Inverse problems arise in practical situations such as medical imaging, geophysical exploration, and nondestructive evaluation where measurements made on the exterior of a body are used to determine properties of the inaccessible interior. In this book leading experts in the theoretical and applied aspects of inverse problems have written extended surveys on some of the main topics of the inverse problems semester held at MSRI during the Fall of 2001. We describe here briefly the chapters of the book.

The chapter by Faridani is an introduction to computed tomography (CT), which is probably the inverse problem best known to the general public. In this imaging method the attenuation in intensity of an X-ray beam is measured, and the information from many X-rays from different sources is assembled and analyzed on a computer. Mathematically it is a problem of recovering a function from the set of its line integrals (or the set of its plane integrals). Radon found in the early part of the twentieth century a formula to recover a function from this information. The application to diagnostic radiology did not happen until the late 1960s with the aid of the increasing calculating power of the computer. In 1970 the first computer tomograph that could be used in clinical work was developed by G. N. Hounsfield. He and Allan M. Cormack, who independently proposed some of the algorithms, were jointly awarded the 1979 Nobel prize in medicine. In practice only integrals from finitely many lines can be measured, and the distribution of these lines is sometimes restricted.