Solution to the long object problem by convolutions with spatially variant 1-D Hilbert transforms in spiral cone-beam computed tomography

In the long object problem it is intended to reconstruct exactly a region-of-interest (ROI) of an object from spiral cone-beam data which covers the ROI and its immediate vicinity. One possible solution is the previously published local ROI technique. The Radon derivative data are computed for different local ROI's which are adapted to the scan path such that the contributing cone-beams are not contaminated by object information outside the local ROI. The common intersection of all local ROI's is reconstructed. The method was implemented in the filtered backprojection based 4-step algorithm. It mainly consists of explicit calculations of line integrals and their backprojection on the detector. Inside the ROI the same good image quality is achieved as in the reference case where the complete object is sampled. However, the 4-step algorithm suffers from long computation time. It is found that the demanding filtering operations are equivalent to a number of spatially variant 1-D Hilbert transforms. Thus filtering can be performed by 1-D convolutions. To optimize the convolution kernels with respect to numerical stability, the empirical point spread function corresponding to the filtering of the 4-step algorithm is analyzed. Modifications of the theoretical filter kernels are derived and discussed.