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) log P), when backprojecting an N x N 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.