Shear-Based Fast Hierarchical Backprojection for Parallel-Beam Tomography

We introduce a family of fast algorithms for 2-D parallel-beam tomographic backprojection. They aggregate the projections in a hierarchical structure involving the shearing and addition of sparsely sampled images. The algorithms achieve a computational cost of O(N
²
logP), when backprojecting an NtimesN pixel image from P projections. The algorithms provide a systematic means, guided by a Fourier-domain interpretation, to adjust and optimize the tradeoff between computational cost and accuracy. In an example with N=512 and P=1458 the algorithms provide high accuracy, with more than an order of magnitude reduction in operation counts