Dept. of Radiol., Univ. of Chicago Med. Centre, Chicago, IL.
IEEE Trans Med Imaging. 1995;14(4):643-58. doi: 10.1109/42.476106.
Exact methods of inverting the two-dimensional (2-D) exponential Radon transform have been proposed by Bellini et al. (1979) and by Inouye et al. (1989), both of whom worked in the spatial-frequency domain to estimate the 2-D Fourier transform of the unattenuated sinogram; by Hawkins et al. (1988), who worked with circularly harmonic Bessel transforms; and by Tretiak and Metz (1980), who followed filtering of appropriately-modified projections by exponentially-weighted backprojection. With perfect sampling, all four of these methods are exact in the absence of projection-data noise, but empirical studies have shown that they propagate noise differently, and no underlying theoretical relationship among the methods has been evident. Here, an analysis of the 2-D Fourier transform of the modified sinogram reveals that all previously-proposed linear methods can be interpreted as special cases of a broad class of methods, and that each method in the class can be implemented, in principle, by any one of four distinct techniques. Moreover, the analysis suggests a new member of the class that Is predicted to have noise properties better than those of previously-proposed members.
贝尔尼尼等人（1979 年）和猪饲等人（1989 年）曾提出过二维（2-D）指数型雷登变换的精确反演方法，他们均在空间频率域中工作，以估计未经衰减的正弦图的二维傅里叶变换；霍金斯等人（1988 年）采用的是圆谐贝塞尔变换；特里蒂克和梅茨（1980 年）采用的是经指数加权反投影滤波后的适当修正投影。在完全采样的情况下，这四种方法在没有投影数据噪声的情况下都是精确的，但经验研究表明，它们传播噪声的方式不同，而且这些方法之间没有明显的理论关系。在这里，对修正后正弦图的二维傅里叶变换进行分析后发现，以前提出的所有线性方法都可以解释为一个广泛类方法的特例，而且该类中的每个方法原则上都可以通过四种不同技术中的任意一种来实现。此外，该分析还提出了一个新的类成员，据预测，它的噪声特性比以前提出的成员更好。
