Image Approximation by Self Aane Fractals

This thesis describes a novel method for approximating digital images and similar data. The work is based on deterministic fractal geometry, which for some time has promised to provide new data compression techniques which may challenge existing industrial standards. However, few robust algorithms for fractal compression have hitherto been published. The present work provides enabling techniques for the eventual development of commercially applicable compression algorithms. A self-a ne fractal is a geometrically complex shape which may be speci ed with relatively little information. The problem of constructing a fractal from its de ning equations is identi ed as a forward problem, with the harder inverse of determining which equations de ne a given fractal. A new solution of the forward problem is proposed, which constructs an approximation to the fractal in optimal time. A novel scheme for assigning attributes to the elements of a fractal is then developed, which permits the representation of greytone or colour information on a fractal. This is possible through the concept of an invariant function, which has a self-similarity property analogous to that of a fractal. A method is proposed for nding invariant approximations to functions de ned on fractals. The particular case of the least-squares self-a ne approximant is derived and analysed. The technique is applied to the approximation of a greytone rectangular image block, which allows the blockwise approximation of an entire image. A novel scheme for constructing a partition of an image into adjoint rectangular blocks is proposed, based on the observation that the edge-value of a block is related to the accuracy of its fractal approximation. The partition scheme combined with the fractal approximation technique gives a prototype image compression algorithm. A brief comparison with existing algorithms con rms our methods as a feasible subject for continued development. 1 Acknowledgements I am indebted to many people for their contributions towards the success of this work. The help of my colleagues in the William Penney Lab has been much appreciated, especially that of Henry Young for his early assistance, and of Dave Wilson whose copious help and advice has been invaluable in preparing the thesis. Discussions with Dave Sharp, Alan Wilson, Frank Devlin and David Fung were of considerable help in clarifying the various ideas. The test images and their ADCT compressions were provided by the Independent Broadcasting Authority of Great Britain. Some equipment was provided by SERC Grant no. GR/H13819, \Image compression by fractal sets"; my maintenance was provided by SERC studentship number 88303797. The project was supervised by Dr D. Monro at Imperial College London. My deepest thanks are reserved for my parents, who have continued to provide unceasing support for all my activities. Finally I would like to thank the many friends I have made during the last three years, in particular the Hill eld Harriers, The Little Swing Band, Dr Ron Holder and Laura Dudbridge, all of whom have helped to make this a personally memorable and rewarding period of study. 2 Chapter