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

The incremental backprojection algorithm is a fast backprojection approach based an restructuring the Shepp and Logan algorithm. By exploiting interdependency (position and values) of adjacent pixels, the incremental algorithm requires only O(N) and O(N²) multiplications in contrast to O(N²) and O(N³) multiplications for the Shepp and Logan algorithm in two-dimensional (2-D) and three dimensional (3-D) backprojections, respectively, for each view, where N is the size of the image in each dimension. In addition, it may reduce the number of additions for each pixel computation. The improvement achieved by the incremental algorithm in practice was not, however, as significant as expected. One of the main reasons is 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 backprojecting, 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 backprojection algorithm, the proposed scheme may achieve five-eight and three times speedups for 2-D and 3-D backprojections, respectively