Discrete Tomography: Foundations, Algorithms, and Applications (Applied and Numerical Harmonic Analysis)

Goals of the Book Overthelast thirty yearsthere has been arevolutionindiagnostic radiology as a result oftheemergenceofcomputerized tomography (CT), which is the process of obtaining the density distribution within the human body from multiple x-ray projections. Since an enormous variety of possible density values may occur in the body, a large number of projections are necessary to ensure the accurate reconstruction oftheir distribution. There are other situations in which we desire to reconstruct an object from its projections, but in which we know that the object to be recon­ structed has only a small number of possible values. For example, a large fraction of objects scanned in industrial CT (for the purpose of nonde­ structive testing or reverse engineering) are made of a single material and so the ideal reconstruction should contain only two values: zero for air and the value associated with the material composing the object. Similar as­ sumptions may even be made for some specific medical applications; for example, in angiography ofthe heart chambers the value is either zero (in­ dicating the absence of dye) or the value associated with the dye in the chamber. Another example arises in the electron microscopy of biological macromolecules, where we may assume that the object to be reconstructed is composed of ice, protein, and RNA. One can also apply electron mi­ croscopy to determine the presenceor absence ofatoms in crystallinestruc­ tures, which is again a two-valued situation.

The visualization, construction, and reconstruction of multidimensional images are of intense interest in science and engineering today, and discrete tomography―which deals with the special case in which the object to be reconstructed has a small number of possible values―offers some significant new analytical and computational tools.

Discrete Tomography: Foundations, Algorithms, and Applications provides a critical survey of new methods, algorithms, and select applications that are the foundations of multidimensional image construction and reconstruction. The survey chapters, written by leading international authorities, are self-contained adn present the latest research and results in the field. The book covers three main areas: important theoretical results, available algorithms to utilize for reconstruction, and key applications where new results are indicative of greater utility. Following a thorough historical overview of the field, the book provides a journey through the various mathematical and computational problems of discrete tomography. This is followed by a section on numerous algorithmic techniques that can be used to achieve real reconstructions from image projections.

Topics and Features:

* historical overview and summary chapter

* uniqueness and complexity in discrete tomography

* probabilistic modeling  of discrete images

* binary tomography using Gibb priors

* discrete tomography on the 3-D torus and crystals

* binary steering

* 3-D tomographic reconstruction from sparse radiographic data

* symbolic projections

The book is an essential resource for the latest developments and tools in discrete tomography. Professionals, researchers, and practitioners in mathematics, computer imaging, scientific visualization, computer engineering, and multidimensional image processing will find the book an authoritative guide and reference to current research, methods, and applications.