The Radon Transform over Cones with Vertices on the Sphere and Orthogonal Axes

Abstract

Recovering a function from its integrals over circular cones recently gained significance because of its relevance to novel medical imaging technologies such emission tomography using Compton cameras. In this paper we investigate the case where the vertices of the cones of integration are restricted to a sphere in n-dimensional space and symmetry axes are orthogonal to the sphere. We show invertibility of the considered transform and develop an inversion method based on series expansion and reduction to a system of one-dimensional integral equations of generalized Abel type. Because the arising kernels do not satisfy standard assumptions, we also develop a uniqueness result for generalized Abel equations where the kernel has zeros on the diagonal. Finally, we demonstrate how to numerically implement our inversion method and present numerical results.

Variational regularization of the weighted conical Radon transform

“…Let C denote the Gegenbauer polynomials normalized in such a way that C @IA a I. As in [19], we derive different relations between f`; k and @gfA`; k in the form of Abelian integral equations. Theorem 2.4 (Generalized Abel equation for f`; k ).…”

“…In this section, we show uniqueness of recovering the function by its conical radon transform and thus the injectivity of g by showing solution uniqueness of the Abelian integral equations in Theorem 2.7. Since the kernel function of the Abelian integral equation (2.10) has zeros on the diagonal, the proof of the uniqueness relies on the uniqueness result derived in [19], which we briefly state at this point: For a; b P R with a < b we set ¡@a; bA Xa f@t; sA P R P j a s t bg.…”

“…fs P a; bA j F @s; sA a Hg is finite and consists of simple roots. (F3) For every s P N F , the gradient @ I ; P A Xa rF@s; sA satisfies See[19, Theorem 3.4].Theorem 2.7 (Uniqueness of recovering f`; k ). Suppose the function U is real-analytic and satisfies nCI P C p s U H @ p sA U@ p sA > H for all s P H; P. For any f P D X and any @`; kA P I@nA, the spherical harmonic coefficient f`; k of f can be recovered as the unique solution of…”

The V-line transform with some generalizations and cone differentiation

“…Many researchers have obtained interesting results on the CRT and the VLT for such setups (e.g. see [15,16,18,23,24,27,28,33,34,35,36,37,41,43,45,46,47]). A nice survey of this field was recently published in [44].…”

Inversion of the attenuated conical Radon transform with a fixed opening angle

“…During the last decade, the interest towards such transforms was triggered by the connection between the conical Radon transform and the mathematical models of many novel imaging modalities. [1][2][3][4][5][6][7][8][9][10][11][12][13][14] In all the above works, the attenuation phenomena was neglected. However, in many medical imaging techniques, ignoring the effect of the attenuation of photon can significantly degrade the quality of the reconstruction image.…”