GENERALIZED RADON TRANSFORMS & IMAGING BY SCATTER PARTICLES

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.

Contents:

• Preface
• About the Author
• Particle Transport and Scattering Tomography:
  • Radiation Transport Equation
  • Scattered Particle Tomography
• Generalized Radon Transforms 'With a Vertex':
  • V-line and Conical Radon Transforms in Slab Geometry
  • V-line Transforms in Curvilinear Geometry
  • Star Transform
  • Transforms on Vector Fields
• Appendix A: Mathematical Tools
• Bibliography
• Index

Readership: Advanced undergraduate and graduate students, mathematicians, engineers and physicists interested in mathematical models of image reconstruction using scattered particles.

'This well-written book provides important information on a timely topic, Compton tomography, in a very readable form. The author focuses on models for Compton tomography that are generalized Radon transforms on broken rays (V-s or stars) in the plane or cones in space. The reader will come away with a solid understanding of the applications and physical background, along with the mathematical models and the properties of the associated generalized Radon transforms.' - Todd QuintoRobinson Professor of MathematicsTufts University, USA

Key Features:

• A self-contained book covering the state of the art in the subject is timely and can be of great value for both researchers and practitioners
• Substantial amount of work has been done in these fields, the results are scattered across journal articles in various disciplines, some of which are hard to read for non-experts. This book bridges all these parts
• The technical level and exposition style of the material are appropriate for advanced undergraduate and graduate students. Thus, parts of the book can be used in special topics courses on tomography, as well as in capstone projects or REU programs on that subject