Mathematics Of Medical Imaging.pdf

The Mathematics of Medical
Imaging
Charles L. Epstein
November 6, 2001
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Charles L. Epstein
Department of Mathematics
University of Pennsylvania
Philadelphia, PA 19104
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c Charles L. Epstein, 2001. All rights reserved.
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This book is dedicated to my wife, Jane
and our children, Leo and Sylvia. They
make it all worthwhile.
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Preface
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 tele-
phones 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 advances in
medical imaging and in no technology is the role of mathematics more pronounced. X-ray
tomography, ultrasound, positron emission tomography and ic 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 recon-
struct 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 which arise in medical imaging. Our treatment steers
a course midway between plete, rigorous, abstract mathematical discussion and a
cookbook engineering approach.
The target audience for this book is junior or senior math undergraduates with a firm
command of calculus and linear algebra. The book is written in the language of mathemat-
ics, which, as I have learned is quite distinct from the language of physics or the language
of engineering. heless, the discussion of every topic begins at an el