At the heart of every medical imaging technology is a sophisticated mathematical model of the measurement process and an algorithm to reconstruct an image from the measured data. This book provides a firm foundation in the mathematical tools used to model the measurements and derive the reconstruction algorithms used in most of these modalities. The text uses X-ray computed tomography (X-ray CT) as a pedagogical machine' to illustrate important ideas and its extensive discussion of background material makes the more advanced mathematical topics accessible to people with a less formal mathematical education. This new edition contains a chapter on magnetic resonance imaging (MRI), a revised section on the relationship between the continuum and discrete Fourier transforms, an improved description of the gridding method, and new sections on both Grangreat's formula and noise analysis in MR-imaging. Mathematical concepts are illuminated with over 200 illustrations and numerous exercises.
Contents
Preface to the second edition; Preface; How to use this nook; Notational conventions; 1. Measurements and modeling; 2. Linear models and linear equations; 3. A basic model for tomography; 4. Introduction to the Fourier transform; 5. Convolution; 6. The radon transform; 7. Introduction to Fourier series; 8. Sampling; 9. Filters; 10. Implementing shift invariant filters; 11. Reconstruction in X-ray tomography; 12. Imaging artifacts in X-ray tomography; 13. Algebraic reconstruction techniques; 14. Magnetic resonance imaging; 15. Probability and random variables; 16. Applications of probability; 17. Random processes; A. Background material; B. Basic analysis; Bibliography; Index.