Improved numerical integration for analytical photon distribution calculation in SPECT

Computational methods are a fundamental component of many image reconstruction and data processing techniques used in medical imaging. Given several possible approaches for accomplishing the same task, an understanding of the underlying mathematics for each method can allow one to make the best choice for a particular situation, resulting in a dramatic improvement in the efficiency and/or accuracy of a particular computation. In this study we consider the choice of numerical integration method used in an analytical photon distribution (APD) calculation for single photon emission computed tomography (SPECT). This calculation is extremely time-consuming in the original implementation of the method, taking several weeks to complete on a single processor. Fast computation of propagation and detection probabilities is especially challenging for scattered photons, due to the large number of possible photon paths and the high dimension of the integration problem. The choice of integration method, however, can significantly improve the speed of the calculation. Replacing Romberg integration, which was used in the original implementation of APD, with a Gaussian quadrature method allowed us to speed up the numerical integration steps in the calculation by up to 40 times. The photon distributions generated using the improved quadrature method show excellent agreement with those calculated using the original integration scheme.