Robust source coding with generalised T-Codes

This thesis presents a range of novel and improved results in the area of source coding with T-Codes, based on a thorough review of the recursive structure of the T-Codes. T-Codes were introduced by Mark Titchener in 1984. They are variable-length codes, similar to the well-known Hu man codes and may thus be used for compression by source coding. T-Codes have also been noted for their self-synchronisation properties, which result from their recursive construction. This thesis reviews and formalises the theory of generalised T-Codes and their recursive construction. It extends an existing recursive storage concept, the binary depletion codes, as the \T-depletion codes". These are shown to provide a unique xed-length representation for T-Codes, which may be used in a recursive encoder/decoder scheme for T-Codes. It is shown that the T-depletion code format accommodates unique representations for all proper pre xes of T-Code codewords, thus covering the complete state space of an encoder or decoder. For use in decoder applications, the thesis derives a recursive conversion between T-depletion codes and a contiguous integer index. A new simpli ed method for the calculation of the expected synchronisation delay (ESD) is introduced. Finally, the thesis develops a recursive search algorithm to nd the T-Code set that most e ciently encodes a source. 5 Preface and Acknowledgements The present thesis is, if the author has counted correctly, the fourth PhD thesis in the area of T-Codes, and the rst in the area of generalised T-Codes. While a PhD thesis is always an individual's research record, few if any are written in isolation. This thesis is no exception. In fact, given the circumstances under which the project was conducted, I owe thanks to rather more people than the average PhD student. The list should perhaps start with Gavin Higgie, who rst introduced me to TCodes in a graduate class that he was teaching. Then it was Gary Bold who alerted me to the fact that Mark Titchener had returned to New Zealand and was teaching at Auckland University's Tamaki Campus. Mark was very supportive from the start and right throughout the thesis process, and his support was not restricted to university matters alone. I hope very much that this thesis rewards his con dence in me. Throughout my time at Tamaki, Mark has been a continuous source of new ideas. While he often had a hard time convincing a sceptic like me, I must say that his \gut feel" has often proved to be more reliable than my (and other's) doubts. Many of the new and extended results in this thesis have their origin in our brainstorming sessions. Radu Nicolescu helped a great deal in formalising the notation used in this thesis, and his uniqueness theorem proved to be an extremely useful tool for this thesis. His interest in my work was always appreciated. Peter Hertling deserves a special mention for his suggestions for cleaning up some of the proofs in Chapters 4 and 6. Professor Cris Calude deserves thanks for his ongoing interest in my work, and his help and feedback regarding several of the publications that this thesis is based on. Professor Reinhard Klette was kind enough to read some early chapters of this thesis his comments on style and structure have had a profound in uence on the shape of this thesis. Professor Clark Thomborson also gave helpful comments | his assistance with administrative and policy matters and an always open ear for PhD students' problems have been a great reassurance. Thanks go to the Department of Computer Science, in particular its present head Professor Peter Gibbons, for providing me with a departmental stipend and with the necessary hardware to keep my tutoring load under control so I could write this thesis. Penny Barry also deserves a mention for making the administrative side of this run so smoothly. Bruce Benson, Paul Burkimsher, Paul Bonnington, Rob 6Burrowes, Mano Manoharan, Clare West, Gary Wong, and Edouard Poor have had their part in saving the computational side from the brink of disaster a few times. Working at Tamaki Campus has been a great experience. Its interdisciplinary family atmosphere has been a source of ideas. My special thanks go to Alastair McNaughton whose comments made me re-think aspects of the search algorithm in Chapter 10. As a result, the algorithm's usability for real-world problems has improved signi cantly. Many other sta from computer science, mathematics, physics, and statistics have had input into the notation used. Being one of the rst PhD students at Tamaki campus meant that I often did not t into established categories. Professor Chris de Freitas o ered a lot of support during the early stages of my research, while he was Head of Division. Steve Chaney and others from ITSS often put in unscheduled jobs to keep me online. The administrative sta at Tamaki have also su ered much under me and deserve a mention. Library support is vital for any research, especially when most of the relevant literature is located on another campus, and one's eld of research spans several disciplines. Lynley Stone and her team at the Tamaki Library have bent over backwards many times to make the physical distance disappear. Thanks here go especially to Brenda Dwane, Danielle Carter, Hester Mounti eld, and Russell Tu ery for going the extra mile on so many occasions. Thanks also go to the many sta and fellow students who have helped to keep me sane. My motivation also owed a lot to my friends in amateur radio and elsewhere. Finally, I am indebted to my family for their continuous love and support over time and distance. Auckland, February 1998 Ulrich G unther