I would like to thank my supervisors Dr. Neil Roberts and Dr. Ian Porteous, who have provided help and encouragement on numerous occasions throughout the course of this Ph.D. I started at the University of Liverpool in 1993, studying for an M.Sc. in Pure Mathematics under the supervision of Ian. After the M.Sc., Ian introduced me to a colleague of his, Dr. Neil Roberts, who had previously given a seminar at the Department of Pure Mathematics on the geometry of surfaces obtained by magnetic resonance imaging (MRI). Following this meeting, I joined the Magnetic Resonance and Image Analysis Research Centre (MARIARC) as a Ph.D. student. I would especially like to thank Neil for his assistance over the last year, during which time I have juggled Ph.D. and full-time job. Without his guidance, this thesis would almost certainly not have reached fruition.

I am also grateful to my colleagues and friends at MARIARC, many of whom have been involved (directly or indirectly) with this Ph.D. They include (in order of acquaintance) Jon Brooks, Clare Mackay, Dib Betal, Bill Bimpson, Jocasta Webb, Sue Emery, Enis Cezayirli, Qiyong Gong, Vicky Spark, Matt Howard, Guillaume Calmon and Tom Barrick. Chapter 3 (which describes basic principles of MRI) is, in part, based on work by Bill, Enis and Qiyong, while many of the experimental results in Chapter 8 were obtained by Vicky using the software described in this thesis. I would also like to thank Chris Taylor, who I lived next door to in Philharmonic Court (1993/1994) and shared an office with in the Department of Pure Mathematics (1995/1997). We probably spent too much time playing Stunt Racer 2000!

Finally, I would like to say a special thank you to my family and girlfriend, Maria. They have been inspirational, offering continued advice and encouragement. I could not have done any of this without them.

I gratefully acknowledge financial support from Shell Research for part of my studies at MARIARC.