A HIERARCHICAL ALGORITHM FOR FAST BACKPROJECTION IN HELICAL CONE-BEAM TOMOGRAPHY Yoram Bresler and Jeffrey Brokish Coordinated Science Laboratory and Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign ybresler@uiuc.edu, brokish@uiuc.edu ABSTRACT Existing algorithms for exact helical cone beam (HCB) to- mographic reconstruction involve a 3-D backprojection step, which dominates the the computational cost of the algo- rithm. We present a fast hierarchical 3-D backprojection algorithm, generalizing fast 2-D parallel beam and fan beam algorithms, which reduces the complexity of this step from O(N 4 ) to O(N 3 log N ), greatly accelerating the reconstruc- tion process. 1. INTRODUCTION Helical cone-beam tomography has several advantages over traditional two dimensional tomographic imaging, includ- ing decreased scanning times and increased x-ray source utilization. However, image reconstruction from cone beam projections relies on inversion formulas [3], [2] of higher complexity than those found in two dimensional tomogra- phy. These algorithms consist of individually “filtering” the cone beam projections followed by a backprojection over the image volume. This 3-D backprojection has complexity of O(N 3 P ) for reconstruction of an N ×N ×N voxel image from P projections. Generally P = O(N ), which results in an O(N 4 ) operation and accounts for a large amount of the computation in the reconstruction process. Several fast algorithms for backprojection in two dimen- sional tomography exist. Algorithms based on hierarchi- cal decomposition reduce the complexity of the backprojec- tion operation by successively subdividing the reconstruc- tion area into smaller nonoverlapping regions. As the re- gion size decreases, the number of projections necessary for accurate reconstruction also decreases. The number of pro- jections can then be reduced, which reduces the computa- tional complexity. This hierarchical decomposition of the backprojection operation initially developed for 2-D paral- lel beam [1], was extended to fan beam [4] and 3-D cone This material is based upon work supported under a National Science Foundation Graduate Research Fellowship and by NSF grants Nos. CCR 99-72980 and CCR 02-09203. . . . DETECTOR PLANE Fig. 1. Helical Cone-Beam Acquisition beam with a circular trajectory (CCB) [5]. Here we extend the method to the more general divergent-beam geometry of a helical trajectory. HCB does not suffer from the inherent artifacts present in CCB but carries a higher computational cost. The fast CCB method decomposes the volume only in the x and y dimensions. Here we fully decompose the volume along all three dimensions. This is important for accurate reconstruction with a helical trajectory, and for ex- tensions to an arbitrary trajectory. 2. ALGORITHM DESCRIPTION 2.1. 3-D Cone Beam Backprojection Figure 1 shows the setup for helical cone beam projection data acquisition. An x-ray source is placed at equally spaced intervals along a helical trajectory parameterized by a(λ)= (R cosλ, R sinλ, hλ 2π ) where R is the distance between the source and the z-axis and h is the pitch of the helix. The de- tector plane is assumed to contain the z-axis and be perpen- dicular to a(λ). After the filtering step is completed for each cone beam projection, the filtered projection data ˜ f [u, v, λ] 1420 0-7803-8388-5/04/$20.00 ©2004 IEEE