Reconstruction from projections has revolutionized radiology and has now become one of the most important tools of medical diagnosis The E. M. I. Scanner is one example. In this text, some fundamental theoretical and practical questions are resolved. Despite recent research activity in the area, the crucial subject of the uniqueness of the reconstruction and the effect of noise in the data posed some unsettled fundamental questions. In particular, Kennan Smith proved that if we describe an object by a C^inf_o function, i.e., infinitely differentiable with compact support, then there are other objects with the same shape, i.e., support, which can differ almost arbitrarily and still have the same projections in finitely many directions. On the other hand, he proved that objects in finite dimensional function spaces are uniquely determined by a single projection for almost all angles, i.e., except on a set of measure zero. Along these lines, Herman and Rowland in "Three Methods for reconstructing objects from x-rays: a comparative study" (1973) showed that reconstructions obtained from the commonly used algorithms can grossly misrepresent the object and that the algorithm which produced the best reconstruction when using noiseless data gave unsatisfactory results with noisy data. Equally important are reports in Science, and personal communications by radiologists indicating that in medical practice failure rates of reconstruction vary from four to twenty percent. within this work, the mathematical dilemma posed by Kennan Smith's result is discussed and clarified.

The first part of this new volume in the Inverse and Ill-Posed Problems Series studies uniqeness questions for recovering the shapes of the convex and more complicated bodies from shapes of their projections onto planes of low dimension. Some stability estimates of the solutions to these inverse problems are given. The second part deals with inverse problems with projection data directly connected to tomography, in partcular to apparent contours of smooth surfaces, which have practical interpretations such as thin cracks in continuous media which are studied in industrial defectoscopy, caustic surfaces which are studies in wave optics, etc. New results on reconstruction of smooth surfaces from observations of the wave fronts generated by these surfaces are obtained.

The Inverse and Ill-Posed Problems Series is a series of monographs publishing postgraduate level information on inverse and ill-posed problems for an international readership of professional scientists and researchers. The series aims to publish works which involve both theory and applications in, e.g., physics, medicine, geophysics, acoustics, electrodynamics, tomography, and ecology.

This book constitutes the refereed proceedings of the 12th International Conference on Discrete Geometry for Computer Imagery, DGCI 2005, held in Poitiers, France in April 2005. The 36 revised full papers presented together with an invited paper were carefully reviewed and selected from 53 submissions. The papers are organized in topical sections on applications, discrete hierarchical geometry, discrete tomography, discrete topology, object properties, reconstruction and recognition, uncertain geometry, and visualization.

Applied sciences in the 20th century have developed and used unitary transforms for concentrating energy. Now, the challenge lies in the expression and use of redundancy to build redundant systems. The Mojette transform is a very simple transform using only additions but with strong properties that break this challenge. The first part of the book gives the basics of the Mojette transform both mathematically and the corresponding optimal algorithms. The second part exemplifies its use through different fields: image representation, watermarking, medical imaging, distributed storage, information and cryptography. This book about a discrete exact Radon transform explains how to usefully produce and cope with redundancy for solving 21st century problems

This book constitutes the refereed proceedings of the 13th International Conference on Discrete Geometry for Computer Imagery, DGCI 2006, held in Szeged, Hungary in October 2006. The 28 revised full papers and 27 revised poster papers presented together with two invited papers were carefully reviewed and selected from 99 submissions.

This book constitutes the refereed proceedings of the 19th IAPR International Conference on Discrete Geometry for Computer Imagery, DGCI 2016, held in Nantes, France, in April 2016. The 32 revised full papers presented together with 2 invited talks were carefully selected from 51 submissions. The papers are organized in topical sections on combinatorial tools; discretization; discrete tomography; discrete and combinatorial topology; shape descriptors; models for discrete geometry; circle drawing; morphological analysis; geometric transforms; and discrete shape representation, recognition and analysis.