We introduce a family of fast algorithms for backprojecting images from tomographic projections. They aggregate the projections in a hierarchical structure and achieve a computational cost of O(N2 log P), when backprojecting an N × N pixel image from P projections. The images in the hierarchy are formed by the rotation and the adding together of other images made up of fewer projections. While these algorithms are related to existing fast algorithms, this work places them within the signal processing framework, providing systematic means to optimize and adjust the trade off between computational cost and accuracy. Rotations are performed separably in order that higher-order interpolators may be used with low computational cost. The same ideas can be applied to tomographically project an N × N pixel image onto P view-angles with a cost of O(N2 log P).