Preface This volume brings together six articles on the mathematical aspects of to- mography and related inverse problems. They are based on the lectures in the Short Course, The Radon Transform and Applications to Inverse Problems, at the American Mathematical Society meeting in Atlanta, GA, January 3-4, 2005. They covered introductory material, theoretical problems, and practical issues in 3-D to- mography, impedance imaging, local tomography, wavelet methods, regularization and approximate inverse, sampling, and emission tomography. All contributions are written for a general audience, and the authors have included references for further reading. Tomography and inverse problems are active and important fields combining pure and applied mathematics with strong interplay between applications and the diverse mathematical problems that have emerged since the first article in the field appeared almost a century ago. The applied side is best known for medical and scientific applications, in particular, medical imaging, radiotherapy, and industrial non-destructive testing. Doctors use tomography to see the internal structure of the body or to find functional information, such as metabolic processes, noninvasively. Scientists discover defects in objects, the topography of the ocean floor, and geo- logical information using X-rays, geophysical measurements, sonar, or other data. Thus, tomography consists of a broad range of inverse problems. These are called inverse problems because information about an object is obtained from indirect data. X-ray tomography is the most basic modality, and it can be described in the following way: a beam of X-rays is emitted with a known intensity from a source outside the material to be scanned, usually some part of the human body. A detector on the other side of the body picks up the intensity after the ray has traveled along a straight line segment, L, through the body. Some X-rays are lost due to scattering and absorption because of the attenuation effects of the material. Let / be the linear attenuation coefficient of the body. If the X-rays are monochromatic, then the attenuation coefficient is proportional to the density of the object (the proportionality depends on the energy of the photons). Choosing units so the proportionality is 1 we can view / as the density function of the object. Then, a simple derivation (see e.g., [8, (2.1)]) shows that the logarithm of the intensity ratio is proportional to the line integral of the attenuation function, so in appropriate units, J f(x)dx=:Rf(L). (1) In /(source) /(detector) vri
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