Helical scanning configuration has been used widely in diagnostic cone-beam computed tomography (CBCT) for acquiring data sufficient for exact image reconstruction over an extended volume. In image-guided radiation therapy (IGRT) and other applications of CBCT, it can be difficult, if not impossible, to implement mechanically a multiple-turn helical trajectory on the imaging systems due to hardware constraints. However, imaging systems in these applications often allow for the implementation of a reverse helical trajectory in which the rotation direction changes between two consecutive turns. Because the reverse helical trajectory satisfies Tuy’s condition, when projections of the imaged object are nontruncated, it yields data sufficient for exact image reconstruction within the reverse helix volume. The recently developed chord-based algorithms such as the backprojection filtration (BPF) algorithm can readily be applied to reconstructing images on chords of a reverse helical trajectory, and they can thus reconstruct an image within a volume covered by the chords. Conversely, the chord-based algorithms cannot reconstruct images within regions that are not intersected by chords. In a reverse helix volume, as shown below, chordless regions exist in which no images can thus be reconstructed by use of the chord-based algorithms. In this work, based upon Pack–Noo’s formula, a shift-invariant filtered backprojection (FBP) algorithm is derived for exact image reconstruction within the reverse helix volume, including the chordless region. Numerical studies have also been conducted to demonstrate the chordless region in a reverse helix volume and to validate the FBP algorithm for image reconstruction within the chordless region. Results of the numerical studies confirm that the FBP algorithm can exactly reconstruct an image within the entire reverse helix volume, including the chordless region. It is relatively straightforward to extend the FBP algorithm to reconstruct images for general trajectories, including reverse helical trajectories with variable pitch, tilted axis, and∕or additional segments between turns.
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