Multi-sheet surface rebinning methods for reconstruction from asymmetrically truncated cone beam projections: II. Axial deconvolution

In airport baggage scanning it is desirable to have a system that can scan baggage moving at standard conveyor belt speeds. One way to achieve this is to use multiple electronically switched sources rather than a single source on a mechanically rotated gantry. In such a system placing the detectors opposite the sources would obstruct the beam, so they have to be offset (hence offset multi-source geometry). This results in asymmetrical axial truncation of the cone beam projections. As such projections do not constitute complete data in the sense of integral geometry, the standard cone beam reconstruction algorithms do not apply. In this series of papers we introduce a new family of rebinning methods for reconstruction from axially asymmetrically truncated cone beam projections. In the first paper we discussed the approximation of the data on the multi-sheet surface with the truncated projection data obtained from offset multi-source geometries. In this second paper we focus on the recovery of the volumetric image from the reconstruction of data rebinned to multi-sheet surfaces. Multi-sheet rebinning effects an implicit relation between the fan beam transforms on the individual sheets and the rebinned data. This relation in conjunction with the linearity of the ray transform allows us to formulate the deconvolution problem for the recovery of the volume from a stack of reconstructed images on multi-sheet surfaces. We discuss the errors in the right-hand side of the deconvolution problem (reconstruction on multi-sheet surfaces) resulting from rebinning approximation. We introduce convolution matrix models based on the distribution of the distances of the rays from the multi-sheet surface, which considerably improve the data model fit and in turn lead to a superior solution. Multiple strategies for solution of the deconvolution problem are discussed and an efficient and robust implementation is presented, which makes the method capable of real time reconstruction. We conclude with some reconstruction results from simulated as well as real data collected with a Rapiscan RTT80 scanner.