Image reconstruction from truncated fan-beam projections along a circular trajectory using the virtual fan-beam method

Abstract

Abstract Objective. In this paper, we investigate how the virtual fan-beam (VFB) method can be used to perform mathematically correct 2D reconstruction in a region-of-interest (ROI), using truncated fan-beam projections acquired on a circular scan, for truncation that only occurs on one side of the object. Approach. We start by choosing a virtual fan-beam trajectory and specifying how to obtain the corresponding virtual projections. Then, three VFB formulas are obtained by applying known super-short-scan (SSS) formulas to this virtual trajectory. Two of them perform the backprojection in a virtual parallel geometry and the third in the virtual fan-beam geometry. Next, we develop two VFB formulas that perform the backprojection step in the fan-beam acquisition geometry. Main results. We present five VFB reconstruction formulas for this truncation setting. To our knowledge, the two VFB formulas performing the backprojection in the fan-beam acquisition geometry are new. Moreover, the five VFB formulas presented here obtain accurate reconstruction in a larger ROI than what has been previously reported in the literature in the same setting. A complete mathematical derivation of these five VFB formulas is given, and their implementation is described step by step. Numerical simulations, using the Forbild head and thorax phantoms, demonstrate the efficacy of these formulas. A spatial resolution analysis and a variance study indicate minor differences between these five VFB formulas. Significance. This work shows that many different VFB formulas can be applied to perform mathematically correct 2D reconstruction in a ROI, in case of truncated fan-beam projections acquired on a circular scan. Moreover, the two new VFB formulas, with backprojection in the acquisition geometry, may open the path for an extension of the VFB method to 3D reconstruction from transversely truncated cone-beam projection acquired on a circular scan.