A mathematical model incorporating the effects of detector width in 2D PET

Abstract

For decades, the Radon transform has been used as an approximate model for two-dimensional (2D) positron emission tomography (PET). Since this model assumes that detector tubes are represented by lines (hence have no area), PET reconstruction algorithms need to be modified to account for the nonzero width of detectors. To date, these modifications have been obtained by computational methods, so fail to exhibit any inherent mathematical structure of the PET transform which takes emission intensity to detector tube means. This paper contains a precise mathematical representation of this PET transform and exploits this representation to propose a new method for reconstructing PET images. This representation is achieved by expressing the probability that an emission at a point is detected in a detector tube, in terms of the Green function and Poisson kernel for Laplace's equation on the unit disc. This new PET transform involves four weighted line integrals of the emission intensity function, instead of the single unweighted line integral defining the 2D Radon transform. Despite the complexity of this model, a reconstruction method is obtained by using classical orthogonal series representations of the emission intensity and detection means in terms of circular harmonics, Bessel functions and Chebyshev polynomials.