New properties of the V-line Radon transform and their imaging applications

Abstract

This paper reports on new aspects of the so-called V-line Radon transforms complementing those reported in an earlier work. These new properties are nicely uncovered and described with Cartesian coordinates. In particular, we show that the V-line Radon transform belongs to the class of Radon transforms on curves in the plane which can be mapped onto the standard Radon transform on straight lines and thereby are fully characterizable and invertible. Next, we show that the effect of geometric inversion on the V-line Radon transform is to produce a new Radon transform on a pair of supplementary circular arcs, which provides a new access to image reconstruction in the so-called Norton's modality of Compton scatter tomography, a front runner in the race for alternatives to current emission imaging.

Inversion of weighted divergent beam and cone transforms

“…For the discretization of the sphere, we used a uniform mesh for both the azimuthal and the polar angles. Figure 8 shows the three cross sections of the spherical phantom and of its reconstructions from the cone data obtained via (33). The comparison of the phantom and the reconstruction given in Figure 8 in terms of their coordinate axis profiles is provided in Figure 9.…”

“…The 3D ball phantom with radius 0.5, center (0,0,0.25) and unit density (left), and 90 × 90 image reconstructed via(33) from weighted cone data simulated using 1800 counts for vertices u on the unit sphere, 1800 counts for directions β and 200 counts for opening angles ψ (right). The cross sections by the planes x = 0, y = 0 and z = 0.25 are shown.…”

The Development of Radon Transforms Associated to Compton Scatter Imaging Concepts

“…Further properties of the V-line Radon transform have been uncovered recently in [33]. It is well known that under geometric inversion of center O (origin of coordinates) and modulus q…”

“…Under a geometric inversion of center S and arbitrary modulus q, this circular arc is mapped onto a half-line in the upper half-plane starting at site D on the line SD such that SD.SD = q 2 . Thus the "Norton Radon circular arc transform" is nicely mapped onto the Radon transform on half-lines of the upper-plane, which was shown to be invertible [33].…”

Three-Dimensional Image Reconstruction from Compton Camera Data

“…One thus could restrict the set of cones, in order to get a non-over-determined problem (e.g. [3,4,6,8,16,17,[19][20][21][22]31,37,42, and references therein]). In most of these considerations only a subset of cones with vertex at a given scattering detector is used.…”