Ultra - Fast Convolution Approximations for Computerized Tomography

The amount of computation required to convolve projection data with a filter array may be reduced by implementing the required multiplications with reduced precision or by approximating the filter with a function which is piecewise constant over intervals several times longer than the projection sampling increment. We investigate an extreme form of the above approximations, for which multiplication by any filter element (except the central one) requires only a simple binary shift. Using this approximation, a projection of M samples may be filtered in an extremely straightforward manner using only M full-precision multiplications, representing a significant advantage over convolution implementations using Fourier or number-theoretic transforms. Simulations are presented which show that in most cases only an insignificant amount of error in the reconstructed image results from the use of this form of convolution approximation.