Multi-sheet surface rebinning methods for reconstruction from asymmetrically truncated cone beam projections: I. Approximation and optimality

Abstract

The mechanical motion of the gantry in conventional cone beam CT scanners restricts the speed of data acquisition in applications with near real time requirements. A possible resolution of this problem is to replace the moving source detector assembly with static parts that are electronically activated. An example of such a system is the Rapiscan Systems RTT80 real time tomography scanner, with a static ring of sources and axially offset static cylinder of detectors. A consequence of such a design is asymmetrical axial truncation of the cone beam projections resulting, in the sense of integral geometry, in severely incomplete data. In particular we collect data only in a fraction of the Tam–Danielsson window, hence the standard cone beam reconstruction techniques do not apply. In this work we propose a family of multi-sheet surface rebinning methods for reconstruction from such truncated projections. The proposed methods combine analytical and numerical ideas utilizing linearity of the ray transform to reconstruct data on multi-sheet surfaces, from which the volumetric image is obtained through deconvolution. In this first paper in the series, we discuss the rebinning to multi-sheet surfaces. In particular we concentrate on the underlying transforms on multi-sheet surfaces and their approximation with data collected by offset multi-source scanning geometries like the RTT. The optimal multi-sheet surface and the corresponding rebinning function are found as a solution of a variational problem. In the case of the quadratic objective, the variational problem for the optimal rebinning pair can be solved by a globally convergent iteration. Examples of optimal rebinning pairs are computed for different trajectories. We formulate the axial deconvolution problem for the recovery of the volumetric image from the reconstructions on multi-sheet surfaces. Efficient and stable solution of the deconvolution problem is the subject of the second paper in this series (Betcke and Lionheart 2013 Inverse Problems 29 115004).