Zhao Shiying, Yu Hengyong, Wang Ge
CT/Micro-CT Laboratory, Department of Radiology, University of Iowa, 200 Hawkins Drive, Iowa City, Iowa 52242, USA.
Med Phys. 2005 Jun;32(6):1712-21. doi: 10.1118/1.1869632.
In this paper, we present concise proofs of several recently developed exact cone-beam reconstruction methods in the Tuy inversion framework, including both filtered-backprojection and backprojection-filtration formulas in the cases of standard spiral, nonstandard spiral, and more general scanning loci. While a similar proof of the Katsevich formula was previously reported, we present a new proof of the Zou and Pan backprojection-filtration formula. Our proof combines both odd and even data extensions so that only the cone-beam transform itself is utilized in the backprojection-filtration inversion. More importantly, our formulation is valid for general smooth scanning curves, in agreement with an earlier paper from our group [Ye, Zhao, Yu, and Wang, Proc. SPIE 5535, 293-300 (Aug. 6 2004)]. As a consequence of that proof, we obtain a new inversion formula, which is in a two-dimensional filtering backprojection format. A possibility for generalization of the Katsevich filtered-backprojection reconstruction method is also discussed from the viewpoint of this framework.
在本文中，我们给出了Tuy反演框架下几种最近开发的精确锥束重建方法的简洁证明，包括标准螺旋、非标准螺旋以及更一般扫描轨迹情况下的滤波反投影和反投影滤波公式。虽然之前已有关于Katsevich公式的类似证明，但我们给出了邹和潘反投影滤波公式的新证明。我们的证明结合了奇数和偶数数据扩展，使得在反投影滤波反演中仅使用锥束变换本身。更重要的是，我们的公式对于一般光滑扫描曲线是有效的，这与我们团队早期的一篇论文[Ye, Zhao, Yu, and Wang, Proc. SPIE 5535, 293 - 300 (Aug. 6 2004)]一致。作为该证明的一个结果，我们得到了一个新的反演公式，它是二维滤波反投影格式。还从这个框架的角度讨论了Katsevich滤波反投影重建方法的推广可能性。
