Line plus arc source trajectories and their R-line coverage for long-object cone-beam imaging with a C-arm system

Cone-beam imaging with C-arm systems has become a valuable tool in interventional radiology. Currently, a simple circular trajectory is used, but future applications should use more sophisticated source trajectories, not only to avoid cone-beam artifacts but also to allow extended volume imaging. One attractive strategy to achieve these two goals is to use a source trajectory that consists of two parallel circular arcs connected by a line segment, possibly with repetition. In this work, we address the question of R-line coverage for such a trajectory. More specifically, we examine to what extent R-lines for such a trajectory cover a central cylindrical region of interest (ROI). An R-line is a line segment connecting any two points on the source trajectory. Knowledge of R-line coverage is crucial because a general theory for theoretically exact and stable image reconstruction from axially truncated data is only known for the points in the scanned object that lie on R-lines. Our analysis starts by examining the R-line coverage for the elemental trajectories consisting of (i) two parallel circular arcs and (ii) a circular arc connected orthogonally to a line segment. Next, we utilize our understanding of the R-lines for the aforementioned elemental trajectories to determine the R-line coverage for the trajectory consisting of two parallel circular arcs connected by a tightly fit line segment. For this trajectory, we find that the R-line coverage is insufficient to completely cover any central ROI. Because extension of the line segment beyond the circular arcs helps to increase the R-line coverage, we subsequently propose a trajectory composed of two parallel circular arcs connected by an extended line. We show that the R-lines for this trajectory can fully cover a central ROI if the line extension is long enough. Our presentation includes a formula for the minimum line extension needed to achieve full R-line coverage of an ROI with a specified size, and also includes a preliminary study on the required detector size, showing that the R-lines added by the line extension are not constraining.