14 research outputs found
On the V-Line Radon Transform and Its Imaging Applications
Radon transforms defined on smooth curves are well known and extensively studied in the literature. In this paper, we consider a Radon transform defined on a discontinuous curve formed by a pair of half-lines forming the vertical letter V. If the classical two-dimensional Radon transform has served as a work horse for tomographic transmission and/or emission imaging, we show that this V-line Radon transform is the backbone of scattered radiation imaging in two dimensions. We establish its analytic inverse formula as well as a corresponding filtered back projection reconstruction procedure. These theoretical results allow the reconstruction of two-dimensional images from Compton scattered radiation collected on a one-dimensional collimated camera. We illustrate the working principles of this imaging modality by presenting numerical simulation results
Broken ray transform on a Riemann surface with a convex obstacle
We consider the broken ray transform on Riemann surfaces in the presence of
an obstacle, following earlier work of Mukhometov. If the surface has
nonpositive curvature and the obstacle is strictly convex, we show that a
function is determined by its integrals over broken geodesic rays that reflect
on the boundary of the obstacle. Our proof is based on a Pestov identity with
boundary terms, and it involves Jacobi fields on broken rays. We also discuss
applications of the broken ray transform.Comment: 24 pages, 2 figure
A reflection approach to the broken ray transform
We reduce the broken ray transform on some Riemannian manifolds (with
corners) to the geodesic ray transform on another manifold, which is obtained
from the original one by reflection. We give examples of this idea and present
injectivity results for the broken ray transform using corresponding earlier
results for the geodesic ray transform. Examples of manifolds where the broken
ray transform is injective include Euclidean cones and parts of the spheres
. In addition, we introduce the periodic broken ray transform and use the
reflection argument to produce examples of manifolds where it is injective. We
also give counterexamples to both periodic and nonperiodic cases. The broken
ray transform arises in Calder\'on's problem with partial data, and we give
implications of our results for this application.Comment: 29 pages, 6 figures; final versio
Scattered Radiation Emission Imaging: Principles and Applications
Imaging processes built on the Compton scattering effect have been under continuing investigation since it was first suggested in the 50s. However, despite many innovative contributions, there are still formidable theoretical and technical challenges to overcome. In this paper, we review the state-of-the-art principles of the so-called scattered radiation emission imaging. Basically, it consists of using the cleverly collected scattered radiation from a radiating object to reconstruct its inner structure. Image formation is based on the mathematical concept of compounded conical projection. It entails a Radon transform defined on circular cone surfaces in order to express the scattered radiation flux density on a detecting pixel. We discuss in particular invertible cases of such conical Radon transforms which form a mathematical basis for image reconstruction methods. Numerical simulations performed in two and three space dimensions speak in favor of the viability of this imaging principle and its potential applications in various fields