Introduction to the Mathematics of Medical Imaging

Over the past several decades, advanced mathematics has quietly insinuated itself into many facets of our day-to-day life. Mathematics is at the heart of technologies from cellular telephones and satellite positioning systems to online banking and metal detectors. Arguably no technology has had a more positive and profound effect on our lives than medical imaging, and in no technology is the role of mathematics more pronounced or less appreciated. X-ray tomography, ultrasound, positron emission tomography, and magnetic resonance imaging have fundamentally altered the practice of medicine. At the core of each modality is a mathematical model to interpret the measurements and a numerical algorithm to reconstruct an image. While each modality operates on a different physical principle and probes a different aspect of our anatomy or physiology, there is a large overlap in the mathematics used to model the measurements, design reconstruction algorithms, and analyze the effects of noise. In this text we provide a tool kit, with detailed operating instructions, to work on the sorts of mathematical problems that arise in medical imaging. Our treatment steers a course midway between a complete, rigorous mathematical discussion and a cookbook engineering approach. The target audience for this book is junior or senior math undergraduates with a firm command of multivariable calculus, linear algebra over the real and complex numbers, and the basic facts of mathematical analysis. Some familiarity with basic physics would also be useful. The book is written in the language of mathematics, which, as I have learned, is quite distinct from the language of physics or the language of engineering. Nonetheless, the discussion of every topic begins at an elementary level and the book should, with a little translation, be usable by advanced science and engineering students with some mathematical sophistication. A large part of the mathematical background material is provided in two appendices. X-ray tomography is employed as apedagogical machine,similar in spirit to the elaborate devices used to illustrate the principles of Newtonian mechanics. Thephysical principlesused in x-ray tomography are simple to describe and require little formal background in physics to understand. This is not the case in any of the other modalities listed nor in less developed modalities like infrared imaging or impedance tomography. Themathematical problemsthat arise in x-ray tomography and the tools used to solve them have a great deal in common with those used in the other imaging modalities. This is why our title isIntroduction to the Mathematics of Medical Imaginginstead ofIntroduction to the Mathematics of X-Ray Tomography.A student with a thorough understanding of the material in this book should be mathematically prepared for further investigations in most subfields of medical imaging. Very good treatments of the physical principles underlying the other modalities can be found inRadiological Imagingby Harrison H. Barrett and William Swindell, 4,Principles of Computerized Tomographic Imagingby Avinash C. Kak and Malcolm Slaney, 56,Foundations of Medical Imagingby Cho, Jones, Singh, 14,Image Reconstruction from Projectionsby Gabor T. Herman, 35, andMagnetic Resonance Imagingby E. Mark Haacke, Robert W Brown, Michael R. Thompson, Ramesh Venkatesan, 33. Indeed these books were invaluable sources as I learned the subject myself. My treatment of many topics owes a great deal to these books as well as to the papers of Larry Shepp and Peter Joseph and their collaborators. More advanced treatments of the mathematics and algorithms introduced here can be found inThe Mathematics of Computerized Tomographyby Frank Natterbr, 72, andMathematical Methods in Image Reconstructionby Frank Natterer and Frank Wubbelling, 73. The order and presentation of topics is somewhat nEpstein, Charles L. is the author of 'Introduction to the Mathematics of Medical Imaging' with ISBN 9780130675484 and ISBN 0130675482.