X-Ray Reconstruction by Finite Field Transforms

Abstract

The reconstruction of two-dimensional density distributions, obtained by scanning a source of radiation through various angles, was suggested originally by Bracewell and Riddle. Recently Ramachandran and Lakshminarayanan proposed that a direct convolution technique might be used to reconstruct three-dimensional objects from projections obtained from x-ray scanning. They showed that the reconstruction of the sections of an object in the spatial domain was the average of the convolutions of an appropriate filter function with these projections over all possible projection angles. The best filter function presently known for this method was given by Shepp and Logan. The new tomograph machines are now using substantially larger sets of data than considered originally by Shepp and Logan. As a consequence, the fast Fourier traisform (FFT) method of convolution now competes favorably inspeed with the direct convolution method, described by Shepp and Logan. In this paper a fast transform method is used to perform the convolution of the Shepp and Logan filter with the projection data. To perform these convolutions at optimal computational speed and accuracy, a fast Fourier transform (FFT) algorithm is developed over a finite or Galois field GF(q) of integers instead of the usual field of complex numbers. GF(q) is the field of integers modulo q where q is a prime of the form k. 2n+ 1. Important features of this computational technique to tomography are both a high computational speed and an absence of round-off errors.