<title>Orthogonal series approach to reconstructing 2D PET images using data obtained from detector tubes of arbitrary width</title>

Abstract

This paper describes a method for reconstructing two dimensional profiles from positron emission tomography (PET) emission data. Unlike the usual methods which assume the detectors are small, the method is based on an accurate system response model which is valid for PET detectors of arbitrary size. The emission profile is represented as an orthogonal series of basis functions (on the circular region inside the scanner ring) which are tensor products of Bessel functions in the radial direction, and classical harmonics in the angular direction. By applying a simple linear transformation to the emission sinogram, it can be viewed as data corresponding to detector arcs, and represented by basis functions involving Chebyshev polynomials and classical harmonics. The coefficients in the orthogonal series for the emission intensity are obtained by solving a block diagonal linear system in which the data vector consists of the coefficients of the orthogonal series of the transformed sinogram. The coefficient matrix of this system is obtained from an orthogonal series representation of the probability that an emission is detected in one of the detector arcs. The mathematical description of this probability and the reconstruction method was developed in a recent paper by the authors. In this paper we discuss details of the numerical implementation and present numerical results obtained from applying this method to simulated data. The results indicate that our method produces reconstructions which are comparable in quality to the maximum likelihood expectation maximization (MLEM) method wiht a speed which is similar that of the filtered back projection (FBP) method.