New classes of helical weighting algorithms with applications to fast CT reconstruction

The focus of this paper is on CT helical weighting algorithms using one source rotation, or 2 pi, worth of projection data. Currently known 2 pi helical weighting algorithms include a fan-angle dependency, and do not lend themselves to fast reconstruction, for two reasons. First, it can be shown that the weight distributions present a line of discontinuity across the sinogram (projection space), which defines two separate sinogram regions. Second, the expressions for the weighting functions differ for those two regions. Accordingly, reconstruction of P different image planes (all using a given projection) requires P weightings and filterings of that projection. In this paper, it is shown that, by first generalizing the concept of the interpolation/extrapolation function used in the weighting, to the concept of distance function, and second by selecting particular classes of such distance functions, the discontinuity across the sinogram can be eliminated. By imposing specific sufficient conditions on such distance functions, single analytical expressions across the entire 2 pi sinogram are obtained. Decomposition of these particular "single" distance functions leads to two-filtering reconstruction algorithms, for which a given projection needs to be filtered only two times for an arbitrary number P of reconstruction planes. Finally, another generalization of the concept of helical weighting leads to one-filtering weight functions that depend only on the sum of the projection--and fan angles. Accordingly, after rebinning the fan-beam projections to parallel projections, the corresponding 2 pi helical weighting algorithms do not include a dependency over the ray parameter. Equivalently, for these algorithms, weighting commutes with filtering, and reconstruction of an arbitrary number P of image planes requires only one filtering per projection. These algorithms are shown to be consistent with the hypothesis of a linear z variation of the projections.