Truncated Hilbert transform and image reconstruction from limited tomographic data
Truncated Hilbert transform and image reconstruction from limited tomographic data
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DOI：
10.1088/0266-5611/22/3/019
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发表时间：
2006-06-01
期刊：
影响因子：
2.1
通讯作者：
Kudo, Hiroyuki
中科院分区：
文献类型：
--
作者：
Defrise, Michel;Noo, Frederic;Kudo, Hiroyuki
利用数据导数的反投影和图像沿覆盖ROI的沿着某些线段的希尔伯特变换之间的关系，最近引入了用于从有限族线积分重建2D或3D感兴趣区域（ROI）的数据充分性条件。本文概括了这一充分条件表明，独特的和稳定的重建，可以实现从一个更有限的家庭的数据集，或者，相反，甚至更大的ROI可以从一个给定的数据集重建。该条件是通过分析截断希尔伯特变换的逆导出的，这里定义为从其希尔伯特变换的知识沿着仅部分覆盖函数的支持但至少有一个端点在该支持之外的段沿着恢复一个真实的变量的函数的问题。给出了该问题的唯一性证明和稳定性估计。数值模拟的2D胸部幻影，以说明新的数据充分性条件和良好的稳定性，在噪声的存在下的ROI重建。
A data sufficiency condition for 2D or 3D region-of-interest (ROI) reconstruction from a limited family of line integrals has recently been introduced using the relation between the backprojection of a derivative of the data and the Hilbert transform of the image along certain segments of lines covering the ROI. This paper generalizes this sufficiency condition by showing that unique and stable reconstruction can be achieved from an even more restricted family of data sets, or, conversely, that even larger ROIs can be reconstructed from a given data set. The condition is derived by analysing the inversion of the truncated Hilbert transform, here defined as the problem of recovering a function of one real variable from the knowledge of its Hilbert transform along a segment which only partially covers the support of the function but has at least one end point outside that support. A proof of uniqueness and a stability estimate are given for this problem. Numerical simulations of a 2D thorax phantom are presented to illustrate the new data sufficiency condition and the good stability of the ROI reconstruction in the presence of noise.