Frank Natterer - Böcker

Since the advent of computerized tomography in radiology, many imaging techniques have been introduced in medicine, science, and technology. This book describes the state of the art of the mathematical theory and numerical analysis of imaging. The authors survey and provide a unified view of imaging techniques, provide the necessary mathematical background and common framework, and give a detailed analysis of the numerical algorithms. This book not only reflects the theoretical progress and the growth of the field in the last 10 years but also serves as an excellent reference. It will provide readers with a superior understanding of the mathematical principles behind imaging and will enable them to write state-of-the-art software as a result.Mathematical Methods in Image Reconstruction provides a very detailed description of two-dimensional algorithms. For three-dimensional algorithms, the authors derive exact and approximate inversion formulas for specific imaging devices and describe their algorithmic implementation (which by and large parallels the two-dimensional algorithms). Integral geometry is surveyed as far as is necessary for imaging purposes; imaging techniques based on or related to integral geometry are briefly described in the section on tomography.Some of the applications covered in the book include computerized tomography, magnetic resonance imaging, emission tomography, electron microscopy, ultrasound transmission tomography, industrial tomography, seismic tomography, impedance tomography, and NIR imaging. The authors provide the necessary mathematical background and common mathematical framework needed to understand the book. Knowledge of tomography literature from the 1980s will be useful to the reader.

The articles collected in this volume are based on lectures given at the IMA Workshop, "Computational Radiology and Imaging: Therapy and Diagnostics", March 17-21, 1997. Introductory articles by the editors have been added. The focus is on inverse problems involving electromagnetic radiation and particle beams, with applications to X-ray tomography, nuclear medicine, near-infrared imaging, microwave imaging, electron microscopy, and radiation therapy planning. Mathematical and computational tools and models which play important roles in this volume include the X-ray transform and other integral transforms, the linear Boltzmann equation and, for near-infrared imaging, its diffusion approximation, iterative methods for large linear and non-linear least-squares problems, iterative methods for linear feasibility problems, and optimization methods. The volume is intended not only for mathematical scientists and engineers working on these and related problems, but also for non-specialists. It contains much introductory expository material, and a large number of references. Many unsolved computational and mathematical problems of substantial practical importance are pointed out.