Ye Yangbo, Wang Ge
Department of Radiology and Mathematics, The University of Iowa, Iowa City, Iowa 52242-1419, USA.
Med Phys. 2005 Jan;32(1):42-8. doi: 10.1118/1.1828673.
Recently, Katsevich proved a filtered backprojection formula for exact image reconstruction from cone-beam data along a helical scanning locus, which is an important breakthrough since 1991 when the spiral cone-beam scanning mode was proposed. In this paper, we prove a generalized Katsevich's formula for exact image reconstruction from cone-beam data collected along a rather flexible curve. We will also give a general condition on filtering directions. Based on this condition, we suggest a natural choice of filtering directions, which is more convenient than Katsevich's choice and can be applied to general scanning curves. In the derivation, we use analytical techniques instead of geometric arguments. As a result, we do not need the uniqueness of the PI lines. In fact, our formula can be used to reconstruct images on any chord as long as a scanning curve runs from one endpoint of the chord to the other endpoint. This can be considered as a generalization of Orlov's classical theorem. Specifically, our formula can be applied to (i) nonstandard spirals of variable radii and pitches (with PI- or n-PI-windows), and (ii) saddlelike curves.
最近，卡特塞维奇证明了一个用于从沿螺旋扫描轨迹的锥束数据进行精确图像重建的滤波反投影公式，这是自1991年提出螺旋锥束扫描模式以来的一项重要突破。在本文中，我们证明了一个广义的卡特塞维奇公式，用于从沿相当灵活的曲线收集的锥束数据进行精确图像重建。我们还将给出滤波方向的一般条件。基于此条件，我们提出了一种自然的滤波方向选择，它比卡特塞维奇的选择更方便，并且可以应用于一般的扫描曲线。在推导过程中，我们使用解析技术而非几何论证。因此，我们不需要PI线的唯一性。事实上，只要扫描曲线从弦的一个端点延伸到另一个端点，我们的公式就可用于在任何弦上重建图像。这可被视为奥洛夫经典定理的一种推广。具体而言，我们的公式可应用于：（i）具有可变半径和螺距的非标准螺旋（具有PI或n - PI窗口），以及（ii）鞍状曲线。
