On new {\mathfrak V}-line Radon transforms in \mathbb {R}^{2} and their inversion

Abstract

Radon transforms on piecewise smooth curves in R 2 are rather unfamiliar and have not been so far widely investigated. In this paper we consider three types of Radon transforms defined on a pair of half-lines in the shape a V-letter, with a fixed axis direction. These three Radon transforms arise from recently suggested tomographic procedures. Our main result consists in obtaining their analytic inverse formulas, which may serve as mathematical foundation for new imaging systems in engineering and physics.

Combined Modalities of Compton Scattering Tomography

“…This is why our modeling is more complex than the one shown by Norton. Then, we present the functional examination that was proposed by Truong and Nguyen [10] and that is modeled by a compounded V-line Radon transform.…”

“…With all these considerations, the measured photon flux density at under a scattering angle can be written as (2) with and Equation (2) stands for image formation, but this form does not permit the inversion procedure without correcting the additional physical factors through a correction algorithm. Without this factor, we can use the inversion proposed in [10]. Nevertheless, to avoid singularities, we consider that the object of interest is placed at a minimal distance from the detector.…”

“…Let us define The inversion of the CVRT has been established in [10], and the inversion formula is (6) with putting , and . As this expression can be rewritten as a simple backprojection over corresponding broken lines convolved with a filter, it does not need a computational regularization.…”

“…The crux of the matter is that one can show that from this data set, a three-dimensional object reconstruction is possible. Moreover, a two-dimensional version of this scattered radiation has been shown to be feasible in the last few years, referred to as V-line emission imaging (VEI) [9], [10]. Thus, it is tempting (and logical) to combine Compton scattering tomography with V-line emission imaging as the first bimodal scatter radiation imaging, which is the topic of this work.…”

“…Section II deals with the concept of our Compton scattering tomography bimodality through the transmission one as proposed by Norton in 1994 [8] and the emission one as proposed by Nguyen and Truong in 2011 [10]. Image formation is also introduced.…”