Inversion for the Attenuated Radon Transform with Constant Attenuation

An exact form of the inversion formula for the attenuated Radon transform with constant attenuation in a convex domain for use in Single-Photon Computerized Tomography is presented. This problem is reduced to solving a generalized Abel integral equation and the conditions for the existence of a unique continuous solution are given. Implementation of this method involves a preprocessing step (modified attenuated Radon transform), a convolution by an attenuation-dependent function and a weighted backprojection. Therefore, only slight modifications of existing reconstruction algorithms are needed. If the attenuation is zero, this formula reduces to Radon´s original inversion formula. When attenuation is not constant, the conditions for a unique continuous solution can be established with a similar approach. Many results found empirically by previous authors are consistent with this theory.