Mallinckrodt Inst. of Radiol., Washington Univ., St. Louis, MO.
IEEE Trans Image Process. 1993;2(4):543-7. doi: 10.1109/83.242365.
In order to perform fan-beam reconstruction using projection data collected from a noncircular scanning locus, existing noncircular fan-beam formulas require a derivative of the scanning locus with respect to the rotation angle. A derivative-free noncircular fan-beam reconstruction formula that is based on a geometrical explanation of the circular equispatial fan-beam reconstruction formula is obtained here. A mathematical proof is then provided under the conditions that the source-to-origin distance is symmetric with respect to the origin of the reconstruction coordinate system, is differentiable almost everywhere, and does not change too fast with respect to the rotation angle. The derivative-free noncircular fan-beam reconstruction formula is the same as the circular one, except that the source-to-origin distance is a function of the rotation angle. A typical simulation result for the noncircular fan-beam formula is given.
为了使用从非圆形扫描轨迹中收集的投影数据进行扇形束重建，现有的非圆形扇形束公式需要对旋转角度求导。本文提出了一种基于对圆形等距扇形束重建公式的几何解释的无导数非圆形扇形束重建公式。然后，在源到原点距离关于重建坐标系原点对称、几乎处处可微且相对于旋转角度变化不太快的条件下，提供了一个数学证明。无导数非圆形扇形束重建公式与圆形公式相同，只是源到原点的距离是旋转角度的函数。给出了非圆形扇形束公式的一个典型模拟结果。
