任意不完整X射线CT数据的重建伪影分析
Analyzing Reconstruction Artifacts from Arbitrary Incomplete X-ray CT Data
摘要：
本文提供了一个数学分析的奇异（非光滑）伪影添加到重建的过滤反投影（FBP）型算法的X射线CT与任意不完整的数据。我们证明，这些奇异的文物出现在数据集的边界点。我们的研究结果表明，根据这个边界的几何形状，两种类型的文物可能会出现：对象相关和对象无关的文物。对象相关伪影是由被扫描对象的奇点产生的，并且这些伪影可以沿着线延伸。他们概括了条纹文物中观察到的有限角度断层扫描。另一方面，与对象无关的伪影基本上与对象无关，并采取两种形式之一：如果数据集的边界在某个点处不平滑，则为线条上的条纹;如果边界局部平滑，则为弯曲伪影。我们证明，这些条纹和曲线文物是唯一的奇异文物，可以发生在连续的情况下，为FBP。除了工件的几何描述，文章提供了在某些情况下，在索伯列夫尺度的强度的表征。本文的结果适用于众所周知的不完整数据问题，包括有限角度和感兴趣的区域断层扫描，以及在新的实际应用中出现的非常规X射线CT成像设置。从模拟和真实数据的重建进行了分析，以说明我们的定理，包括重建，激发这项工作-一个同步数据集，其中文物出现在线上，没有关系的对象。
英文摘要：
This article provides a mathematical analysis of singular (nonsmooth) artifacts added to reconstructions by filtered backprojection (FBP) type algorithms for X-ray CT with arbitrary incomplete data. We prove that these singular artifacts arise from points at the boundary of the data set. Our results show that, depending on the geometry of this boundary, two types of artifacts can arise: object-dependent and object-independent artifacts. Object-dependent artifacts are generated by singularities of the object being scanned and these artifacts can extend along lines. They generalize the streak artifacts observed in limited-angle tomography. Object-independent artifacts, on the other hand, are essentially independent of the object and take one of two forms: streaks on lines if the boundary of the data set is not smooth at a point and curved artifacts if the boundary is smooth locally. We prove that these streak and curve artifacts are the only singular artifacts that can occur for FBP in the continuous case. In addition to the geometric description of artifacts, the article provides characterizations of their strength in Sobolev scale in certain cases. The results of this article apply to the well-known incomplete data problems, including limited-angle and region-of-interest tomography, as well as to unconventional X-ray CT imaging setups that arise in new practical applications. Reconstructions from simulated and real data are analyzed to illustrate our theorems, including the reconstruction that motivated this work---a synchrotron data set in which artifacts appear on lines that have no relation to the object.
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