Reconstruction From Integral Data

Reconstruction of a function from data of integrals is used for problems arising in diagnostics, including x-ray, positron radiography, ultrasound, scattering, sonar, seismic, impedance, wave tomography, crystallography, photo-thermo-acoustics, photoelastics, and strain tomography. Reconstruction from Integral Data presents both long-standing and recent mathematical results from this field in a uniform way. The book focuses on exact analytic formulas for reconstructing a function or a vector field from data of integrals over lines, rays, circles, arcs, parabolas, hyperbolas, planes, hyperplanes, spheres, and paraboloids. It also addresses range characterizations. Coverage is motivated by both applications and pure mathematics. The book first presents known facts on the classical and attenuated Radon transform. It then deals with reconstructions from data of ray (circle) integrals. The author goes on to cover reconstructions in classical and new geometries. The final chapter collects necessary definitions and elementary facts from geometry and analysis that are not always included in textbooks.

This book provides easy-to-understand explanations to systematically and comprehensively describe the X-ray CT technologies, techniques, and skills used for industrial and scientific purposes. Included are many references along with photographs, figures, and equations prepared by the author. These features all facilitate the reader's gaining a deeper understanding of the topics being discussed. The book presents expertise not only on fundamentals but also about hardware, software, and analytical methods for the benefit of technical users. The book targets engineers, researchers, and students who are involved in research, development, design, and quality assurance in industry and academia.

A generalized Radon transform (GRT) maps a function to its weighted integrals along a family of curves or surfaces. Such operators appear in mathematical models of various imaging modalities. The GRTs integrating along smooth curves and surfaces (lines, planes, circles, spheres, amongst others) have been studied at great lengths for decades, but relatively little attention has been paid to transforms integrating along non-smooth trajectories. Recently, an interesting new class of GRTs emerged at the forefront of research in integral geometry. The two common features of these transforms are the presence of a 'vertex' in their paths of integration (broken rays, cones, and stars) and their relation to imaging techniques based on physics of scattered particles (Compton camera imaging, single scattering tomography, etc).This book covers the relevant imaging modalities, their mathematical models, and the related GRTs. The discussion of the latter comprises a thorough exploration of their known mathematical properties, including injectivity, inversion, range description and microlocal analysis. The mathematical background required for reading most of the book is at the level of an advanced undergraduate student, which should make its content attractive for a large audience of specialists interested in imaging. Mathematicians may appreciate certain parts of the theory that are particularly elegant with connections to functional analysis, PDEs and algebraic geometry.

In 1917, Johann Radon published his fundamental work, where he introduced what is now called the Radon transform. Including important contributions by several experts, this book reports on ground-breaking developments related to the Radon transform throughout these years, and also discusses novel mathematical research topics and applications for the next century.

This book covers facts and methods for the reconstruction of a function in a real affine or projective space from data of integrals, particularly over lines, planes, and spheres. Recent results stress explicit analytic methods. Coverage includes the relations between algebraic integral geometry and partial differential equations. The first half of the book includes the ray, the spherical mean transforms in the plane or in 3-space, and inversion from incomplete data.

The continuing growth of computed tomography (CT) and other imaging techniques motivated us to bring together a comprehensive review of the state of the art in diagnostic imaging. Twelve years after the first appearance of x-ray CT, computerized diagnostic imaging has grown so rapidly in sophistication that it is difficult to follow current developments in this diversified field. In this book, we have attempted to cover the basic developments in several areas. The first part includes some of the fundamental aspects of computerized diagnostic imaging such as algorithms and detectors. Specific applications in emission tomography, digital radiography, ultrasound and nuclear magnetic resonance imaging are dealt with in the secondpart. The contributed papers are by experts currently in the field, whom we feel would certainly enlighten the subject matter and possibly suggest directions for future development. We would like to express our sincere thanks to those who have contributed to this volume. We are sure that their original papers will be beneficial for readers and will also remain as an important reference for researchers in the years to come. We would also like to thank Betty Trent for her expert and patient typing of the entire book. Finally, special thanks are due to Mrs. Ingeborg Mayer of Springer-Verlag for her encouragements, support and patience throughout the preparation of this book.

Driven by the requirements of a large number of practical and commercially - portant applications, the last decade has witnessed considerable advances in p- tern recognition. Better understanding of the design issues and new paradigms, such as the Support Vector Machine, have contributed to the development of - proved methods of pattern classi cation. However, while any performance gains are welcome, and often extremely signi cant from the practical point of view, it is increasingly more challenging to reach the point of perfection as de ned by the theoretical optimality of decision making in a given decision framework. The asymptoticity of gains that can be made for a single classi er is a re?- tion of the fact that any particular design, regardless of how good it is, simply provides just one estimate of the optimal decision rule. This observation has motivated the recent interest in Multiple Classi er Systems , which aim to make use of several designs jointly to obtain a better estimate of the optimal decision boundary and thus improve the system performance. This volume contains the proceedings of the international workshop on Multiple Classi er Systems held at Robinson College, Cambridge, United Kingdom (July 2{4, 2001), which was organized to provide a forum for researchers in this subject area to exchange views and report their latest results.

PET and SPECT are two of today’s most important medical-imaging methods, providing images that reveal subtle information about physiological processes in humans and animals. Emission Tomography: The Fundamentals of PET and SPECT explains the physics and engineering principles of these important functional-imaging methods. The technology of emission tomography is covered in detail, including historical origins, scientific and mathematical foundations, imaging systems and their components, image reconstruction and analysis, simulation techniques, and clinical and laboratory applications. The book describes the state of the art of emission tomography, including all facets of conventional SPECT and PET, as well as contemporary topics such as iterative image reconstruction, small-animal imaging, and PET/CT systems. This book is intended as a textbook and reference resource for graduate students, researchers, medical physicists, biomedical engineers, and professional engineers and physicists in the medical-imaging industry. Thorough tutorials of fundamental and advanced topics are presented by dozens of the leading researchers in PET and SPECT. SPECT has long been a mainstay of clinical imaging, and PET is now one of the world’s fastest growing medical imaging techniques, owing to its dramatic contributions to cancer imaging and other applications. Emission Tomography: The Fundamentals of PET and SPECT is an essential resource for understanding the technology of SPECT and PET, the most widely used forms of molecular imaging. *Contains thorough tutorial treatments, coupled with coverage of advanced topics *Three of the four holders of the prestigious Institute of Electrical and Electronics Engineers Medical Imaging Scientist Award are chapter contributors *Include color artwork