hausdorff distance .................. -15 2.5 Barnsley's fern with Sierpinski triangle leaves .......................................... 19 3.1 The three dimensional graph of Lema and the original Lerma image ...............-22 3.2 The flow chart of the brute Force approach for colour images ........................ -25 3.3 The range blocks of an arbitrary image ................................................. 27 3.4 A few of the possible domain blocks of the same arbituary image ...................-27 4.1 (a) An example of the quadtree partitioning of an image ............................ -33 4.1 (b) The quadtree coresponding to the partitioning in Figure 4.la ...................-33 4.1 (c)A representation of the Lenna image. partitioned using the quatree algorit hm .................................................................................... 34 4.1 (d) The decoded image of a do~,which was encoded with the quadtree partitioning algorithm .................................................................... -34 4.2 (a) The original image ..................................................................... 36 4.2 (b) The fist partitioning of the image ..................................................-36 4.2 (c) The next two partitions of the image ...............................................-36 4.3 (a) The seed (solid Line) and a possible extension (dashed line) ..................... -38 4.3 (b) A range region and its coresponding domûin region .. .. .. -.. .. -.. .. .. 38 4.4 Which show hro regions competing to include a range block and two blocks which should not be considered as candidates to belong to another range region: respectively .. - - - .- .. - . .. - - - .. - .- .. - .. - .- .. - .. - . - - - -39 4.3 A representation of the Lenna image, partitioned using the tnangular partitioning method ..-.-. - - - -.. _. ,. .. -. _, .. - - -.-. .-,- -. .- - - .. -- -.- .. .. -... .. -- -- 40 4.6 (a) The graph of the number of domain blocks vs variance: For a quadtree partitioning of Lenna .. -. .. - - .. .. - ... .. .. .. .. - - .. .. - .. -. - - 42 4.6 (b) The actual domain blocks used (in black) for the encoding of the Lenna 4.7
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