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

Abstract

Since Compton cameras were introduced in the use of single photon emission computed tomography, various types of conical Radon transforms, which integrate the emission distribution over circular cones, have been studied. Most of previous works did not address the attenuation factor, which may lead to significant degradation of image quality. In this paper, we consider the problem of recovering an unknown function from conical projections affected by a known constant attenuation coefficient called an attenuated conical Radon transform.In the case of a fixed opening angle and vertical central axis, new explicit inversion formula is derived. Two-dimensional numerical simulations were performed to demonstrate the efficiency of the suggested algorithm.

Analysis of the weighted conical and v-line Radon transform

“…In this section, we study the setup of the attenuated conical Radon transform, represented by equation (3). Additionally, in the past, an explicit inversion formula was introduced in [29] for specifically cases n is an odd integer. This formula is expressed as follows…”

“…This attenuation leads to a degradation in the accuracy of the reconstructed image. Despite the existence of some studies on this topic (see, e.g., [22,29,32]), the attenuated conical Radon transform has received relatively little attention.…”

“…For simplicity, we set g * (x, y) := e −µy cos ψ V µ f (x, y) and f * (x, y) := e −µy cos ψ f (x, y), then g * (x ν , y ν ) is the v-line Radon transform of f * on the non-attenuated case. To find the inversion formula, we utilize the results presented in [29].…”

“…Theorem 4 [29] For f ∈ C ∞ (D), let g(x ν , y ν ) := V µ f (x ν , y ν ). An exact solution of the inversion problem for the v-line Radon transform is given by the formula…”

Variational regularization of the weighted conical Radon transform

“…In the context of SPECT using the classical Anger cameras where the exponential and the attenuated Radon transform appear this issue is well investigated; see for example [25][26][27][28][29][30][31][32][33][34]. For the conical Radon transform, attenuation is investigated much less and we are only aware of the works [20,35] considering the special case U(r) = e −λr .…”