Generalized Radon Transforms And Imaging By Scattered Particles: Broken Rays, Cones, And Stars In Tomography

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, for example, X-ray computed tomography, positron emission tomography, single photon emission computed tomography, ultrasound reflection tomography, photo-acoustic tomography, among others. The GRTs integrating along smooth curves and surfaces (lines, planes, circles, spheres, etc) have been studied at great length 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 (for example, 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 will cover, in a detailed manner, the relevant imaging modalities, their mathematical models, and the related generalized Radon transforms. The discussion of the latter will comprise a thorough exploration of their known mathematical properties, including injectivity, inversion, range description and microlocal analysis, as well as numerical examples, and description of open problems. 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 other fields of mathematics, including functional analysis, PDEs and algebraic geometry.