A unified analysis of FBP-based algorithms in helical cone-beam and circular cone- and fan-beam scans

Abstract

A circular scanning trajectory is and will likely remain a popular choice of trajectory in computed tomography (CT) imaging because it is easy to implement and control. Filtered-backprojection (FBP)-based algorithms have been developed previously for approximate and exact reconstruction of the entire image or a region of interest within the image in circular cone-beam and fan-beam cases. Recently, we have developed a 3D FBP-based algorithm for image reconstruction on PI-line segments in a helical cone-beam scan. In this work, we demonstrated that the 3D FBP-based algorithm indeed provided a rather general formulation for image reconstruction from divergent projections (such as cone-beam and fan-beam projections). On the basis of this formulation we derived new approximate or exact algorithms for image reconstruction in circular cone-beam or fan-beam scans, which can be interpreted as special cases of the helical scan. Existing algorithms corresponding to the derived algorithms were identified. We also performed a preliminary numerical study to verify our theoretical results in each of the cases. The results in the work can readily be generalized to other non-circular trajectories.

Recent Advances in CT Image Reconstruction

“…Reprinted from [24], with permission category of FBP with a shift-invariant one-dimensional filter, and the flow of this reconstruction algorithm is similar to that of the fan-beam FBP. Since its invention, many new algorithms have been proposed and developed to allow more general trajectories, more efficient use of the projection samples, and region-of-interest reconstruction [14,[35][36][37][38][39]. The advantage of these algorithms is of course their mathematical exactness.…”

Region‐of‐interest image reconstruction with intensity weighting in circular cone‐beam CT for image‐guided radiation therapy

“…Approximate image reconstruction, therefore, can be performed by use of the BPF algorithm for those off midplanes. 17,18 Derivatives of the cone-beam data acquired by a circular scan are backprojected first onto each chord, followed by the filtering along each chord in one dimension. In the FDK algorithm the filtering occurs in the projection data, thus transverse truncation can produce artifacts in the reconstructed images.…”

Noise Properties of Chord-Image Reconstruction

“…The BPF, MDFBP, and FBP algorithms described above can be applied to reconstructing chord images from parallel-, fan-, and cone-beam data [17]. Algorithms analogous to the BPF algorithm that are capable of reconstructing 2-D ROI images from truncation data have also previously been proposed [8], [13], [18].…”

“…Furthermore, we can write the variance of as (29) For the sake of simplifying the estimation of , we have ignored the correlation between and and assumed that . We select as the average over all of the possible s, which can be computed as (30) Finally, with a substitution of and into (17) and (24), we obtain the variances of the backprojection images on the chords for the parallel-beam and fan-beam projections, respectively, as (31) (32)…”

“…3. Using the noiseless data as the Poisson-noise variance in (17), one can readily determine ; and using the determined in (16), (36), and (38), one can compute analytical image variances, which are displayed in the middle row of Fig. 3.…”