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A generalized |ω|-filter for 3-D reconstruction

The 3-D reconstruction of a density function is based on a direct convolution algorithm developed first by Ramachandran and Lakshiminarayanan. Their method adopts a particular choice of weighting function or filter which is called here an |ω|-filter. In some cases this choice of filter had an undesirable oscillatory response. To remedy this problem Shepp and Logan found a weighting function which produced a better reconstruction of a head section. The filter functions of Ramachandran and Lakshminarayanan and Shepp and Logan are only two of many choices for an |ω|-filter. Shepp and Logan's filter was the best for the early tomographic machines. Their filter function provided both a damped response to the cut-off frequency and a low sensitivity to noise. For the new tomographic machines, however, it is desirable to find filters that are not sensitive to counting noise, sample size and sample spacing as the previous filters. Here a study and generalization is made of the previous |ω|-filters. It extends the important filters of Ramachandran and Lakshiminarayanan, and Shepp and Logan to a class of generalized |ω|-filters. A generalized |ω|-filter can be chosen to have both good accuracy and a flexibility to cope with noise. A detailed comparison is made among the different possible filter shapes with respect to their responses to simulated data and noise. Finally in this paper it is demonstrated that a substantial reduction in the x-ray exposure time can be accomplished by choosing the appropriate generalized |ω|-filter.