Exact image reconstruction from limited projection data has been a central topic in the computed tomography (CT) field. In this paper, we present a general region-of-interest/volume-of-interest (ROI/VOI) reconstruction approach using a truly truncated Hilbert transform on a line-segment inside a compactly supported object aided by partial knowledge on one or both neighboring intervals of that segment. Our approach and associated new data sufficient condition allows the most flexible ROI/VOI image reconstruction from the minimum account of data in both the fan-beam and cone-beam geometry. We also report primary numerical simulation results to demonstrate the correctness and merits of our finding. Our work has major theoretical potentials and innovative practical applications.

Using the backprojection filtration (BPF) and filtered backprojection (FBP) approaches, respectively, we prove that with cone-beam CT the interior problem can be exactly solved by analytic continuation. The prior knowledge we assume is that a volume of interest (VOI) in an object to be reconstructed is known in a subregion of the VOI. Our derivations are based on the so-called generalized PI-segment (chord). The available projection onto convex set (POCS) algorithm and singular value decomposition (SVD) method can be applied to perform the exact interior reconstruction. These results have many implications in the CT field and can be extended to other tomographic modalities, such as SPECT/PET, MRI.

Recently, Katsevich proved a filtered backprojection formula for exact image reconstruction from cone-beam data along a helical scanning locus, which is an important breakthrough since 1991 when the spiral cone-beam scanning mode was proposed. In this paper, we prove a generalized Katsevich's formula for exact image reconstruction from cone-beam data collected along a rather flexible curve. We will also give a general condition on filtering directions. Based on this condition, we suggest a natural choice of filtering directions, which is more convenient than Katsevich's choice and can be applied to general scanning curves. In the derivation, we use analytical techniques instead of geometric arguments. As a result, we do not need the uniqueness of the PI lines. In fact, our formula can be used to reconstruct images on any chord as long as a scanning curve runs from one endpoint of the chord to the other endpoint. This can be considered as a generalization of Orlov's classical theorem. Specifically, our formula can be applied to (i) nonstandard spirals of variable radii and pitches (with PI- or n-PI-windows), and (ii) saddlelike curves.

In this article we consider cone-beam CT projections along a nonstandard 3-D spiral with variable radius and variable pitch. Specifically, we generalize an exact image reconstruction formula by Pan (2004a) and to the case of nonstandard spirals, by giving a new, analytic proof of the reconstruction formula. Our proof is independent of the shape of the spiral, as long as the object is contained in a region inside the spiral, where there is a PI line passing through any interior point. Our generalized reconstruction formula can also be applied to much more general situations, including cone-beam scanning along standard (Pack, et al. 2004) and nonstandard saddle curves, and any smooth curve from one endpoint of a line segment to the other endpoint, for image reconstruction of that line segment. In other words, our results can be regarded as a generalization of Orlov's classical papers (1975) to cone-beam scanning.

A hot topic in cone-beam CT research is exact cone-beam reconstruction from a general scanning trajectory. Particularly, a nonstandard saddle curve attracts attention, as this construct allows the continuous periodic scanning of a volume-of-interest (VOI). Here we evaluate two algorithms for reconstruction from data collected along a nonstandard saddle curve, which are in the filtered backprojection (FBP) and backprojection filtration (BPF) formats, respectively. Both the algorithms are implemented in a chord-based coordinate system. Then, a rebinning procedure is utilized to transform the reconstructed results into the natural coordinate system. The simulation results demonstrate that the FBP algorithm produces better image quality than the BPF algorithm, while both the algorithms exhibit similar noise characteristics.