An Improved Extrapolation Scheme for Truncated CT Data Using 2D Fourier-Based Helgason-Ludwig Consistency Conditions.

Yan Xia,Andreas Maier,Sebastian Bauer,Martin Berger,Andre Aichert,Shiyang Hu

doi:10.1155/2017/1867025

Abstract

We improve data extrapolation for truncated computed tomography (CT) projections by using Helgason-Ludwig (HL) consistency conditions that mathematically describe the overlap of information between projections. First, we theoretically derive a 2D Fourier representation of the HL consistency conditions from their original formulation (projection moment theorem), for both parallel-beam and fan-beam imaging geometry. The derivation result indicates that there is a zero energy region forming a double-wedge shape in 2D Fourier domain. This observation is also referred to as the Fourier property of a sinogram in the previous literature. The major benefit of this representation is that the consistency conditions can be efficiently evaluated via 2D fast Fourier transform (FFT). Then, we suggest a method that extrapolates the truncated projections with data from a uniform ellipse of which the parameters are determined by optimizing these consistency conditions. The forward projection of the optimized ellipse can be used to complete the truncation data. The proposed algorithm is evaluated using simulated data and reprojections of clinical data. Results show that the root mean square error (RMSE) is reduced substantially, compared to a state-of-the-art extrapolation method.

Highlights

It is known that traditional computed tomography (CT) reconstruction algorithms, for example, filtered backprojection methods, are not compatible to laterally truncated projection data, which appears often in the case of either (1) when the object extends outside of the field of view (FOV) or (2) X-ray beam collimation for the purpose of dose reduction
This restriction is expressed by Helgason-Ludwig (HL) consistency conditions [6, 7], which are a mathematical expression to precisely describe the overlap of information between different projections
In PET/SPECT, the HL consistency conditions were used for attenuation correction if no transmission data is available [12]

Summary

Introduction

It is known that traditional computed tomography (CT) reconstruction algorithms, for example, filtered backprojection methods, are not compatible to laterally truncated projection data, which appears often in the case of either (1) when the object extends outside of the field of view (FOV) or (2) X-ray beam collimation for the purpose of dose reduction. A typical approach to reduce truncation artifacts is to perform extrapolation, for example, with the symmetric mirroring method [1], water cylinder extrapolation method [2], optimization-based extrapolation scheme [3], or implicit extrapolation method performed in the second-order derivative domain [4]. These heuristic extrapolation methods typically rely on techniques that complete the truncated data by means of a continuity assumption and appear to be ad hoc. In PET/SPECT, the HL consistency conditions were used for attenuation correction if no transmission data is available [12]

Results

Discussion

Conclusion