Error analysis and performance optimization of fast hierarchical backprojection algorithms

We recently proposed a novel fast backprojection algorithm for reconstruction of an N × N pixel object from O(N) projections in O(N ² log ₂ N) operations. In this paper, we analyze a simplified version of that algorithm, to determine the effects of various parameter choices on the algorithm's theoretical performance. We derive a bound on the variance of the per-pixel error introduced by using the hierarchical backprojection. This bound is with respect to an ensemble of input sinograms, and allows us to construct confidence intervals (for any specified level) for the per-pixel errors. The bound has a simple form, and we show how to use it to select algorithm parameters for different cost versus error tradeoffs. Simulation results show that the bound accurately predicts the performance of the algorithm over a wide range of parameter choices. These results are verified for different images, including a tomographic reconstruction from the visual human dataset (VHD). The analysis therefore provides an effective tools for the selection of parameters and operating point for the fast hierarchical backprojection algorithm.