A fast implementation of the incremental backprojection algorithms for parallel beam geometries

The incremental backprojection algorithm is a fast backprojection approach requiring only O(N) and O(N²) multiplications in contrast to O(N²) and O(N³) multiplications for the Shepp and Logan´s backprojection algorithm in 2D and 3D backprojections, respectively, for each view. However, the improvement achieved by the incremental algorithm in practice was not so significant as expected in theory due to inevitably visiting pixels outside the beam in the searching flow scheme originally developed for the Incremental algorithm. To optimize implementation of the incremental algorithm, an efficient scheme, namely, coded searching flow scheme, is proposed in this paper to minimize the overhead caused by searching for all pixels in a beam. The key idea of this scheme is to encode the searching flow for all pixels inside each beam. While backprojection, all pixels may be visited without any overhead due to using the coded searching flow as the a priori information. The proposed coded searching flow scheme has been implemented on a Sun Sparc 10 and a Sun Sparc 20 workstations. The implementation results show that the proposed scheme is 1.45-2.0 times faster than the original searching flow scheme for most cases tested. Compared to the conventional Shepp and Logan´s backprojection algorithm, the proposed scheme may achieve 5-8 and 3 times speed-ups for 2D and 3D backprojection, respectively